We study traveling waves in mass and spring dimer Fermi–Pasta–Ulam–Tsingou (FPUT) lattices in the long wave limit.
Such lattices are known to possess nanopteron traveling waves in relative displacement coordinates.
These nanopteron profiles consist of the superposition of an exponentially localized “core,” which is close to a KdV solitary wave, and a periodic “ripple,” whose amplitude is small beyond all algebraic orders of the long wave parameter, although a zero amplitude is not precluded.
Here we deploy techniques of spatial dynamics, inspired by results of Iooss and Kirchgässner, Iooss and James, and Venney and Zimmer, to construct mass and spring dimer nanopterons whose ripples are both exponentially small and also nonvanishing.
We first obtain “growing front” traveling waves in the original position coordinates and then pass to relative displacement.
To study position, we recast its traveling wave problem as a first-order equation on an infinite-dimensional Banach space; then we develop hypotheses that, when met, allow us to reduce such a first-order problem to one solved by Lombardi. A key part of our analysis is then the passage back from the reduced problem to the original one.
Our hypotheses free us from working strictly with lattices but are easily checked for FPUT mass and spring dimers.
We also give a detailed exposition and reinterpretation of Lombardi's methods, to illustrate how our hypotheses work in concert with his techniques, and we provide a dialogue with prior methods of constructing FPUT nanopterons, to expose similarities and differences with the present approach.

§ INTRODUCTION

§.§ The polyatomic FPUT lattice

A polyatomic, or polymer, Fermi–Pasta–Ulam–Tsingou (FPUT) lattice [22, 13, 28, 70, 82] is an infinite, one-dimensional chain of particles, connected on the left and right to their nearest neighbors by springs, with motion constrained to a horizontal axis.
We index the particles and springs by integers $j \in \Z$; the $j$th spring connects the $j$th particle on the left to the $(j+1)$st particle on the right.
Let $m_j$ be the mass of the $j$th particle, let $\V_j$ be the potential of the $j$th spring, and let $\ell_j$ be the equilibrium length of the $j$th spring, so that the $j$th spring exerts the force $\V_j'(r-\ell_j)$ when stretched a distance $r$ along this horizontal axis.

If $y_j$ denotes the position of the $j$th particle along the axis, relative to some fixed origin point, then Newton's second law gives the lattice's equations of motion:

\begin{equation}\label{eqn: original equations of motion with ell-j}
= \V_j'(y_{j+1}-y_j-\ell_j) - \V_{j-1}'(y_j-y_{j-1}-\ell_{j-1}).
\end{equation}

We may assume that the equilibrium lengths are all 0 by making the change of variables

\begin{equation}\label{eqn: equilibrium length cov}
y_j = \begin{cases}
u_j + \medsum_{k=0}^{j-1} \ell_k, & j \ge 1 \\
u_0, & j = 0 \\
u_j-\medsum_{k=j}^{-1} \ell_k, & j \le -1
\end{cases}
\end{equation}

to find

\begin{equation}\label{eqn: rel disp y and u}
y_{j+1}-y_j - \ell_j
= u_{j+1}-u_j.
\end{equation}

Then $y_j$ satisfies (<ref>) if and only if $u_j$ satisfies

\begin{equation}\label{eqn: original equations of motion}
= \V_j'(r_j) - \V_{j-1}'(r_{j-1}),
\end{equation}

\begin{equation}\label{eqn: rel disp coords}
:= u_{j+1} - u_j
\end{equation}

are the classical relative displacement coordinates.
In terms of relative displacement, the equations of motion are

\begin{equation}\label{eqn: rel disp eqns}
\ddot{r}_j
= \frac{1}{m_{j+1}}\V_{j+1}'(r_{j+1}) - \left(\frac{1}{m_j}+\frac{1}{m_{j+1}}\right)\V_j'(r_j) + \frac{1}{m_j}\V_{j-1}'(r_{j-1}).
\end{equation}

While most studies of lattice dynamics concentrate on relative displacement, we will construct solutions both in position coordinates (<ref>) and relative displacement (<ref>).
We state these main results in Theorems <ref> and <ref>, discuss the prior history of our problems of interest in Section <ref>, present our primary motivations for the current research in Section <ref>, and give an overview of our methods in Section <ref>.

What we call “position” coordinates $u_j$ are really “displacement from equilibrium” coordinates, due to the change of variables (<ref>).
Nonetheless, we will retain the “position” terminology so as not to overwork the noun “displacement.”

We now impose further structure on our material data by assuming that the masses of the particles and the potentials of the springs repeat with some finite periodicity.
That is, for some integer $N \ge 1$ we have

\begin{equation}\label{eqn: N-periodic data}
m_{j+N} = m_j,
\qquad
\V_{j+N} = \V_j,
\quadword{and}
\ell_{j+N} = \ell_j, \ j \in \Z,
\end{equation}

in which case we might say that the lattice is $N$-polyatomic.
These material values will be fixed throughout our analysis; other studies permit the masses or springs to vary in certain limits, which we briefly discuss in Section <ref>.
If $N = 1$, then the lattice is called monatomic, and if $N = 2$, then the lattice is a (general) dimer.

From now on we will focus on dimers, and we will eventually specialize to two species of dimer: the mass dimer (or diatomic lattice), in which $\V_j = \V$ for some single potential function $\V$ and $m_1 \ne m_2$ are distinct, and the spring dimer, in which $m_j = m$ for some single mass value $m > 0$ and $\V_1 \ne \V_2$ are distinct.
We sketch the “general” dimer in Figure <ref>.

\[
\input{fig-general_dimer}
\]

An FPUT general dimer (masses normalized to 1 and to $m \ne 1$)

We assume that the spring potentials $\V_1$ and $\V_2$ are real analytic with $\V_j'(0) > 0$ for $j=1$, 2 and $\V_j''(0) \ne 0$ for at least one $j$.
Routine nondimensionalizations (see <cit.> and <cit.>) allow us to write

\begin{equation}\label{eqn: dimer masses}
= \begin{cases}
1, &j \text{ is odd} \\
1/w, &j \text{ is even}
\end{cases}
\end{equation}

for some $w > 0$ and

\begin{equation}\label{eqn: dimer springs}
\V_j'(r)
= \begin{cases}
r + r^2 + \Vscr_1(r), &j \text{ is odd} \\
\kappa{r} + \beta{r}^2 + \Vscr_2(r), &j \text{ is even},
\end{cases}
\qquad
\lim_{r \to 0} \frac{\Vscr_1(r)}{r^2} = \lim_{r \to 0} \frac{\Vscr_2(r)}{r^2} = 0.
\end{equation}

We require $\kappa > 0$ but do not (yet) place any restrictions on $\beta$.
To ensure that the lattice is a dimer, then, we will always take $w \ne 1$ or $\kappa \ne 1$.
Moreover, by relabeling the original nondimensionalized lattice, we may always assume that at least one of $w$ or $\kappa$ is larger than 1.
Finally, when discussing a mass dimer, we will write the potential as

\[
\V(r)
= r+r^2+\Vscr(r), \qquad \lim_{r \to 0} \frac{\Vscr(r)}{r^2} = 0.
\]

Our convention with the dimer's material data in (<ref>) and (<ref>) has been to normalize the “odd” parameters to 1, while the “even” parameters vary.
This is how [21] treated the mass dimer, but [16] used the reverse convention for the spring dimer.
One should keep this in mind when comparing our set-up, in particular our relative displacement traveling wave problem (<ref>), and results to theirs.

The assumption of real analytic spring potentials is not so stringent a restriction as it might seem when we consider that most studies assume a high order of regularity for the potentials.
For example, in the solitary wave papers [23, 18] the potentials are at least $\Cal^5$, while in the nanopteron papers [21, 16, 19] they are $\Cal^{\infty}$.
We discuss in Remark <ref> the utility of $\Cal^{\infty}$-potentials for these latter papers and in Remark <ref> the technically essential reason why we demand real analytic potentials in the current approach.

§.§ Traveling waves in dimers

We will work in two coordinate systems.
First, we can remain in position coordinates and make the traveling wave ansatz

\begin{equation}\label{eqn: position tw ansatz}
= \begin{cases}
p_1(j-ct), &j \text{ is odd} \\
p_2(j-ct), &j \text{ is even.}
\end{cases}
\end{equation}

Here the profiles $p_1$ and $p_2$ are functions of a single real variable and $c \in \R$ is the wave speed.
Then we obtain the system of advance-delay differential equations

\begin{equation}\label{eqn: position tw eqns}
\begin{cases}
c^2p_1'' = \V_1'(S^1p_2-p_1)-\V_2'(p_1-S^{-1}p_2) \\
c^2p_2'' = w\V_2'(S^1p_1-p_2)-w\V_1'(p_2-S^{-1}p_1).
\end{cases}
\end{equation}

For $d \in \R$, the operator $S^d$ is the “shift-by-$d$” operator $(S^dp)(x) := p(x+d)$.

Although we will work primarily in the position traveling wave coordinates (<ref>), this is not the typical framework in the literature, which instead largely studies traveling waves in relative displacements.
For that, put

\begin{equation}\label{eqn: rel disp tw ansatz}
= \begin{cases}
\varrho_1(j-ct), &j \text{ is odd} \\
\varrho_2(j-ct), &j \text{ is even}
\end{cases}
\end{equation}

to find that from the relative displacement equations (<ref>), the new profiles $\varrho_1$ and $\varrho_2$ must satisfy

\begin{equation}\label{eqn: rel disp tw eqns}
\begin{cases}
c^2\varrho_1'' = -(1+w)\V_1'(\varrho_1)+(wS^1+S^{-1})\V_2'(\varrho_2) \\
c^2\varrho_2'' = (S^1+wS^{-1})\V_1'(\varrho_1)-(1+w)\V_2'(\varrho_2).
\end{cases}
\end{equation}

Any solution to the position traveling wave problem (<ref>) yields a solution to the relative displacement traveling wave problem (<ref>) via the definition of relative displacement.
Specifically, if the pair $(p_1,p_2)$ solves (<ref>), then the pair $(\varrho_1,\varrho_2)$ defined by

\begin{equation}\label{eqn: tw profile rels}
\varrho_1 := S^1p_2 - p_1
\quadword{and}
\varrho_2 := S^1p_1-p_2
\end{equation}

solves (<ref>).
This follows by factoring $S^{-1}$ out of the second term on the right in each equation in (<ref>), e.g., $\V_2'(p_1-S^{-1}p_2) = S^{-1}\V_2'(S^1p_1-p_2)$.
However, a relative displacement solution does not necessarily guarantee a position solution; see Question <ref> below.

§.§ Prior nanopteron results for dimers

Now that we possess an adequate vocabulary and notation for traveling wave problems in dimers, we can discuss our historical motivation and then, in the proper context, at last state our results.
Our interest in traveling waves in dimers arises from the fact that over long times, (solitary wave) solutions to certain KdV equations are very good approximations to solutions in relative displacement coordinates of (<ref>).
More precisely, Gaison, Moskow, Wright, and Zhang [28] used techniques from homogenization theory to prove the estimate

\begin{equation}\label{eqn: gmwz est}
\sup_{|t| \le T_0\ep^{-3}} \big|r_j(t) - \ep^2\Ksf_+(\ep(j+\cs{t})) - \ep^2\Ksf_-(\ep(j-\cs{t}))\big|
\le C\ep^{5/2}
\end{equation}

for solutions $r_j$ to (<ref>).
The functions $\Ksf_{\pm}$ are solutions of certain KdV equations whose coefficients are derived from the material data (<ref>) of the lattice, and $C$ and $T_0$ are (material-dependent) constants.
The $\ep$-dependent scaling $\ep^2\Ksf_{\pm}(\ep\cdot)$ is the classical “long wave” scaling; see <cit.> for a concise historical overview.
The wave speed $\cs = \cs(\kappa,w)$ is the lattice's “speed of sound,” and we define it precisely for the dimer in terms of $w$ and $\kappa$ in (<ref>).

The estimate (<ref>) is actually valid for all “$N$-polyatomic” lattices satisfying the $N$-periodicity conditions of (<ref>) and some additional technical hypotheses; in particular, (<ref>) holds for monatomic lattices and dimers without any further hypotheses.
Chirilus-Bruckner, Chong, Prill, and Schneider [10] obtained a similar estimate using Bloch wave transforms.
A version of this estimate was first proved by Schneider and Wayne in [73] for monatomic lattices.
See <cit.> and <cit.> for proofs in the framework of differential equations on $\ell^2(\Z)$ that such relative displacement solutions $r_j$ exist in the first place.

A natural question, then, is if these KdV approximate “solutions” extend to solitary wave solutions of (<ref>) for all time.
Friesecke and Pego [23] answered this question affirmatively for the monatomic lattice.
Their solutions have the form

\begin{equation}\label{eqn: intro mono sol}
= \ep^2\varsigma(\ep(j-\cep{t})) + \ep^3\hsf_{\ep}(\ep(j-\cep{t})),
\end{equation}

where $\varsigma$ is a rescaled $\sech^2$-type solution to a certain KdV equation that naturally arises as the monatomic lattice's “continuum limit.”
The higher-order remainder $\hsf_{\ep} = \hsf_{\ep}(X)$ is exponentially localized in $X$; that is, there exist $C$, $q > 0$ such that $|\hsf_{\ep}(X)| \le Ce^{-q|X|}$ for all $X \in \R$.
The wave speed $\cep$ is “near-sonic” in the sense that $\cep^2 = \cs^2 + \O(\ep^2)$.
Friesecke and Wattis [27] also obtained solitary waves in monatomic lattices using variational techniques.

The results for dimers are rather different.
Via an ansatz established by Beale [3] for the capillary-gravity water wave problem and techniques refined by Amick and Toland [2] for a singularly perturbed KdV-type equation, originally derived in [37], Faver and Wright proved in [21] that the mass dimer has relative displacement traveling waves of the form

\begin{equation}\label{eqn: intro dimer nano}
= \ep^2\varsigma_w(\ep(j-\cep{t}))
+ \ep^2a_{\ep}\psf_j^{\ep}(\ep(j-\cep{t})).
\end{equation}

As before, $\varsigma_w$ is a scaled $\sech^2$-type solution to a KdV equation that is the dimer's “continuum limit”; we give the precise formula for $\varsigma_w$ in (<ref>).
The wave speed is $\cep^2 = \cs^2+\O(\ep^2)$, where now $\cs$ is the dimer's speed of sound.
Both $\hsf_j^{\ep} = \hsf_j^{\ep}(X)$ and the new term $\psf_j^{\ep} = \psf_j^{\ep}(X)$ are smooth in $X$ and $2$-periodic in $j$.

Although the remainder term $\hsf_j^{\ep}$ is exponentially localized in $X$ as in the Friesecke-Pego solution, the new term $\psf_j^{\ep}$ is periodic in $X$ with frequency $\O(\ep^{-1})$.
Moreover, taking

\begin{equation}\label{eqn: per profs FW}
= \ep^2a_{\ep}\psf_j^{\ep}(\ep(j-\cep{t}))
\end{equation}

independently solves the relative displacement traveling wave problem (<ref>).

The “amplitude” coefficient $a_{\ep}$ is small beyond all algebraic orders of $\ep$ in the sense that

\begin{equation}\label{eqn: small BAAO}
\lim_{\ep \to 0} \ep^{-r}a_{\ep}
= 0
\end{equation}

for all $r > 0$.
Thus the traveling wave (<ref>) is a nanopteron, not a solitary wave like (<ref>), unless $a_{\ep} = 0$.
Boyd [5] coined the term “nanopteron” to denote a “nonlocal” or “generalized” solitary wave whose profile consists of the superposition of an exponentially localized “core” and extremely small asymptotic  “ripples.”
We sketch the nanopteron from (<ref>) in Figure <ref>.

A largely similar result holds for relative displacement traveling waves in spring dimers [16]; see Section <ref> for a more detailed leading order expansion.
For the sake of contrast with our methods in this paper, we outline in Appendix <ref> the “Beale's ansatz” method used in [21, 16].

Beale's ansatz has facilitated the construction of traveling wave solutions to a variety of problems beyond the dimer long wave scenario.
Hoffman and Wright [36] used it to construct nanopterons in the dimer “small mass” limit, in which $c$ is now fixed while $m \to 0^+$, thereby making the dimer monatomic.
Faver and Hupkes [19] subsequently produced micropterons (the superposition of a localized core and an algebraically, but not exponentially, small periodic ripple) in the dimer “equal mass” limit, in which $c$ is again fixed and $m \to 1$.
See <cit.> for a succinct comparison of the long wave, small mass, and equal mass scenarios.
Faver [17] studied a related small mass limit for mass-in-mass (MiM) lattices, which are, roughly, monatomic FPUT lattices in which each particle is paired with an internal resonator [11].
There are numerous technical differences among these problems, on which we do not dwell here, but all three share an obvious difference from the long wave results in [21, 16]: the exponentially localized core of the traveling waves in these three “material” limits is always $\O(1)$ in the problem's relevant small parameter.
In contrast, as (<ref>) specifies, the dimers' long wave core is $\O(\ep^2)$.
However, Johnson and Wright [52] used Beale's ansatz to study the gravity-capillary Whitham equation in the long wave limit, and their core was also $\O(\ep^2)$.
This small core naturally arises from the long wave scaling, which is, of course, not present in the material limits.
It is quite interesting to note that Johnson, Truong, and Wheeler [51] also studied the gravity-capillary Whitham equation using a nonlocal center manifold reduction and found, effectively, micropterons; they obtained an $\O(\ep^2)$ core but an explicit nonzero $\O(\ep)$ leading order term for their periodic ripples.

Exact traveling waves are far from the only area of interest for FPUT lattices.
We mention just a handful of related results here and discuss others in the context of future problems in Section <ref>.
Pelinovsky-Schneider [72] considered the dimer small mass limit as an initial value problem in $\ell^2(\Z)$-type sequence spaces and found that if the initial data is close to a solution of the limiting monatomic lattice and the mass ratio is sufficiently small, then the dimer solution remains close to that monatomic solution.
McGinnis and Wright [67] moved well beyond the polyatomic regime to find that the classical wave equation is a good approximation for linear FPUT lattices with “random” material data, i.e., their potentials are $\V_j'(r) = \kappa_jr$, where $\kappa_j$ and the masses $m_j$ are all random variables.
Carmichael [8] proved a KdV approximation result like (<ref>) for monatomic lattices with “planar” motion, where the lattice is not constrained to the horizontal like ours but can move in two dimensions.

\[
\input{fig-nanopteron}
\]

The nanopteron traveling wave profile from (<ref>) for the mass dimer (inspired by <cit.>).
In the inset, we write $\O(\ep^{\infty})$ to indicate that the amplitude of the periodic ripples is small beyond all orders of $\ep$.

§.§ Guiding questions about nanopterons in FPUT dimers

The nanopteron results in [21, 16] suggest a number of interesting questions for FPUT dimers.

Is it possible that the periodic amplitude coefficient $a_{\ep}$ is 0?
In this case, the nanopteron reduces to a solitary wave.
There is a large body of numerical evidence in mass dimers suggesting that for a discrete sequence of mass ratios that accumulates at 0, the traveling wave problem (<ref>) has solitary wave solutions [83, 77, 65, 66, 20].
The dimer small mass limit [36] must exclude such a sequence of mass ratios from its nanopteron constructions, although nonexistence of nanopterons is not established; a similar phenomenon occurs in [17], although there solitary waves can be constructed at the excluded mass ratios [18].
However, it is less clear if, given a fixed mass ratio, the long wave solution (<ref>) can be a solitary wave for some value(s) of $\ep$.
This has been resolved for other long wave problems; Sun [79] proved nonexistence of solitary waves for the water wave problem, as did Amick and McLeod [1] for the Amick–Toland model KdV equation.

Is it possible to obtain a sharper upper bound on the periodic amplitude in the form of an exponential estimate like $|a_{\ep}| \le A_1e^{-A_2/\ep}$?
Here $A_1$ and $A_2$ should be $\ep$-independent constants.
Boyd [5] actually terms nanopterons only those nonlocal solitary waves whose ripples are exponentially small in a relevant small parameter, not small beyond all algebraic orders as in (<ref>).
Thus our terminology is somewhat of an abuse of Boyd's.

The ripples in Beale's water wave nanopterons, as well as those of Amick and Toland, are “only” small beyond all algebraic orders, but Sun and Shen [81, 80] proved that both families of ripples are indeed exponentially small.
Conversely, Champneys and Lord [9] gave numerical evidence for extending the Amick–Toland nanopterons both to larger-amplitude periodic solutions and for larger values of the small parameter than the theory indicates.

The relative displacement problem is translation invariant in the sense that if the family $\{r_j\}_{j \in \Z}$ solves (<ref>), then so does $\{r_{j+d}\}_{j \in \Z}$ for any $d \in \Z$.
Thus, per (<ref>), taking

\[
= \ep^2a_{\ep}\psf_j^{\ep}(\ep(j-\cep{t})+\ep{d})
\]

also solves the mass (or spring) dimer version of (<ref>).
Can we then replace $\psf_j^{\ep}$ in (<ref>) with this shifted version?
More precisely, are there relative displacement solutions $r_j$ with

\[
\lim_{t \to \pm\infty} \big|r_j(t) -\ep^2\varsigma_w(\ep(j-\cep{t}))- \ep^2a_{\ep}\psf_j^{\ep}(\ep(j-ct \pm d))\big|
= 0
\]

exponentially fast, for some — or all — shifts $d$?
For symmetry concerns, we allow a sign change in the phase shift from the asymptotics at $-\infty$ to $+\infty$.

Amick and Toland [2] achieved this for their model equation; Sun's results in [78] further elaborate on the consequent dependence of the ripple amplitude on the phase shift.
A related construction of phase-shifted nanopterons is proposed, though not executed, for the mass dimer in <cit.>.
Since the proofs in [2, 21, 16] hinge in no small part on symmetries (evenness/oddness of various terms in the nanopteron ansatz), and since shifts destroy symmetries (e.g., if $f$ is even and periodic, then $S^df$ is typically not even), this will require some quite delicate attention.

What, if anything, can be deduced about position traveling waves from these relative displacement results?
We can formally solve for position from relative displacement via identities like

\[
u_j = \sum_{k=-\infty}^{j-1} r_k
\quadword{or}
u_j = \begin{cases}
u_0+\medsum_{k=0}^{j-1} r_k, \ j \ge 1 \\
u_0-\medsum_{k=j}^{-1} r_k, \ j \le -1.
\end{cases}
\]

However, barring some decay conditions on $r_k$, there is no guarantee that the series will converge, while analysis of the finite sums requires at least knowledge of $u_0$ (or $u_{j_*}$ for some fixed $j_* \in \Z$).
For example, the exponential localization of the monatomic solitary waves provides such sufficient decay, see <cit.>, while the Friesecke–Wattis results [27] were developed in position coordinates.
However, if $r_k$ is a nanopteron, then the periodic tails will almost surely prevent convergence of the series, at least for some time values.

Working at the level of the traveling wave profiles $p_k$ from (<ref>) and $\varrho_k$ from (<ref>) is no more enlightening.
We might be tempted to try to solve for $p_k$ in terms of $\varrho_k$ via (<ref>).
Doing so pointwise is simply fruitless.
Given the reliance of [23, 21, 16] on Fourier analysis, we might try to express (<ref>) using Fourier transforms.
There is no guarantee that all of $p_k$ and $\varrho_k$ have well-defined transforms, and even if they do, we uncover the relation

\[
\begin{pmatrix*}
\hat{\varrho}_1(k) \\
\hat{\varrho}_2(k)
\end{pmatrix*}
= \begin{bmatrix*}[r]
-1 &e^{ik} \\
e^{ik} &-1
\end{bmatrix*}
\begin{pmatrix*}
\hat{p}_1(k) \\
\hat{p}_2(k)
\end{pmatrix*}.
\]

The matrix above is, of course, singular for $k \in 2\pi\Z$, and so there is no clear way to invert the system to solve for $\hat{p}_1$ and $\hat{p}_2$.

All of this is to say that we are not optimistic about working backwards from relative displacement results to position results.

§.§ The main results

Motivated by these four questions, we approach the long wave problem in mass and spring dimers from a substantially different perspective from that of [21, 16]: spatial dynamics.
We present and interpret our results below and then give an overview of the spatial dynamics method in Section <ref>.
We defer a comparison of these techniques to the prior Beale's ansatz method until Appendix <ref>, by which point we will have presented sufficient technical detail for both methods to make worthwhile remarks.

Here are our results for the mass dimer, which we prove in Section <ref>.

Suppose $w > 1$ in (<ref>) and $\V_1 = \V_2$ in (<ref>).

\begin{equation}\label{eqn: qw}
:= \left(\frac{6w(1+w)}{w^2-w+1}\right)^{1/2}
\end{equation}

and let $q \in (0,q_w)$.
There are constants $\ep_*$, $\Alpha_0$, $\Alpha_1$, $\Alpha_{\infty}$, $\omega_w > 0$ such that if $0 < \ep < \ep_*$ and

\begin{equation}\label{eqn: cep intro}
\cep
:= \left(\frac{1+w}{2w}-\ep^2\right)^{-1/2},
\end{equation}

then the following hold.

Position “periodic + growing” solutions
For $0 < \ep < \ep_*$ and $0 \le \alpha \le \Alpha_1$, there are real analytic maps $\varphi_{j,\ep}^{\alpha}$ and $\Gsf_{\ep}^{\alpha}$ such that

\begin{equation}\label{eqn: md pos per soln}
= \alpha\ep^2\varphi_{j,\ep}^{\alpha}(\ep(j-\cep{t}))+ \alpha\ep(\alpha+\ep^2)\Gsf_{\ep}^{\alpha}(\ep(j-\cep{t}))
\end{equation}

solves the equations of motion (<ref>) in position coordinates.
These maps satisfy the estimate

\begin{equation}\label{eqn: varphi G est md}
\sup_{\substack{0 < \ep < \ep_* \\ 0 \le \alpha \le \Alpha_1}} \left(\sup_{X \in \R} |\varphi_{j,\ep}^{\alpha}(X)| + \sup_{X \in \R\setminus\{0\}} \frac{|\Gsf_{\ep}^{\alpha}(X)|}{|X|}\right)
< \infty.
\end{equation}

The maps $\varphi_{j,\ep}^{\alpha}$ are $2$-periodic in $j$ and $2\pi\ep(\omega_w+\ep\varpi_{\ep}^{\alpha})$-periodic in $X$, where

\[
\sup_{\substack{0 < \ep < \ep_* \\ 0 \le \alpha \le \Alpha_1}} |\varpi_{\ep}^{\alpha}|
< \infty.
\]

Position “growing front” solutions
For $0 < \ep < \ep_*$ and $\Alpha_0\ep{e}^{-\Alpha_{\infty}/\ep} \le \alpha \le \Alpha_1$, there are real analytic maps $\eta_{j,\ep}^{\alpha}$ and real numbers $\Lsf_{\ep}^{\alpha}$ and $\theta_{\ep}^{\alpha}$ such that if

\begin{equation}\label{eqn: Xi-ep-alpha intro}
\Tsf_{\ep}^{\alpha}(X)
= X+\ep^2\theta_{\ep}^{\alpha}\tanh\left(\frac{q_wX}{2}\right)
\end{equation}

\begin{multline}\label{eqn: md pos F}
\Fsf_{j,\ep}^{\alpha}(X)
= \left[\left(\frac{6w(w^2-w+1)}{(1+w)^3}\right)^{1/2}+\ep\Lsf_{\ep}^{\alpha}\right]\tanh\left(\frac{q_wX}{2}\right)
+ \ep\eta_{j,\ep}^{\alpha}(X)
+ \alpha\ep\varphi_{j,\ep}^{\alpha}(\Tsf_{\ep}^{\alpha}(X)) \\
+ \alpha(\alpha+\ep^2)\Gsf_{\ep}^{\alpha}(\Tsf_{\ep}^{\alpha}(X)),
\end{multline}

with $\varphi_{j,\ep}^{\alpha}$ and $\Gsf_{\ep}^{\alpha}$ defined in part <ref>, then

\begin{equation}\label{eqn: md pos tw final}
= \ep\Fsf_{j,\ep}^{\alpha}(\ep(j-\cep{t}))
\end{equation}

solves (<ref>).
The scaled profile $\ep\Fsf_{j,\ep}^{\alpha}(\ep\cdot)$ is sketched in Figure <ref>.

There are constants $C_{\pm} > 0$ such that

\[
C_- < \theta_{\ep}^{\alpha} < C_+
\quadword{and}
|\Lsf_{\ep}^{\alpha}| < C_+
\]

for all $\ep$ and $\alpha$, while $\eta_{j,\ep}^{\alpha}$ is $2$-periodic in $j$ and satisfies

\begin{equation}\label{eqn: md pos est}
\sup_{\substack{0 < \ep < \ep_* \\ \Alpha_0\ep{e}^{-\Alpha_{\infty}/\ep} \le \alpha \le \Alpha_1 \\ X \in \R}} e^{q|X|}|\eta_{j,\ep}^{\alpha}(X)|
< \infty.
\end{equation}

Relative displacement exact periodic solutions
For $0 < \ep < \ep_*$ and $0 \le \alpha \le \Alpha_1$, there are real analytic maps $\tvarphi_{j,\ep}^{\alpha}$ such that

\begin{equation}\label{eqn: md rel disp per soln}
= \alpha\ep^2\tvarphi_{j,\ep}^{\alpha}(\ep(j-\cep{t}))
\end{equation}

solves the equations of motion (<ref>) in relative displacement coordinates.
These maps do not vanish identically and satisfy the estimates

\begin{equation}\label{eqn: md rel disp per nonvanishing}
\inf_{\substack{0 < \ep < \ep_* \\ 0 \le \alpha \le \Alpha_1}} \sup_{X \in \R} |\tvarphi_{j,\ep}^{\alpha}(X)| > 0
\quadword{and}
\sup_{\substack{0 < \ep < \ep_* \\ 0 \le \alpha \le \Alpha_1 \\ X \in \R}} |\tvarphi_{j,\ep}^{\alpha}(X)| < 0.
\end{equation}

They are periodic in $X$, with the same period as $\varphi_{j,\ep}^{\alpha}$, and $2$-periodic in $j$.

Relative displacement nanopteron solutions
For $0 < \ep < \ep_*$ and $\Alpha_0\ep{e}^{-\Alpha_{\infty}/\ep} \le \alpha \le \Alpha_1$, there are real analytic maps $\teta_{j,\ep}^{\alpha}$ such that if

\begin{equation}\label{eqn: md rel disp N}
\Nsf_{j,\ep}^{\alpha}(X)
= \frac{3w}{1+w}\sech^2\left(\frac{q_wX}{2}\right)
+ \alpha\tvarphi_{j,\ep}^{\alpha}(\Tsf_{\ep}^{\alpha}(X))
+ \ep\teta_{j,\ep}^{\alpha}(X),
\end{equation}

with $\Tsf_{\ep}^{\alpha}$ defined in (<ref>) and $\tvarphi_{j,\ep}^{\alpha}$ in part <ref>, then

\begin{equation}\label{eqn: md rel disp tw final}
= \ep^2\Nsf_{j,\ep}^{\alpha}(\ep(j-\cep{t}))
\end{equation}

solves (<ref>).
The maps $\teta_{j,\ep}^{\alpha}$ are $2$-periodic in $j$ with

\[
\sup_{\substack{0 < \ep < \ep_* \\ \Alpha_0\ep{e}^{-\Alpha_{\infty}/\ep} \le \alpha \le \Alpha_1 \\ X \in \R}}
< \infty.
\]

Next, we present our position and relative displacement solutions for spring dimers; we prove the following theorem in Section <ref>.

Suppose $w = 1$ in (<ref>) and, in (<ref>), take $\kappa > 1$ and $\beta \in \R$ with $\kappa +\beta^3 \ne 0$.

\begin{equation}\label{eqn: qkappa}
:= \left(\frac{6\kappa(1+\kappa)}{\kappa^2-\kappa+1}\right)^{1/2}
\end{equation}

and let $q \in (0,q_{\kappa})$.
There are constants $\ep_*$, $\Alpha_0$, $\Alpha_1$, $\Alpha_{\infty}$, $\omega_{\kappa} > 0$ such that if $0 < \ep < \ep_*$ and

\[
\cep
:= \left(\frac{1+\kappa}{2\kappa}-\ep^2\right)^{-1/2},
\]

then the following hold.

* Position “periodic + growing” solutions
For $0 < \ep < \ep_*$ and $0 \le \alpha \le \Alpha_1$, there are real analytic maps $\varphi_{j,\ep}^{\alpha}$ and $\Gsf_{\ep}^{\alpha}$ such that

\begin{equation}\label{eqn: sd pos per soln}
= \alpha\ep^2\varphi_{j,\ep}^{\alpha}(\ep(j-\cep{t}))+ \alpha\ep(\alpha+\ep^2)\Gsf_{\ep}^{\alpha}(\ep(j-\cep{t}))
\end{equation}

solves the equations of motion (<ref>) in position coordinates.

* Position “growing front” solutions
For $0 < \ep < \ep_*$ and $\Alpha_0\ep{e}^{-\Alpha_{\infty}/\ep} \le \alpha \le \Alpha_1$, there are real analytic maps $\eta_{j,\ep}^{\alpha}$ and real numbers $\Lsf_{\ep}^{\alpha}$ and $\theta_{\ep}^{\alpha}$ such that if

\begin{multline}\label{eqn: sd pos F}
\Fsf_{j,\ep}^{\alpha}(X)
= \left(\frac{[6\kappa^3(1+\kappa)(\kappa^2-\kappa+1)]^{1/2}}{2(\beta+\kappa^3)} + \ep\Lsf_{\ep}^{\alpha}\right)\tanh\left(\frac{q_{\kappa}X}{2}\right)
+ \ep\eta_{j,\ep}^{\alpha}(X)
+ \alpha\ep\varphi_{j,\ep}^{\alpha}(\Tsf_{\ep}^{\alpha}(\ep{x})) \\
+ \alpha(\alpha+\ep^2)\Gsf_{\ep}^{\alpha}(\Tsf_{\ep}^{\alpha}(\ep{x})).
\end{multline}

\begin{equation}\label{eqn: sd pos tw final}
= \ep\Fsf_{j,\ep}^{\alpha}(\ep(j-\cep{t}))
\end{equation}

solves (<ref>).

* Relative displacement exact periodic solutions
For $0 < \ep < \ep_*$ and $0 \le \alpha \le \Alpha_1$, there are real analytic maps $\tvarphi_{j,\ep}^{\alpha}$ such that

\begin{equation}\label{eqn: sd rel disp per soln}
= \alpha\ep^2\tvarphi_{j,\ep}^{\alpha}(\ep(j-\cep{t}))
\end{equation}

solves the equations of motion (<ref>) in relative displacement coordinates.

* Relative displacement nanopteron-stegoton solutions
For $0 < \ep < \ep_*$ and $\Alpha_0\ep{e}^{-\Alpha_{\infty}/\ep} \le \alpha \le \Alpha_1$, there are real analytic maps $\teta_{j,\ep}^{\alpha}$ such that if

\begin{equation}\label{eqn: sd rel disp N}
\Nsf_{j,\ep}^{\alpha}(X)
= \kappa^{[(-1)^j+1]/2}\frac{3\kappa^2}{\beta+\kappa^3}\sech^2\left(\frac{q_{\kappa}X}{2}\right)
+ \alpha\tvarphi_{j,\ep}^{\alpha}(\Tsf_{\ep}^{\alpha}(X))
+ \ep\teta_{j,\ep}^{\alpha}(X).
\end{equation}

\begin{equation}\label{eqn: sd rel disp tw final}
= \ep^2\Nsf_{j,\ep}^{\alpha}(\ep(j-\cep{t}))
\end{equation}

solves (<ref>).

The functions $\varphi_{j,\ep}^{\alpha}$, $\tvarphi_{j,\ep}^{\alpha}$, $\Gsf_{\ep}^{\alpha}$, $\eta_{j,\ep}^{\alpha}$, $\teta_{j,\ep}^{\alpha}$, and $\Tsf_{\ep}^{\alpha}$ and the scalars $\theta_{\ep}^{\alpha}$ and $\Lsf_{\ep}^{\alpha}$ have the same properties as their counterparts throughout Theorem <ref>.
In particular, $\varphi_{j,\ep}^{\alpha}$ and $\tvarphi_{j,\ep}^{\alpha}$ are $2\pi\ep(\omega_{\kappa}+\ep\varpi_{\ep}^{\alpha})$-periodic in $X$.

In the following sections we analyze and interpret various consequences of these two theorems.
We address the extent to which our results answer the four motivating questions above, and we compare our conclusions to the prior dimer nanopteron results.

§.§.§ Growing fronts for position and nanopterons for relative displacement

We have answered Question <ref> on position traveling waves: the position profiles $\Fsf_{j,\ep}^{\alpha}$ in (<ref>) and (<ref>) are “growing fronts” with ripples.
We sketch $\Fsf_{j,\ep}^{\alpha}$ in Figure <ref>.
To leading order it is dominated by the $\tanh$-term and then at higher order exhibits small ripples via $\varphi_{j,\ep}^{\alpha}$ and (possibly) linear growth in $\Gsf_{\ep}^{\alpha}$, due to the estimate (<ref>).
We discuss below in Remark <ref> why we do not necessarily rule out this linear growth.

\[
\input{fig-front}
\]

The slowly linearly growing front profile for position traveling waves.

Also, we do not see quite the classical long wave scaling in the position solutions (<ref>) and (<ref>).
In each case, the leading order term from this scaling is only $\O(\ep)$, not $\O(\ep^2)$.

However, we do recover both the long wave scaling and nanopterons at the level of relative displacement.
The relative displacement profiles (<ref>) and (<ref>) are each the superposition of an exponentially localized function and a periodic function, and the relative displacement solutions (<ref>) and (<ref>) are $\O(\ep^2)$ at leading order.

§.§.§ The periodic terms

The periodic terms $\varphi_{j,\ep}^{\alpha}$ in the position front and $\tvarphi_{j,\ep}^{\alpha}$ in the relative displacement nanopteron exist for all $\alpha$ in the interval $[0,\Alpha_1]$, even though the fronts and nanopterons are only defined for $\alpha$ in the range $[\Alpha_0\ep{e}^{-\Alpha_{\infty}/\ep},\Alpha_1]$.
Inspired by <cit.>, we sketch in Figure <ref> the different parameter ranges in the $(\ep,\alpha)$-plane for which the periodic terms and the fronts/nanopterons exist.

We remark that the auxiliary position solutions (<ref>) and (<ref>), which include the periodic terms $\varphi_{j,\ep}^{\alpha}$, need not be periodic themselves, due to the presence of the $\Gsf_{\ep}^{\alpha}$ term.
This is a marked contrast to the exact periodic solutions (<ref>) and (<ref>) for relative displacement.

The periodic terms in the relative displacement nanopteron profiles are definitely not identically zero, due to the estimates (<ref>).
While the amplitude $\alpha$ can be taken to be exponentially small in $\ep$, it can also extend to a small $\O(1)$ threshold independent of $\ep$.
This control over $\alpha$ addresses Questions <ref> and <ref>.
However, while none of the relative displacement solutions that we construct are solitary waves, we did not prove the nonexistence of supersonic solitary traveling waves in dimers; see Section <ref> for further discussion of this issue.

Finally, we are not free to pick the phase shift $\theta_{\ep}^{\alpha}$ to be an arbitrary number; instead, as we discuss in Appendix <ref>, we must accept the very particular value $\ep^2\theta_{\ep}^{\alpha}$.
Nonetheless, since this is $\O(\ep^2)$, our phase shift is not particularly large.
Because our results do not permit an arbitrary phase shift, we feel that the modifications to Beale's method for periodic phase shifts, as discussed in [21, 2], will offer a more satisfactory introduction of shifts into the solutions, and a fuller resolution of Question <ref> remains to be accomplished.

\[
\input{fig-periodic_range}
\]

Domain of existence for (linearly growing) periodic solutions.

\[
\input{fig-full_range}
\]

Domain of existence for (linearly growing) fronts and nanopterons

Ranges of the long wave parameter $\ep$ and the amplitude parameter $\alpha$ (based on <cit.>)

§.§.§ Comparisons with prior dimer nanopterons

The nanopterons for relative displacement in mass dimers that Faver and Wright constructed via Beale's method <cit.> have the form

\[
= \nu^2\varsigma_w(\nu(j-C_{\nu}{t})) + \O(\nu^3),
\]

pointwise in $j$ and $t$, where

\begin{equation}\label{eqn: varsigma-w}
\varsigma_w(X)
:= \frac{3}{2}\left(\frac{1+w}{2w}\right)\sech^2\left(\left(\frac{1+w}{2w}\right)\frac{q_wX}{2}\right)
\end{equation}

\begin{equation}\label{eqn: C-nu}
:= \left(\frac{2w}{1+w} + \nu^2\right)^{1/2}.
\end{equation}

The factor $q_w$ was defined in (<ref>), while $\varsigma_w$ appeared, albeit obliquely, as the leading order term in (<ref>).
We are now writing $\nu$, not $\ep$, for the long wave small parameter here.

If we take

\begin{equation}\label{eqn: cs rescaling}
\nu
= \frac{2w}{1+w}\ep,
\end{equation}

\[
\nu^2\varsigma_w(\nu{X})
= \frac{3w}{1+w}\ep^2\sech^2\left(\frac{q_w\ep{X}}{2}\right).
\]

A comparison with (<ref>) and (<ref>) shows that this is exactly the leading order $\sech^2$-term of our nanopterons.
Likewise, with $\cep$ defined in (<ref>) and $C_{\nu}$ in (<ref>), we find

\[
\cep^2-C_{\nu}^2
= \left(\frac{1+w}{2w}-\ep^2\right)^{-1}-\left(\frac{2w}{1+w}+\left(\frac{2w}{1+w}\right)^2\ep^2\right)
= \O(\ep^4),
\]

and so the rescaled wave speed from Beale's method agrees with ours to $\O(\ep^4)$.
Finally, we remark that the scaling factor $2w/(1+w)$ in (<ref>) is exactly the speed of sound for the mass dimer, see (<ref>).

A similar rescaling of the small parameter shows the leading order agreement of our spring dimer nanopterons with those constructed by Faver <cit.>.
In that paper, nanopterons have the form

\[
= \nu^2\kappa^{[(-1)^j+1]/2}\varsigma_{\kappa,\beta}(\nu(j-C_{\nu}{t})) + \O(\nu^3),
\]

where now

\[
\varsigma_{\kappa,\beta}(X)
:= \frac{3(1+\kappa)^2}{4(\beta+\kappa^3)}\sech^2\left(\left(\frac{1+\kappa}{2\kappa}\right)\frac{q_{\kappa}X}{2}\right),
\]

\[
:= \left(\frac{2\kappa}{1+\kappa}+\nu^2\right)^{1/2},
\]

and $q_{\kappa}$ was defined in (<ref>).
It is, unsurprisingly, necessary to assume $\beta+\kappa^3\ne0$ here, too.

\[
\nu
= \frac{2\kappa}{1+\kappa}\ep,
\]

we find

\[
\kappa^{[(-1)^j+1]/2}\varsigma_{\kappa,\beta}(X)
= \kappa^{[(-1)^j+1]/2}\frac{3\kappa^2}{\beta+\kappa^3}\ep^2\sech^2\left(\frac{q_{\kappa}\ep{X}}{2}\right),
\]

and this is our leading order $\sech^2$-term in (<ref>).
The scaling factor $2\kappa/(1+\kappa)$ is now the speed of sound for the spring dimer.

§.§.§ Stegotons in the spring dimer

At first glance Theorems <ref> and <ref> may look identical, and they essentially are, up to one key difference.
The spring dimer relative displacement profile, given in (<ref>), contains a factor on its leading-order localized term that is either $\kappa$ or 1, alternating with the parity of lattice site $j$.
Since we assume $\kappa > 1$ for the spring dimer, this means the leading order term is either slightly larger or smaller depending on $j$.
This phenomenon is observed in [16], which uses the parlance “stegoton” of LeVeque and Yong [57, 58] to describe this alternating behavior in relative displacement; our theorem below, like the result in [16], analytically confirms the stegoton's existence.

§.§.§ Some physical interpretations

Here is how we physically interpret Theorem <ref>; the same will hold for Theorem <ref>.
The “growing front” results for position coordinates mean that, over time, a fixed particle drifts further and further away from its equilibrium position.
If the term $\Gsf_{\ep}^{\alpha}$ is indeed bounded and not truly linearly growing, then that drift will asymptote to some constant length with small “wiggles” due to the ripples $\varphi_{j,\ep}^{\alpha}$.

At first glance the nanopteron results for relative displacement many seem strange; after all, while the $\sech^2$-term dominates the nanopteron, as this term decays the periodic ripples could induce negative values in relative displacement (especially if we do not select the amplitude $\alpha$ to be exponentially small).
That, however, would allow pairs of particles to “cross” or “move through” each other, which is physically bizarre.
(To be fair, with the linearly growing front we are allowing the possibility that the springs become arbitrarily long, which is also physically bizarre.)
But recall our first nondimensionalization of the problem in (<ref>) to take the spring equilibrium lengths to be 0.
And so, with $y_j$ as the original position coordinates from (<ref>), we find from (<ref>) that

\[
= r_j + \ell_j,
\]

where $r_j$ is the nanopteron from (<ref>).
By restricting $\ep$ and $\alpha$, we can make this $r_j$ as small as we like uniformly in time, and so over very long times, the distance between pairs of particles just settles down to oscillate around the equilibrium length of the spring connecting the particles.

§.§ The spatial dynamics method

The progenitors of this “spatial dynamics” method for lattice problems were Iooss and Kirchgässner [43], who applied it to the position traveling wave problem for Klein–Gordon lattices.
Such lattices consist, roughly, of a monatomic FPUT lattice with linear nearest-neighbor spring forces and an additional “on-site” potential applied to each particle.
The Iooss–Kirchgässner methods have been adapted to a host of subsequent lattice problems, including [7, 39, 46, 75, 76, 45, 41, 35].
Accessible surveys of their techniques appear in [47] and <cit.>.
In broad strokes, they make a special change of variables to convert their second-order, nonlocal traveling wave equation into a first-order ordinary differential equation on a particular infinite-dimensional Banach space, with the “spatial” variable of the traveling wave profile now taking the role of the “time” variable in this differential equation, hence the term “spatial dynamics” for this overall approach.
A key idea is that instead of shifting with $(S^{\pm1}p)(x) = p(x\pm1)$ as in (<ref>), one replaces the shifts with “continuous variables” by putting $P(x,v) := p(x+v)$, so that the shift operator $S^{\pm1}p$ becomes the “evaluation” operator $P(x,\cdot) \mapsto P(x,\pm1)$.
We perform this change of variables for our problem (<ref>) in Section <ref>.
Prior to studying the Klein–Gordon lattice, Iooss and Kirchgässner had earlier applied the spatial dynamics method to water wave problems [42]; see the introduction to that paper for a detailed historical overview of the method in the water wave context, and also the articles [14, 40, 64].
The myriad underlying techniques date back to Kirchgässner [55] and Mielke [69].

For future reference, let us write the first-order equation that Iooss and Kirchgässner, and successors, would study as

\begin{equation}\label{eqn: sd intro}
= \L_0U(x) + \mu\L_1(\mu)U(x) + \nl(U(x),\mu),
\end{equation}

where $\mu$ is the appropriate expression of the long wave parameter in Iooss–Kirchgässner coordinates.
The operators $\L_0$ and $\L_1(\mu)$ are linear and $\nl$ is nonlinear.
A deft combination of spectral theory, normal form analysis, and center manifold theory can then capture, in different regimes and to different degrees of detail, small solutions to (<ref>) — in particular, nanopterons.
The existence of nanopterons is typically established by invoking the comprehensive results of Lombardi's magisterial monograph [62], which, together with the articles [63, 60, 59, 61], provides a far-reaching set of hypotheses under which a problem like (<ref>) has nanopteron solutions.
These hypotheses are valid in general Banach spaces and not specific to lattice-type problems.
We give a careful overview of Lombardi's method in Appendix <ref> to put our language and methods in better dialogue with his.
The interested reader may wish at this point to consult our phrasing of Lombardi's nanopteron existence results in Theorem <ref>.

We were specifically inspired to deploy the spatial dynamics method on FPUT lattices by Venney and Zimmer [85], who studied monatomic lattices with nearest and next-to-nearest neighbor couplings — a spring dimer, after a fashion, though of a rather different genus from ours.
Venney and Zimmer, in turn, adapted techniques that Iooss and James [41] used to construct traveling breathers in monatomic lattices.
Neither Venney and Zimmer, nor Iooss and James, obtained quite the same problem that Iooss and Kirchgässner encountered, and neither the Venney–Zimmer nor the Iooss–James problem was directly amenable to Lombardi's techniques.
The chief obstacle in [85, 41] to a direct application of Lombardi's results was that the center spectrum of their versions of the operator $\L_0$ from (<ref>) consisted of 0 (with algebraic multiplicity 4 and geometric multiplicity 1) and $\pm{i}\omega$ (both simple eigenvalues) for a certain “critical frequency” $\omega > 0$.
This is the same situation as in our FPUT dimers.
However, Lombardi calls for a double eigenvalue at 0.
By exploiting translation invariance and a conserved first integral inherent to their respective lattice problems, both pairs of authors were able to change variables and work on a “reduced” version of (<ref>) for which the linearization's eigenvalue at 0 does have multiplicity 2.

Building on the work of our spatial dynamics predecessors, here are the chief contributions of this paper to the spatial dynamics method for lattices, in addition to the FPUT-specific results in Theorems <ref> and <ref>.

* We state seven exact hypotheses under which we can solve an equation like (<ref>), posed in an arbitrary Banach space, when $\L_0$ has an eigenvalue of algebraic multiplicity 4 at 0.
We enumerate these hypotheses in Section <ref> and state their consequence for (<ref>) as Theorem <ref>.
The language of our hypotheses is inspired by Lombardi's assumptions in <cit.> and intended to facilitate the invocation of his results on our reduced problem, for which the linearization at $\mu=0$ has a double eigenvalue at 0.
Our arrangement of the hypotheses also arose by observing the common threads among the problems of [41, 85] and our FPUT dimers and distilling them into more general terminology than previously employed (particularly in the cases of our Hypotheses <ref> and <ref>).
Working at the level of these abstract hypotheses allows us to see precisely what the mass and spring dimers have in common, where they differ, and what is true for a general dimer, all without being obscured by the complex dependencies on the four parameters $c$, $w$, $\kappa$, and $\beta$.
We check the hypotheses for the FPUT dimers in Section <ref> and then analyze the consequences of the abstract Theorem <ref> for dimer position traveling waves in Section <ref>.

* We provide a thorough treatment of the reduced problem that converts (<ref>) into the algebraic multiplicity 2 situation.
For reference, write this reduced problem as

\begin{equation}\label{eqn: sd intro reduced}
= \tL_0W + \mu\tL_1(\mu)W + \tnl(W,\mu)
\end{equation}

for some linear operators $\tL_0$ and $\tL_1(\mu)$ and a nonlinear operator $\tnl$.
In Section <ref>, we show how any (small) solution of (<ref>) also solves (<ref>), and, importantly, we explain how any solution to the reduced problem really does yield a solution to the full original problem (<ref>).
This reduced problem is not the finite-dimensional system that would arise from the center manifold theorem, which we do not use, so as to preserve the analyticity of various maps, which in turn permits the application of Lombardi's analytic-dependent hypotheses.
The operators $\tL_0$, $\tL_1(\mu)$, and $\tnl$ in (<ref>) are really $\O(1)$ perturbations of the original operators $\L_0$, $\L_1(\mu)$, and $\nl$ from (<ref>), and as such we cannot capture the properties of these new operators using ordinary perturbation theory.
With an eye toward our eventual nanopteron construction in relative displacement coordinates, we also prove a host of exact estimates on various terms and operators in the reduced problem.
These are all details that are not treated with particular precision in [41, 85].
They are necessary not only for logical completeness but also to allow us to compare the leading order behavior of our spatial dynamics nanopterons against those previously found by Beale's method, as we discussed in Section <ref>.

* In Appendix <ref> we generalize Lombardi's nanopteron program from [62].
Since Lombardi originally aimed his treatment at the water wave problem, we have adapted some of his hypotheses to encompass broader problems.
We feel that our gloss of Lombardi's intricate methods offers an efficient, accessible outline that will be useful not just to work in concert with our hypotheses but for future nanopteron constructions in settings not limited to water waves or lattices.

§.§ Notation

For clarity, we remark on some aspects of our (mostly standard) notation.

* We abbreviate $\R_+ := (0,\infty)$.

* If $\X$ is a vector space, then $\ind_{\X}$ is the identity operator on $\X$.

* If $\X$ and $\Y$ are normed spaces, then $\b(\X,\Y)$ is the space of bounded linear operators from $\X$ to $\Y$.
We write $\b(\X) := \b(\X,\X)$.
We denote the dual space of $\X$ by $\X^* = \b(\X,\R)$.

* If $\X$ and $\Y$ are vector spaces and $\Zcal\subseteq \X$ is a subspace of $\X$, and if $\A \colon \X \to \Y$ is a linear operator, then we write $\restr{\A}{\Zcal}$ for the restriction of $\A$ to $\Zcal$.

* If $\X$ and $\Y$ are normed spaces, $f \colon \X \to \Y$ is differentiable, and $U$, $\grave{U} \in \X$, then we denote the derivative of $f$ at $U$ evaluated at $\grave{U}$ by $Df(U)\grave{U}$.
Likewise, the second derivative of $f$ at $U$ evaluated at $(\grave{U},\breve{U})$ is $D^2f(U)[\grave{U},\breve{U}]$.
Occasionally we will write $D_Uf$ instead of $Df$ if $f$ is a function of more than just $U$.
For a function $f = f(x)$ of a real variable $x$ (complex variable $z$), we sometimes write $\partial_x[f] = f'$ ($\partial_z[f] = f'$).

* If $\X$ is a normed space, then for a function $f \colon \R \to \X$ we put

\[
\Lip(f)
:= \sup_{\substack{x,\grave{x} \in \R \\ x \ne \grave{x}}} \frac{\norm{f(x)-f(\grave{x})}_{\X}}{|x-\grave{x}|}.
\]

* If a quantity $f$ depends on the variable $x$ and some additional “parameters,” say, $\mu$ and $\nu$, we will often write $f(x;\mu,\nu)$; in general, data “after the semi-colon” is not part of the independent variable.
If we are considering the derivative of $f$ just with respect to $x$, we will write $Df(x;\mu,\nu)$.

* We review a number of conventions and notations for spectral theory in Appendix <ref>.

§.§ Acknowledgments

We acknowledge support from the Netherlands Organization for Scientific Research (NWO) (grant 639.032.612).

§ THE ABSTRACT PROBLEM

§.§ The hypotheses

Let $\D$ and $\X$ be Banach spaces with $\D$ continuously embedded in $\X$.
We will denote the norms of these spaces by $\norm{\cdot}_{\D}$ and $\norm{\cdot}_{\X}$, respectively.
For some $\mu_0 > 0$, we consider a family of maps $\F \colon \D \times [0,\mu_0] \to \X$ of the form (<ref>) given below, and we will construct solutions to the differential equation

\begin{equation}\label{eqn: abstract ode}
= \F(U(x),\mu)
\end{equation}

that are the superposition of a front, an exponentially localized term, a small-amplitude periodic term, and a (possibly) linearly growing term.
To be clear, a solution to (<ref>) is a map $U \in \Cal(\R,\D) \cap \Cal^1(\R,\X)$ that satisfies (<ref>) for each $x \in \R$.
We are not making any assumptions on the well-posedness of (<ref>), and in particular we do not treat it as an initial value problem.

We assume a number of hypotheses on the map $\F$ and its constituent terms.
These hypotheses will permit us to convert (<ref>) to the “reduced” problem (<ref>), which we will then solve using Lombardi's nanopteron theory.
We will then undo our reduction procedure and recover solutions to the original problem (<ref>).
We state our main result on solutions to (<ref>) in Theorem <ref>.

[structural properties of $\F$]
The map $\F$ has the form

\begin{equation}\label{eqn: F-mu}
\F(U,\mu)
:= \L_0U + \mu\L_1(\mu)U + \nl(U,\mu),
\end{equation}

where $\L_0 \in \b(\D,\X)$ and $\L_1 \colon [0,\mu_0] \to \b(\D,\X)$ and $\nl \colon \X \times [0,\mu_0] \to \X$ are analytic.
The operator $\nl$ is quadratic in the sense that

\begin{equation}\label{eqn: nl hypo}
\nl(0,\mu) = 0
\quadword{and}
D_U\nl(0,\mu) = 0
\end{equation}

for all $\mu$.

This hypothesis, along with the analyticity of $\nl$, implies that we have the expansion

\begin{equation}\label{eqn: nl expn}
\nl(U,\mu)
= \nl_0(U,U) + \nl_1(U,\mu)
\end{equation}

for some bounded bilinear operator $\nl_0 \colon \X \times \X \to \X$ and analytic map $\nl_1 \colon \X \times [0,\mu_0] \to \X$ satisfying

\begin{equation}\label{eqn: nl ests}
\norm{\nl_1(U,\mu)}_{\X}
\le C\mu\norm{U}_{\X}^2 + C\norm{U}_{\X}^3.
\end{equation}

for some $C > 0$ when $\norm{U}_{\X} \le 1$.
Equivalently, if $\nl$ has the form (<ref>), then of course $\nl$ satisfies (<ref>).

[reversible symmetry]
There exists an operator $\S \in \b(\X)$ such that $\S^2 = \ind_{\X}$, $\norm{\S}_{\b(\X)} = 1$, and

\[
\S\L_0 = -\L_0\S,
\qquad
\S\L_1(\mu) = -\L_1(\mu)\S,
\quadword{and}
\nl(\S{U},\mu) = -\S\nl(U,\mu)
\]

for all $U \in \D$ and $0 \le \mu \le \mu_0$.

A map $f \colon \R \to \X$ is $\S$-reversible if $\S{f}(x) = f(-x)$ for all $x \in \R$ and $\S$-antireversible if $\S{f}(x) = -f(-x)$ for all $x$.

The symmetry $\S$ does not play a particularly large role in our development in this section, beyond serving as a condition to verify for the invocation of Lombardi's nanopteron theorem.
In Lombardi's nanopteron method, however, the symmetry is a key feature to remove certain redundancies and prevent overdetermined systems, and we point out some of its specific uses at various times in Appendix <ref>.

Our next two hypotheses control the center spectrum of $\L_0$, considered now as an operator in $\X$ with domain $\D$, and its spectral projection.
See Appendix <ref> for our spectral theoretic conventions.

[center spectrum of $\L_0$]
There exists $\omega >0$ such that $\sigma(\L_0) \cap i\R = \{0,\pm{i}\omega\}$.
All three points are eigenvalues; 0 has algebraic multiplicity 4 and $\pm{i}\omega$ each has algebraic multiplicity 1.
Each eigenvalue is geometrically simple, and there exists $\lambda_0 > 0$ such that if $\lambda \in \sigma(\L_0)\setminus{i}\R$, then $|\re(\lambda)| \ge \lambda_0$.

[spectral projection for $\L_0$]
Let $\Pi_0$ be the spectral projection for $\L_0$ corresponding to 0.

\begin{equation}\label{eqn: abstract 0 proj}
\Pi_0U
= \sum_{k=0}^3 \chi_k^*[U]\chi_k,
\end{equation}

where $(\chi_0,\chi_1,\chi_2,\chi_3) \in \X^4$ is a Jordan chain for $\L_0$ associated with 0, and $\chi_k^* \in \X^*$ are functionals.
The vectors $\chi_k$ and the functionals $\chi_k^*$ have the following additional properties.

$\L_1(\mu)\chi_0 = 0$ for all $0 \le \mu \le \mu_0$.

(translation invariance) $\nl(U+\gamma\chi_0,\mu) = \nl(U,\mu)$ for all $U \in \D$, $\gamma \in \R$, and $0 \le \mu \le \mu_0$.

$\S\chi_k = (-1)^{k+1}\chi_k$.

Following <cit.>, we say that the nonlinear operator $\F$ has a “$0^{4-}i\omega$ bifurcation at $(0,0)$,” since

* $\F(0,\mu) = 0$ for all $\mu$;

* The center spectrum of $\L_0 = D_U\F(0,0)$ is $\{0,\pm{i}\omega\}$, where 0 is an eigenvalue of algebraic multiplicity 4 and $\pm{i}\omega$ are eigenvalues of algebraic multiplicity 1;

* $\S\chi_0 = -\chi_0$, where $\chi_0$ is an eigenvector of $\L_0$ corresponding to 0.

Our reduction procedure in Section <ref>, modeled on that of [85, 41], will allow us to consider a problem whose linearization has the more well-known “$0^{2+}i\omega$” bifurcation, in which 0 is an eigenvalue of algebraic multiplicity only 2.
See <cit.> and <cit.> for more details on this bifurcation in both finite and infinite dimensions.

Our antepenultimate hypothesis introduces a map $\J_{\mu}$ that serves as first integral for the problem (<ref>); in particular, per part <ref> below, if $U$ solves (<ref>), then $x \mapsto J_{\mu}(U(x))$ is constant.
The other conditions of this hypothesis require $\J_{\mu}$ to interact with the symmetry $\S$, the eigenvectors $\chi_k$, and the eigenfunctional $\chi_3^*$ in suitable ways.

[first integral]
For each $0 \le \mu \le \mu_0$, there is a map $\J_{\mu} \colon \X \to \R$ with the following properties.

The map $\X \times [0,\mu_0] \to \R \colon (U,\mu) \mapsto \J_{\mu}(U)$ is analytic.

$D\J_{\mu}(U)\F(U,\mu) = 0$ for all $U \in \X$, where $\F$ is defined in (<ref>).

$\J_{\mu}(\S{U}) = \J_{\mu}(U)$ for all $U \in \X$.

$\J_0(0) = 0$, and there is a functional $\Jscr_*(\mu) \in \X^*$ such that

\begin{equation}\label{eqn: Jmu J0 quad}
\J_{\mu}(U)-\J_0(U)
= \mu\Jscr_*(\mu)U
\end{equation}

for all $U \in \D$.
The mapping $[0,\mu_0] \to \X^* \colon \mu \mapsto \Jscr_*(\mu)$ is analytic.

$\J_{\mu}(U + \gamma\chi_0) = \J_{\mu}(U)$ for all $U \in \D$, $\gamma \in \R$.

$\chi_3^*[U] = D\J_0(0)U$.

The following two quantities are essential values for the precise expressions of our solutions to (<ref>), and they control the leading order linear and quadratic behavior of the eventual reduction of (<ref>).

\begin{equation}\label{eqn: frak defn}
\Lfrak_0
:= \chi_2^*[\L_1(0)\chi_1] - \Jscr_*(0)\chi_1
\quadword{and}
\Qfrak_0
:= \chi_2^*\big[\nl_0(\chi_1,\chi_1)\big] - \frac{D^2\J_0(0)[\chi_1,\chi_1]}{2}.
\end{equation}

\begin{equation}\label{eqn: nondegen}
\Lfrak_0 > 0
\quadword{and}
\Qfrak_0 \ne 0.
\end{equation}

Our final hypothesis concerns the solutions of certain affine first-order differential equations involving $\L_0$, and our phrasing of this hypothesis is arguably our greatest departure from the abstract structure implicit in [85, 41].
To state this hypothesis, we need the machinery of Appendix <ref>, specifically Definition <ref> for localized optimal regularity, Definition <ref> for periodic optimal regularity, and Definition <ref> for suboptimal regularity.
We do not elaborate on these concepts here, and they do not play a major role in the reduction process of Section <ref>; however, they are critical to invoking Lombardi's results for our final problem.

Let $\Pi$ be the spectral projection for $\L_0$ corresponding to $\{0,\pm{i}\omega\}$; by the remarks at the conclusion of the proof of Theorem 6.17 in [54], we may write

\begin{equation}\label{eqn: Pi spectral proj}
\Pi{U}
= \Pi_0U + \chi_+^*[U]\chi_+ + \chi_-^*[U]\chi_-
\end{equation}

for some functionals $\chi_{\pm}^* \in \X^*$, where $\chi_{\pm}$ are eigenvectors of $\L_0$ corresponding to $\pm{i}\omega$.

\begin{equation}\label{eqn: hyperbolic space X D}
\X_{\hsf} := (\ind_{\X}-\Pi)\X
\quadword{and}
\D_{\hsf} := \D \cap \X_{\hsf}.
\end{equation}

Let $\X_{\hsf}$ and $\D_{\hsf}$ have the norms $\norm{\cdot}_{\X}$ and $\norm{\cdot}_{\D}$, respectively.

[optimal regularity]
There is a subspace $\Y_{\hsf}$ of $\X_{\hsf}$ such that

\[
(\ind_{\X}-\Pi)\L_1(\mu)W \in \Y_{\hsf}
\quadword{and}
(\ind_{\X}-\Pi)\nl(W,\mu) \in \Y_{\hsf}
\]

for all $W \in \D_{\hsf}$.
There exist $b \in (0,\pi)$, $q \in (0,\Lfrak_0^{1/2})$, and $\grave{q} < 0$ such that on the triple $(\D_{\hsf},\Y_{\hsf},\X_{\hsf})$, the operator $\restr{\L_0}{\D_{\hsf}}$ has the localized optimal regularity property with decay rate $q\Lfrak_0^{-1/2}$ and strip width $b$; the periodic optimal regularity property with base frequency $\omega$; and the suboptimal regularity property with growth rate $\grave{q}$.

Before we state the main result in Theorem <ref>, we collect a variety of immediate and useful consequences of the preceding hypotheses in the following lemma.

The following hold for all $U \in \X$ and $0 \le \mu \le \mu_0$.

$\chi_{k+1}^*[\L_0U] = \chi_k^*[U]$ for $k=0,1,2$, while $\chi_3^*[\L_0U]= 0$.

$\chi_k^*[\S{U}] = (-1)^{k+1}\chi_k^*[U]$.

$\chi_3^*[\L_1(\mu)\chi_3] = 0$.

$D\J_{\mu}(U)\chi_0 = 0$.

$D\J_0(0)\chi_1 = 0$ and $D\J_0(0)\chi_3 = 1$.

$\Jscr_*(\mu)\S{U} = \Jscr_*(\mu)U$.

$\Jscr_*(\mu)\chi_0 = 0$.

* This follows from the commutativity of $\L_0$ and the spectral projection $\Pi_0$ defined in (<ref>), the identities $\L_0\chi_0 = 0$ and $\L_0\chi_{k+1} = \chi_k$ for $k=1$, $2$, $3$, and the linear independence of the $\chi_k$.

* Since $\S$ anticommutes with $\L_0$ by Hypothesis <ref>, the spectral projection $\Pi_0$ and $\S$ commute; see the remarks preceding Theorem 2.5 in Section 2.3.3 of [30].
Then the result follows from Hypothesis <ref>–<ref> and the linear independence of the $\chi_k$.

* Since $\S^2 = \ind_{\X}$, we may use Hypothesis <ref>–<ref> to calculate

\[
\L_1(\mu)\chi_3
= \L_1(\mu)\S^2\chi_3
= \big(\L_1(\mu)\S\big)\big(\S\chi_3\big)
= -\S\L_1(\mu)\chi_3.
\]

We combine this with another invocation of Hypothesis <ref>–<ref> to calculate

\[
\chi_3^*[\L_1(\mu)\chi_3]
= -\chi_3^*[\S\L_1(\mu)\chi_3]
= -\chi_3^*[\L_1(\mu)\chi_3],
\]

and so $\chi_3^*[\L_1(\mu)\chi_3] = 0$.

* Use Hypothesis <ref>–<ref> to calculate the (directional) derivative

\[
= \lim_{\gamma \to 0} \frac{\J_{\mu}(U+\gamma\chi_0)-\J_{\mu}(U)}{\gamma}
= \lim_{\gamma \to 0} \frac{\J_{\mu}(U)-\J_{\mu}(U)}{\gamma}
= 0.
\]

* This follows at once from Hypothesis <ref>–<ref>.

* Use the definition of $\Jscr_*$ and Hypothesis <ref>–<ref>.

* Use part <ref> and Hypothesis <ref>–<ref> to calculate

\[
\Jscr_*(\mu)\chi_0
= \Jscr_*(\mu)\S\chi_0
= -\Jscr_*(\mu)\chi_0.
\qedhere
\]

§.§ The main abstract result

Now we can state our result on the existence of certain solutions to the abstract problem (<ref>).
We prove this theorem in Section <ref> after the reduction procedures of Section <ref>.
We think of this theorem as the abstract existence result for near-sonic position traveling waves in dimer FPUT lattices, for we will show in Section <ref> that the position traveling wave problem (<ref>) satisfies, after a fashion, all the hypotheses above.

Assume Hypotheses <ref>, <ref>, <ref>, <ref>, <ref>, <ref>, and <ref>.
Then there are constants $\Alpha_0$, $\Alpha_1$, $\mu_* > 0$ such that if

\begin{equation}\label{eqn: Ascr-mu}
\mu \in (0,\mu_*)
\quadword{and}
\alpha \in \left[\Alpha_0\mu\exp\left(-\frac{b\omega}{\Lfrak_0^{1/2}\mu^{1/2}}\right), \Alpha_1\right] =: \Ascr_{\mu},
\end{equation}

there is a solution to (<ref>) of the form

\begin{multline}\label{eqn: abstract drifting nanopteron}
\Usf_{\mu}^{\alpha}(x)
= \mu^{1/2}\left[
\left(-\frac{3\Lfrak_0^{1/2}}{\Qfrak_0}+\mu^{1/2}\Lup_{\mu}^{\alpha}\right)
\tanh\left(\frac{\Lfrak_0^{1/2}\mu^{1/2}x}{2}\right)
+ \mu^{1/2}\Upsilon_{\mu}^{\alpha,0}(\mu^{1/2}x)
\right]
\chi_0 \\
+ \mu^{3/2}\Upsilon_{\mu}^{\alpha,*}(\mu^{1/2}x) \\
+ \alpha\mu^{1/2}(\alpha+\mu)\left(\int_0^{\Tup_{\mu}^{\alpha}(\mu^{1/2}x)} \Phi_{\mu}^{\alpha,\int}(s)\ds\right)\chi_0,
\end{multline}

\begin{equation}\label{eqn: Tsf-mu-alpha}
\Tup_{\mu}^{\alpha}(X)
:= X + \mu\vartheta_{\mu}^{\alpha}\tanh\left(\frac{\Lfrak_0^{1/2}X}{2}\right)
\end{equation}

and the terms and coefficients above have the following properties.

* The maps $\Upsilon_{\mu}^{\alpha,0} \colon \R \to \R$ and $\Upsilon_{\mu}^{\alpha,*} \colon \R \to \D$ are real analytic and exponentially localized with

\begin{equation}\label{eqn: main expn loc}
\sup_{\substack{0 < \mu < \mu_* \\ \alpha \in \Ascr_{\mu} \\ X \in \R}}
\big(
\big)
< \infty
\end{equation}

and $\chi_0^*[\Upsilon_{\mu}^{\alpha,*}(X)] = 0$ for all $X \in \R$.

* The real coefficient $\Lup_{\mu}^{\alpha}$ is uniformly bounded with

\[
\sup_{\substack{0 < \mu < \mu_* \\ \alpha \in \Ascr_{\mu} \\ X \in \R}}
< \infty.
\]

* There are constants $0 < C_{\vartheta}^- < C_{\vartheta}^+$ such that the phase shift $\vartheta_{\mu}^{\alpha} \in \R$ satisfies

\begin{equation}\label{eqn: main phase shift}
< C_{\vartheta}^-
\le \vartheta_{\mu}^{\alpha}
\le C_{\vartheta}^+
\end{equation}

for all $0 < \mu < \mu_*$ and $\alpha \in \Ascr_{\mu}$.

* The maps $\Phi_{\mu}^{\alpha} \colon \R \to \D$ and $\Phi_{\mu}^{\alpha,\int} \colon \R \to \R$ are defined for all $0 < \mu < \mu_*$ and $0 \le \alpha \le \Alpha_1$.
They are real analytic, periodic, and uniformly bounded with

\begin{equation}\label{eqn: main periodic est}
\sup_{\substack{0 < \mu < \mu_* \\
0 \le \alpha \le \Alpha_1 \\
X \in \R}}
\left(
\norm{\Phi_{\mu}^{\alpha}(X)}_{\D}
+ |\Phi_{\mu}^{\alpha,\int}(X)|
\right)
< \infty
\end{equation}

There is a constant $C_{\Phi} > 0$ such that

\begin{equation}\label{eqn: main periodic lip}
\sup_{0 \le \alpha \le \Alpha_1}
\left(
\Lip(\Phi_{\mu}^{\alpha})
+ \Lip(\Phi_{\mu}^{\alpha,\int})
\right)
< C_{\Phi}\mu^{-1/2}
\end{equation}

for all $0 < \mu < \mu_*$.
The map $\Phi_{\mu}^{\alpha}$ satisfies the additional properties (<ref>), (<ref>), and (<ref>) of part <ref> of Theorem <ref>, as well as $\chi_0^*[\Phi_{\mu}^{\alpha}(X)] = 0$ for all $X \in \R$.

* For each $0 < \mu < \mu_*$ and $0 \le \alpha \le \Alpha_1$, the map

\begin{equation}\label{eqn: Psf-mu-alpha}
\Psf_{\mu}^{\alpha}(x)
:= \alpha\mu\Phi_{\mu}^{\alpha}(\mu^{1/2}x)
+ \alpha\mu^{1/2}(\alpha+\mu)\left(\int_0^{\mu^{1/2}x} \Phi_{\mu}^{\alpha,\int}(s) \ds\right)\chi_0
\end{equation}

also solves (<ref>).

* The solutions $\Usf_{\mu}^{\alpha}$ and $\Psf_{\mu}^{\alpha}$ are $\S$-reversible.

The map $\Usf_{\mu}^{\alpha}$ defined in (<ref>) is a kind of “growing front,” a version of which we previously drew in Figure <ref>.
Its leading-order term is the $\O(\mu^{1/2})$ product of a $\tanh$-type coefficient and the $\chi_0$ eigenvector.
This induces the “front” behavior.
The next higher-order terms are $\O(\mu)$ exponentially localized terms; one of these has the abstract coefficient $\Upsilon_{\mu}^{\alpha,0}$ on $\chi_0$, while the other has the very explicit $\sech^2$-type coefficient on the generalized eigenvector $\chi_1$.
For the purposes of our lattice calculations in Section <ref>, it is important to separate the abstract localized coefficient from the explicit.

The final higher-order terms are the $\O(\mu^{3/2})$ localized term $\Upsilon_{\mu}^{\alpha,*}$, the genuinely periodic term $\Phi_{\mu}^{\alpha}$, and and an integral term.
The integral term may cause the “growing” behavior of the front if it is not also periodic; we discuss in Remark <ref> why we are not more specific about its potential periodicity.

The parameter $\alpha$ controls the amplitude of the genuinely periodic term.
The interval $\Ascr_{\mu}$, defined in (<ref>), to which $\alpha$ belongs, has a nonzero, exponentially-small-in-$\mu$ lower bound and an $\O(1)$ upper bound.
Thus the periodic amplitude can be quite small, but nonvanishing, and also somewhat large.

Although the integral term could grow, at worst, linearly, it bears the factor $\alpha^2\mu^{1/2} + \alpha\mu^{3/2}$.
Consequently, for $\alpha$ small, this integral term is smaller “in $\mu$” than the genuinely periodic term.
That the integral appears as a factor on the eigenvector $\chi_0$ will be important for the lattice calculations in Section <ref>.

We do not have the freedom to choose the phase shift $\vartheta_{\mu}^{\alpha}$ that appears in (<ref>); it is given to us by Lombardi's nanopteron method.
However, the phase shift is $\O(\mu)$, so, at least, it is small.

§.§ Reduction to the $0^{2+}i\omega$ center spectrum

We show that to solve (<ref>), it suffices to solve a simpler “reduced” differential equation, stated in (<ref>), and this simpler equation turns out to possess the $0^{2+}i\omega$ bifurcation from Remark <ref>.
To convert (<ref>) into (<ref>), we make several changes of variables, and we summarize the related notation in Figure <ref>.
We will apply Lombardi's nanopteron theorem (Theorem <ref>) to this simpler problem and then undo the reductions to recover solutions to the original problem.

\[
\input{fig-reduction}
\]

The two reductions of the original system (<ref>) to the simpler problem (<ref>).
The center spectrum notation follows Remark <ref>.

§.§.§ Removal of translation invariance

By Hypothesis <ref>, we can write any $U \in \X$ uniquely as $U = \gamma\chi_0 + V$, where $\gamma \in \R$ and $\chi_0^*[V] = 0$.
This motivates the decompositions

\[
\hX := \set{V \in \X}{\chi_0^*[V]= 0}
\quadword{and}
\hD := \D \cap \hX.
\]

The spaces $\hX$ and $\hD$ retain the norms of $\X$ and $\D$, respectively.

\begin{equation}\label{eqn: hL0}
\hL_0V
:= \L_0V - \chi_1^*[V]\chi_0
\end{equation}

and set

\begin{equation}\label{eqn: hL1 and hnl}
\hL_1(\mu)V := \L_1(\mu)V-\chi_0^*[\L_1(\mu)V]\chi_0
\quadword{and}
\hnl(V,\mu) := \nl(V,\mu)-\chi_0^*[\nl(V,\mu)]\chi_0.
\end{equation}

Last, define

\begin{equation}\label{eqn: Gamma-mu}
\Gamma_{\mu}(V)
:= \chi_1^*[V]+ \Gamma_{\mu}^*(V),
\qquad
\Gamma_{\mu}^*(V) := \mu\chi_0^*[\L_1(\mu)V] + \chi_0^*[\nl(V,\mu)],
\end{equation}

\begin{equation}\label{eqn: hFmu}
\hF(V,\mu)
:= \hL_0V + \mu\hL_1(\mu)V + \hnl(V,\mu).
\end{equation}

Hypotheses <ref>–<ref> and <ref>–<ref> then imply that (<ref>) is equivalent to the decoupled system

γ'(x) = Γ(V(x),μ)

V'(x) = (V(x),μ) .

Given a solution $V$ to (<ref>), we can of course solve (<ref>) for $\gamma$, up to a constant of integration $\gamma_0 \in \R$:

\begin{equation}\label{eqn: mu soln}
\gamma(x)
= \gamma_0 + \int_0^x \Gamma_{\mu}(V(s)) \ds.
\end{equation}

Then the full solution to (<ref>) will have the form

\[
= \gamma_0\chi_0 + \bunderbrace{\left(\int_0^x \Gamma_{\mu}(V(s)) \ds\right)\chi_0 + V(x)}{U_0(x)}.
\]

By Hypotheses <ref>–<ref> and <ref>–<ref>, if a function $U$ of this form solves (<ref>), then so does $U_0$, which depends only on $V$.
Thus it suffices to solve only (<ref>).
Manipulating the integral term that multiplies $\chi_0$ in $U_0$ will be one of our most delicate tasks later, in Sections <ref> and <ref>.

Our work here has followed <cit.>; one difference is that we assume neither $\chi_0^*[\L_1(\mu)V] = 0$ nor $\chi_0^*[\nl(V,\mu)] = 0$, and so we must keep these terms in $\Gamma_1$, and thus in $\Gamma$, unlike <cit.>.
See Remark <ref> for the precise reason, in the context of our dimer FPUT problems, why we do not assume that these functionals vanish on these kinds of inputs.
However, the terms in $\Gamma_1$ are small, and so their effect is negligible.
See also <cit.>, <cit.>, and <cit.> for the same sort of reduction of translation invariance.

Before proceeding, we note that $\J_{\mu}$ remains a first integral for (<ref>), and we also prove some useful properties of $\Gamma$.

With $\hF$ defined in (<ref>), we have $D\J_{\mu}(V)\hF(V,\mu) = 0$ for all $V \in \hD$.

\[
\hF(V,\mu)
= \F(V,\mu) -  \big(\chi_1^*[V]+ \mu\chi_0^*[\L_1(\mu)V] + \chi_0^*[\nl(V,\mu)]\big)\chi_0
\]

and use Hypothesis <ref>–<ref> and part <ref> of Lemma <ref>.

The map $\Gamma_{\mu}$, defined in (<ref>), has the following properties.

$\Gamma_{\mu}(\S{V}) = \Gamma_{\mu}(V)$ and $\Gamma_{\mu}^*(\S{V}) = \Gamma_{\mu}^*(\S{V})$ for all $V \in \X$.

* Suppose that $f \colon \R \to \X$ is $\S$-reversible (in the sense of Definition <ref>).
Then $\Gamma_{\mu}\circ{f}$ and $\Gamma_{\mu}^*\circ{f}$ are even.

The map $\Gamma_{\mu}^*$ satisfies

\[
\sup_{\substack{0 < \mu < \mu_0 \\
0 < \norm{V}_{\X} \le 1 \\
0 < \alpha, \tau_1,\tau_2 < 1}} \big(\alpha\mu^2\big)^{-1} |\Gamma_{\mu}^*(\alpha\mu\tau_1\chi_3 + \alpha\mu\tau_2V)|
< \infty
\]

\[
\sup_{\substack{0 < \mu < \mu_0 \\
0 < \norm{V}_{\X} + \norm{\grave{V}}_{\X} < 1}}
< \infty.
\]

* This is a direct calculation using the definition of $\Gamma_{\mu}$, Hypotheses <ref> and <ref>–<ref>, and part <ref> of Lemma <ref>.

* We have $\Gamma_{\mu}(f(-x)) = \Gamma_{\mu}(\S{f}(x))$ since $f$ is $\S$-reversible, and $\Gamma_{\mu}(\S{f}(x)) = \Gamma_{\mu}(f(x))$ by part <ref>.
The same holds for $\Gamma_{\mu}^*$.

* This estimate follows from the definition of $\Gamma_{\mu}^*$ and the quadraticity of $\nl$ in its first variable.

§.§.§ Fixing a value for the first integral

We can write any $V \in \hD$ uniquely as $V = \tau\chi_3 + W$, where $\tau \in \R$ and $\chi_3^*[W] = 0$, and so (<ref>) is equivalent to

τ'(x) = χ_3^*[(τ(x)χ_3+W(x),μ)]

W'(x) = (τ(x)χ_3+W(x),μ)-χ_3^*[(τ(x)χ_3+W(x),μ)]χ_3 .

Now let

\begin{equation}\label{eqn: tilde spaces}
\tX := \set{W \in \hX}{\chi_3^*[W] = 0}
\quadword{and}
\tD := \hD \cap \tX.
\end{equation}

We will show that the values of the first integral $\J_{\mu}(V)$ and the “orthogonal” component $W$ completely determine $\chi_3^*(V)$, at least when everything is small.
In other words, we can solve $j = \J_{\mu}(\tau\chi_3+W)$ for $\tau$ given small $\mu$, $j \in \R$ and $W \in \tX$.
Since $\J_{\mu}$ is constant on solutions of the coupled system (<ref>)–(<ref>) by Lemma <ref>, this system therefore reduces to a problem in $W$ alone, at least for “small-amplitude” solutions.
This is the same procedure that Iooss and James performed to obtain <cit.> and likewise that Venney and Zimmer began at the start of <cit.>.
However, these papers did not address a certain “consistency” problem with this approach as we do; see Lemma <ref> and the comments preceding it.
Moreover, we need quite precise estimates on how $\tau$ is so “completely determined” by the values of $W$ and $\J_{\mu}(V)$.
We mention that quite a related result appears in <cit.>, albeit with the intention of different estimates and applications; our goals are sufficiently distinct as to warrant the full proof below.

Part <ref> of the following lemma makes precise how $W$ and $J_{\mu}(V)$ completely determine $\chi_3^*(V)$.
We also include several detailed expansions and estimates whose usefulness will only be apparent later.
We will work on the balls

\[
\Bfrak(r)
:= \set{(W,j,\mu) \in \tX \times \R \times [0,\mu_0]}{ \max\{\norm{W}_{\X}, |j|, \mu\} < r}.
\]

There exists $\mu_{\T} \in (0,\min\{\mu_0,1\})$ and an analytic map $\T \colon \Bfrak(\mu_{\T}) \to \R$ with the following properties.

$\T(0,0,\mu) = 0$ for $0 \le \mu \le \mu_{\T}$.

$j = \J_{\mu}(\T(W,j,\mu)\chi_3+W)$ for $(W,j,\mu) \in \Bfrak(\mu_{\T})$.

$\tau = \T(W,\J_{\mu}(\tau\chi_3+W),\mu)$ for $\max\{\norm{W}_{\X},|\tau|, \mu\} < \mu_{\T}$.

$\T(\S{W},j,\mu) = \T(W,j,\mu)$ for $(W,j,\mu) \in \Bfrak(\mu_{\T})$.

$D_W\T(W,j,\mu)\grave{W} = -\frac{D\J_{\mu}(\T(W,j,\mu)\chi_3+W)\grave{W}}{D\J_{\mu}(\T(W,j,\mu)\chi_3+W)\chi_3}$
and $D_W\T(0,0,\mu)\grave{W} = -\frac{\mu\Jscr_*(\mu)\grave{W}}{1+\mu\Jscr_*(\mu)\chi_3}$ for all $\grave{W}$, $\breve{W} \in \tX$ and $(W,j,\mu) \in \Bfrak(\mu_{\T})$.

= -D^2\J_0(0)[\grave{W},\breve{W}] + D^2\J_0(0)[\grave{W},\chi_3]D\J_0(0)\breve{W}$ for all $\grave{W}$, $\breve{W} \in \tX$.

$\sup_{\substack{0 < |\alpha| \le 1 \\ \norm{W}_{\X} \le \mu_{\T} \\ 0 < \mu < \mu_{\T}}} \alpha^{-1}\mu^{-2}|\T(\alpha\mu{W},0,\mu)| < \infty$.

$\sup_{\substack{0 < |\alpha| \le 1 \\ \norm{W}_{\X}+\norm{\grave{W}}_{\X} \le \mu_{\T} \\ 0 < \mu < \mu_{\T}}} \mu^{-2}|\T(\alpha\mu{W}+\mu\grave{W},0,\mu)-\T(\alpha\mu{W},0,\mu)| < \infty$.

We first construct the map $\T$.

\begin{equation}\label{eqn: Jsf}
\Jsf(\tau,W,j,\mu)
:= j-\J_{\mu}(\tau\chi_3+W),
\end{equation}

so $\Jsf$ is real analytic on $\R \times \tX \times \R \times [0,\mu_0]$ by Hypothesis <ref>–<ref>.
Hypothesis <ref>–<ref> gives

\begin{equation}\label{eqn: Jsf-0-0-0-mu}
\Jsf(0,0,0,\mu)
= -\J_0(0)
= 0
\end{equation}

for all $\mu$, and part <ref> of Lemma <ref> implies

\[
= -D\J_0(0)\chi_3
= -1.
\]

The analytic implicit function theorem then provides $\mu_1 \in (0,\mu_0]$ and $\mu_2 > 0$ and an analytic map $\T \colon \Bfrak(\mu_1) \to (-\mu_2,\mu_2)$ such that if $(W,j,\mu) \in \Bfrak(\mu_1)$ and $|\tau| < \mu_2$, then $\Jsf(\tau,W,j,\mu) = 0$ if and only if $\tau = \T(W,j,\mu)$.

Now we prove the more specialized properties of $\T$.

The identity (<ref>) implies that $\T(0,0,\mu) = 0$ for all $\mu$.

The identity $\Jsf(\T(W,j,\mu),W,j,\mu) = 0$ and the definition of $\Jsf$ in (<ref>) give

\[
= \J_{\mu}(\T(W,j,\mu)\chi_3+W).
\]

Since $\J_0(0) = 0$ by Hypothesis <ref>–<ref>, the continuity result from Hypothesis <ref>–<ref> gives $\mu_3 \in (0,\min\{\mu_1,\mu_2\})$ such that if

\begin{equation}\label{eqn: W tau mu cond}
\min\{\norm{W}_{\X}, |\tau|, \mu\}
< \mu_3
\end{equation}

then $|\J_{\mu}(\tau\chi_3+W)| \le \mu_1$.
Assume that  (<ref>) holds.
Then $(W,\J_{\mu}(\tau\chi_3+W),\mu) \in \Bfrak(\mu_1)$ and $|\tau| < \mu_2$, while a direct calculation yields $\Jsf(\tau,W,\J_{\mu}(\tau\chi_3+W),\mu) = 0$.
Consequently, (<ref>) forces

\[
\tau
= \T(W,\J_{\mu}(\tau\chi_3+W),\mu).
\]

First, since $\norm{\S{W}}_{\X} \le \norm{W}_{\X}$ by Hypothesis <ref>, we have $(\S{W},j,\mu) \in \Bfrak(\mu_1)$ whenever $(W,j,\mu) \in \Bfrak(\mu_1)$.
Thus $\T(\S{W},j,\mu)$ is in fact defined for $(W,j,\mu) \in \Bfrak(\mu_1)$.
Next, we already know that $\Jsf(W,\T(W,j,\mu),j,\mu) = 0$.
If we can also show that $\Jsf(W,\T(\S{W},j,\mu),j,\mu) = 0$, then we must have $\T(\S{W},j,\mu) = \T(W,j,\mu)$.

The definition of $\Jsf$ in (<ref>),  Hypotheses <ref>–<ref> and <ref>–<ref>, and part <ref> of Lemma <ref> enable us to calculate $\Jsf(\S{W},\tau,j,\mu) = \Jsf(W,\tau,j,\mu)$ for any $\tau$.

\[
= \Jsf(\S{W},\T(\S{W},j,\mu),j,\mu)
= \Jsf(W,\T(\S{W},j,\mu),j,\mu),
\]

as desired.

Part <ref> of Lemma <ref> tells us that $D\J_0(0)\chi_3 = 1$.
The analyticity of $\J$ and $\T$ then give $\mu_{\T} \in (0,\min\{\mu_3,1\})$ such that if $(W,j,\mu) \in \Bfrak(\mu_{\T})$, then $D\J_{\mu}(\T(W,j,\mu)\chi_3+W)\chi_3 \ne 0$.

Fix $j$, $\mu \in \R$ with $\max\{|j|,\mu\} < \mu_{\T}$.
By part <ref> above, the map $\Jfrak(W) := \J_{\mu}(\T(W,j,\mu)\chi_3+W)$ is constant for $\norm{W}_{\X} < \mu_{\T}$, and so for any $\grave{W} \in \tX$, we have

\begin{equation}\label{eqn: derivT}
= D\Jfrak(W)\grave{W}
= D\J_{\mu}(\T(W,j,\mu)\chi_3+W)\big[(D_W\T(W,j,\mu)\grave{W})\chi_3+\grave{W}\big].
\end{equation}

This implies

\[
= -\frac{D\J_{\mu}(\T(W,j,\mu)\chi_3+W)\grave{W}}{D\J_{\mu}(\T(W,j,\mu)\chi_3+W)\chi_3}.
\]

In the special case $W = j = 0$, we use the identity $\T(0,0,\mu)$ from part <ref> above and the formula (<ref>) to calculate

\[
= -\frac{D\J_{\mu}(0)\grave{W}}{D\J_{\mu}(0)\chi_3}.
\]

By Hypotheses <ref>–<ref> and <ref>–<ref>, we have

\[
= D\J_0(0)U + \big(D\J_{\mu}(0)-D\J_0(0)\big)U
= \chi_3^*[U] + \mu\Jscr_*(\mu)U
\]

for any $U \in \X$.
Since $\chi_3^*[\grave{W}] = 0$ for $\grave{W} \in \tX$ and $\chi_3^*[\chi_3] = 1$, we conclude

\[
= -\frac{\mu\Jscr_*(\mu)U}{1+\mu\Jscr_*(\mu)\chi_3}.
\]

This follows from part <ref> and a lengthy calculation involving the product and chain rule in Banach spaces.

Since $\T(0,0,\mu) = 0$, we may use the fundamental theorem of calculus twice and part <ref> to write

\begin{multline*}
\T(\alpha\mu{W},0,\mu)
= \int_0^1\int_0^1 D_{WW}^2\T(st\alpha\mu{W},0,\mu)[t\alpha\mu{W},\alpha\mu{W}] \ds\dt + D_W\T(0,0,\mu)(\alpha\mu{W}) \\
= \alpha\mu^2\left(\alpha\int_0^1\int_0^1 D_{WW}^2\T(st\alpha\mu{W},0,\mu)[tW,W] \ds\dt - \frac{\Jscr_*(\mu)W}{1+\mu\Jscr_*(\mu)\chi_3}\right).
\end{multline*}

The estimate then follows from the analyticity of $\T$.

Similar to part <ref>, we use the fundamental theorem of calculus twice to rewrite the difference $\T(\alpha\mu{W}+\mu\grave{W},\mu,0)-\T(\alpha\mu{W},\mu,0)$ in the form

\[
\mu^2\left(\int_0^1\int_0^1 D_{WW}^2\T(s(\alpha\mu{W}+t\mu\grave{W}),0,\mu)[\alpha{W}+t\grave{W},\grave{W}] \ds\dt- \frac{\Jscr_*(\mu)\grave{W}}{1+\mu\Jscr_*(\mu)\chi_3}\right).
\qedhere
\]

Now let $\mu_1$ and $\mu_2$ be as in Lemma <ref>.
Suppose that $\tau_{\mu} \colon \R \to \R$ and $W_{\mu} \colon \R \to \tD$ solve the system (<ref>)–(<ref>) with $0 \le \mu  \le \mu_{\T}$.
Put $j_{\mu}(x) := J_{\mu}(\tau_{\mu}(x)\chi_3+W_{\mu}(x))$, so $j_{\mu}$ is constant by Hypothesis <ref>–<ref>.
Suppose also that $\tau_{\mu}$ and $W_{\mu}$ are small enough that $\max\{\norm{W_{\mu}(x)}_{\X},|\tau_{\mu}(x)|,\mu\} < \mu_{\T}$ for all $x \in \R$.
Then part <ref> of Lemma <ref> implies $\tau_{\mu}(x) = \T(W_{\mu}(x),j_{\mu}(0),\mu)$ for all $x$.
That is, the coefficient on $\chi_3$ is completely determined by the part of the solution orthogonal to $\chi_3$ and the first integral — at least if everything is sufficiently small.

Conversely, this suggests a strategy for solving (<ref>), equivalently the coupled system (<ref>)–(<ref>): assume $\tau(x) = \T(W(x),j,\mu)$ for $j \in \R$ fixed and attempt to solve

\begin{equation}\label{eqn: W chi3 no tau}
= \hF(\T(W(x),j,\mu)\chi_3+W(x),\mu)-\chi_3^*\big[\hF(\T(W(x),j,\mu)\chi_3+W(x),\mu)\big]\chi_3.
\end{equation}

This equation is now an ordinary differential equation in $W$ with $\mu$ and $j$ as parameters.
Of course, here we assume that $W$, $j$, and $\mu$ small enough for $\T(W(x),j,\mu)$ to be defined.

This is precisely what we shall do, except there is a nagging question of consistency.
If $W$ solves (<ref>) and is sufficiently small, does putting $\tau(x) := \T(W(x),j,\mu)$ solve (<ref>)?
Happily, this is the case, at least if we pick $j = 0$.
In other words, we are freezing the value of the first integral to be 0.

Suppose that $W \colon \R \to \tD$ solves (<ref>) with $j = 0$ and $\max\{\norm{W(x)}_{\X},\mu\} < \mu_{\T}$ for all $x$.
Then the map $\tau(x) := \T(W(x),0,\mu)$ solves (<ref>).

Since $(W(x),0,\mu) \in \Bfrak(\mu_{\T})$ for all $x \in \R$, we may differentiate $\tau$ directly using part <ref> from Lemma <ref> and find, suppressing all $x$-dependencies,

\[
\tau_{\mu}'
= D_W\T(W,0,\mu)W'
= -\frac{D\J_{\mu}(\T(W,0,\mu)\chi_3+W)W'}{D\J_{\mu}(\T(W,0,\mu)\chi_3+W)\chi_3}.
\]

We use (<ref>) to calculate

\begin{multline*}
D\J_{\mu}(\T(W,0,\mu)\chi_3+W)W' \\
= D\J_{\mu}(\T(W,0,\mu)\chi_3+W)\hF(\T(W,0,\mu)\chi_3+W,\mu)) \\
\end{multline*}

Lemma <ref> implies

\[
= 0,
\]

and so

\begin{multline*}
\tau_{\mu}'
= \frac{\chi_3^*\big[\hF(\T(W,0,\mu)\chi_3+W,\mu))\big]D\J_{\mu}(\T(W,0,\mu)\chi_3+W)\chi_3}{D\J_{\mu}(\T(W,0,\mu)\chi_3+W)\chi_3} \\
= \chi_3^*\big[\hF(\T(W,0,\mu)\chi_3+W,\mu))\big]
= \chi_3^*\big[\hF(\tau_{\mu}\chi_3+W,\mu)\big].
\qedhere
\end{multline*}

§.§.§ The fully reduced system

From now on, we assume $0 \le \mu \le \mu_{\T}$, where $\mu_{\T}$ was defined in Lemma <ref>, and, for $W \in \tX$ with $\norm{\W}_{\X} \le \mu_{\T}$, we put

\[
\T_{\mu}(W)
:= \T(W,0,\mu),
\]

with $\T$ also defined in Lemma <ref>.
We will study the “fully reduced” problem

\begin{equation}\label{eqn: fully reduced}
= \tF(W(x),\mu),
\end{equation}

\begin{equation}\label{eqn: tFmu}
\tF(W,\mu)
:= \hF(\T_{\mu}(W)\chi_3+W,\mu)-\chi_3^*\big[\hF(\T_{\mu}(W)\chi_3+W,\mu)\big]\chi_3.
\end{equation}

The work in the previous two sections implies that if $W$ solves (<ref>) and is sufficiently small, then

\begin{equation}\label{eqn: W-T-int soln to orig}
:= W(x)
+ \T_{\mu}(W(x))\chi_3
+ \left(\int_0^x \Gamma_{\mu}\big(\T_{\mu}(W(s))\chi_3 + W(s)\big) \ds\right)\chi_0
\end{equation}

solves the original problem (<ref>).

We rewrite $\tF$ in the form

\begin{equation}\label{eqn: tFmu2}
\tF(W,\mu)
= \tL_0W
+ \mu\tL_1(\mu)W
+ \tnl(W,\mu),
\end{equation}

where we use the definition of $\hF$ in (<ref>) and its constituent terms from (<ref>) and (<ref>) to expand

\begin{equation}\label{eqn: tL0}
\tL_0W
:= \hL_0W
= \L_0W - \chi_1^*[W]\chi_0,
\end{equation}

\begin{equation}\label{eqn: tL1}
\begin{aligned}
\tL_1(\mu)W
&:= \L_1(\mu)W-\chi_0^*[\L_1(\mu)W]\chi_0-\chi_3^*[\L_1(\mu)W]\chi_3
- \left(\frac{\Jscr_*(\mu)W}{1+\mu\Jscr_*(\mu)\chi_3}\right)\chi_2 \\
\end{aligned}
\end{equation}

\begin{equation}\label{eqn: tnl}
\begin{aligned}
\tnl(W,\mu)
-\chi_0^*\big[\nl(\T_{\mu}(W)\chi_3+W,\mu)\big]\chi_0 \\
+ \nl_{\T}(W,\mu)\chi_2 \\
&+ \mu\nl_{\T}(W,\mu)\big(\L_1(\mu)\chi_3-\chi_0^*[\L_1(\mu)\chi_3]\chi_0\big),
\end{aligned}
\end{equation}

\[
\nl_{\T}(W,\mu)
:= \T_{\mu}(W)-D\T_{\mu}(0)W
= \T_{\mu}(W)+\mu\left(\frac{\Jscr_*(\mu)W}{1+\mu\Jscr_*(\mu)\chi_3}\right).
\]

We have used the various hypotheses, part <ref> of Lemma <ref>, and part <ref> of Lemma <ref> to simplify and cancel some terms to obtain the expansion (<ref>).
Observe that

\[
\chi_k^*[\tL_0W]
= \chi_k^*[\tL_1(\mu)W]
= \chi_k^*[\tnl(W,\mu)]
= 0, \ k = 0, 3
\]

for all $W \in \tD$, and so $\tF$ and its constituent terms do indeed map $\tD$ into $\tX$.

The next lemma shows that $\tF$ satisfies all the properties of Theorem <ref>, which guarantees nanopteron solutions to the reduced problem (<ref>).

For $0 \le \mu \le \mu_1$, the map $\tF$ defined in (<ref>) has the following properties.

* The mappings $\mu \mapsto \tL_1(\mu)$ and $(W,\mu) \mapsto \tnl(W,\mu)$ are analytic with

\[
\sup_{0 \le \mu \le \mu_1} \norm{\tL_1(\mu)}_{\b(\tD,\tX)} \le C
\quadword{and}
\sup_{0 \le \mu \le \mu_1} \norm{\tnl(W,\mu)}_{\X} \le C\norm{W}_{\X}^2.
\]

* The symmetry $\S$ anticommutes with each term in $\tF$:

\[
\S\tL_0 = -\tL_0\S,
\qquad
\S\tL_1(\mu) = -\tL_1(\mu)\S,
\quadword{and}
\S\tnl(W,\mu) = - \tnl(\S{W},\mu).
\]

The center spectrum of $\tL_0$ as an operator in $\tX$ with domain $\tD$ is

\[
\sigma(\tL_0) \cap i\R
= \{0,\pm{i}\omega\},
\]

where $\pm{i}\omega$ are the pure imaginary eigenvalues of $\L_0$ from Hypothesis <ref>.
The point 0 is an eigenvalue of multiplicity 2 with generalized eigenvectors $\chi_1$ and $\chi_2$ satisfying $\tL_0\chi_1 = 0$, $\tL_0\chi_2 = \chi_1$, and $\S\chi_1 = \chi_1$.
The points $\pm{i}\omega$ are eigenvalues of algebraic multiplicity 1.

* The spectral projection for $\tL_0$ corresponding to 0 has the form

\begin{equation}\label{eqn: tPi0}
\tPi_0W
= \chi_1^*[W]\chi_1 + \chi_2^*[W]\chi_2,
\end{equation}

\begin{equation}\label{eqn: tFmu nondegen}
\chi_2^*\big[\tL_1(0)\chi_1\big]
> 0
\quadword{and}
\chi_2^*\big[D_{WW}^2\tF(0,0)[\chi_1,\chi_1]\big]
\ne 0.
\end{equation}

* Let $\tPi$ be the spectral projection for $\tL_0$ corresponding to $\{0,\pm{i}\omega\}$ and put

\begin{equation}\label{eqn: hyperbolic space tX tD}
\tX_{\hsf} := (\ind_{\tX}-\tPi)(\tX)
\quadword{and}
\tD_{\hsf} := \tX_{\hsf} \cap \tD.
\end{equation}

Put $\tY_{\hsf} := \Y_{\hsf}$, where $\Y_{\hsf}$ was defined in Hypothesis <ref>, and take $b \in (0,\pi)$ and $q \in (0,\Lfrak_0^{1/2})$ from that hypothesis as well.
Then on the triple $(\tD_{\hsf},\tY_{\hsf},\tX_{\hsf})$, the operator $\restr{\tL_0}{\D_{\hsf}}$ has the localized optimal regularity property with decay rate $q\Lfrak_0^{-1/2}$ and strip width $b$; the periodic optimal regularity property with base frequency $\omega$; and the suboptimal regularity property with growth rate $\grave{q}$.

* This is obvious from the definitions in (<ref>), (<ref>), and (<ref>).

* This follows from Hypothesis <ref>, Hypothesis <ref>–<ref>, parts <ref> and  <ref> of Lemma <ref>, and part <ref> of Lemma <ref>.

* This follows from Hypothesis <ref> and Lemma <ref>.

* That the spectral projection has this form follows from Lemma <ref>.
For the inequality in (<ref>), we use the definition of $\tL_1$ in (<ref>) and the identities $\chi_2^*[\chi_0] = \chi_2^*[\chi_3] = 0$ to calculate

\[
\chi_2^*[\tL_1(0)\chi_1]
= \chi_2^*[\L_1(0)\chi_1]- \Jscr_*(0)\chi_1
= \Lfrak_0
> 0,
\]

where the positivity is Hypothesis <ref>.

The proof of the nonequality in (<ref>) is slightly more involved.
From the definition of $\tF$ in (<ref>), we calculate

\[
\chi_2^*[\tF(W,0)]
= \chi_2^*[\tL_0W] + \chi_2^*\big[\nl(\T_{\mu}(W)\chi_3 + W,\T_{\mu}(W)\chi_3+W)\big] + \T_{\mu}(W).
\]

We immediately have

\[
\chi_2^*[\tL_0W]
= \chi_2^*[\L_0W]-\chi_1^*[W]\chi_2^*[\chi_0]
= \chi_3^*[W]
= 0.
\]

Next, we rewrite

\[
\nl(\T_{\mu}(W)\chi_3 + W,\T_{\mu}(W)\chi_3+W,0)
= \nl_0(\T_{\mu}(W)\chi_3+W,\T_{\mu}(W)\chi_3+W) + \nl_1(\T_{\mu}(W)\chi_3+W,0)
\]

from the expansion of $\nl$ in (<ref>).

By parts <ref> and <ref> of Lemma <ref>, we have

\begin{equation}\label{eqn: T-quad est}
\le C\norm{W}_{\X}^2
\end{equation}

for $\norm{W}_{\X} < \mu_1$.
The bilinearity of $\nl_0$ implies

\begin{equation}\label{eqn: nl0 with T}
\nl_0(\T_{\mu}(W)\chi_3+W,\T_{\mu}(W)\chi_3+W)
= \T_{\mu}(W)^2\nl_0(\chi_3,\chi_3) + 2\T_{\mu}(W)\nl_0(\chi_3,W) + \nl_0(W,W).
\end{equation}

The estimate (<ref>) and the bilinearity of $\nl_0$ imply that the first two terms in (<ref>) are quartic and cubic in $W$, respectively.
Next, the term $\nl_1(\T_{\mu}(W)\chi_3+W,0)$ is then cubic in $W$ by (<ref>) and the estimate (<ref>).

We conclude

\begin{align*}
\chi_2^*\big[D_{WW}^2\tF(0,0)[\chi_1,\chi_1]\big]
&= 2\chi_2^*\big[\nl_0(\chi_1,\chi_1)\big]+ D_{WW}^2\T(0)[\chi_1,\chi_1] \\
&= 2\chi_2^*\big[\nl_0(\chi_1,\chi_1)\big] - D^2\J_0(0)[\chi_1,\chi_1]
\end{align*}

by part <ref> of Lemma <ref>.
This final quantity is $2\Qfrak_0 \ne 0$, per Hypothesis <ref>.

* First, the spectral projection for $\tL_0$ onto $\{0,\pm{i}\omega\}$ has the form

\[
\tPi{W}
= \tPi_0W + \chi_+^*[W]\chi_+ + \chi_-^*[W]\chi_-, \qquad W \in \tD_{\hsf}.
\]

To verify this, use Lemma <ref> and the uniqueness of the spectral projection discussed in Appendix <ref>.
Then $\tPi{W} = \Pi{W}$ for all $W \in \tD_{\hsf}$, where $\Pi$ was defined in (<ref>).
It then follows from the definition of $\tX$ and $\tD$ in (<ref>) that

\[
\tX_{\hsf} = \X_{\hsf},
\qquad
\tD_{\hsf} = \D_{\hsf},
\quadword{and}
\tY_{\hsf} = \Y_{\hsf} \subseteq \tX_{\hsf}.
\]

The spaces $\X_{\hsf}$ and $\D_{\hsf}$ were defined in (<ref>) and $\tX_{\hsf}$ and $\tD_{\hsf}$ in (<ref>).

We use (<ref>) to calculate

\begin{align*}
&= (\ind_{\X}-\Pi)\tL_1(\mu)W \\
&= (\ind_{\X}-\Pi)\left(\L_1(\mu)W-\mu\left(\frac{\Jscr_*(\mu)W}{1+\mu\Jscr_*(\mu)\chi_3}\right)\L_1(\mu)\chi_3\right)
\in \Y_{\hsf}
\end{align*}

and (<ref>) to calculate

\begin{align*}
&= (\ind_{\X}-\Pi)\tnl(W,\mu) \\
&= (\ind_{\X}-\Pi)\big(\nl(\T_{\mu}(W)\chi_3+W,\mu) + \mu\nl_{\T}(W,\mu)\L_1(\mu)\chi_3\big)
\in \Y_{\hsf},
\end{align*}

with both inclusions from Hypothesis <ref>.

Finally, $\restr{\tL_0}{\tD_{\hsf}} = \restr{\L_0}{\D_{\hsf}}$, so we can just directly import the assumptions of Hypothesis <ref> on the triple $(\tD_{\hsf}, \tY_{\hsf},\tX_{\hsf}) = (\D_{\hsf},\X_{\hsf},\Y_{\hsf})$.

§.§ The proof of Theorem <ref>

We will use the following term for brevity.

Let $\Y$ be a Banach space.
A family of functions $f_{\mu}^{\alpha} \colon \R \to \Y$ defined for $\mu \in \mathscr{I}_1 \subseteq (0,1)$ and $\alpha \in \mathscr{I}_2 \subseteq \R$ is uniformly bounded if

\[
\sup_{\substack{\mu \in \mathscr{I}_1 \\ \alpha \in \mathscr{I}_2 \\ X \in \R}}
\norm{f_{\mu}^{\alpha}(X)}_{\Y}
< \infty.
\]

We say that the family is uniformly bounded and $q$-localized if the estimate above is true when $\norm{f_{\mu}^{\alpha}(X)}_{\Y}$ is replaced by $e^{q|X|}\norm{f_{\mu}^{\alpha}(X)}_{\Y}$, with $q \in \R$.

Now we begin the proof of Theorem <ref>.
Lemma <ref> allow us to invoke Theorem <ref> to obtain nanopteron solutions to the fully reduced problem (<ref>).
Specifically, these solutions have the form $W = \Wsf_{\mu}^{\alpha}$, where

\begin{equation}\label{eqn: W nano}
\Wsf_{\mu}^{\alpha}(x)
= -\frac{3\Lfrak_0}{2\Qfrak_0}\mu\sech^2\left(\frac{\Lfrak_0^{1/2}\mu^{1/2}x}{2}\right)\chi_1
+ \mu^{3/2}\tUpsilon_{\mu}^{\alpha}(\mu^{1/2}x)
+ \alpha\mu\tPhi_{\mu}^{\alpha}(\Tup_{\mu}^{\alpha}(\mu^{1/2}x))
\end{equation}

and $\Tup_{\mu}^{\alpha}$ is defined in (<ref>).
Taking $W = \tPsf_{\mu}^{\alpha}$, where

\begin{equation}\label{eqn: tPsf-mu-alpha}
\tPsf_{\mu}^{\alpha}(x)
:= \alpha\mu\tPhi_{\mu}^{\alpha}(\mu^{1/2}x),
\end{equation}

also solves (<ref>).

The maps $\tUpsilon_{\mu}^{\alpha}$ and $\tPhi_{\mu}^{\alpha}$ have the same properties as (their tilde-denuded counterparts) in Theorem <ref>.
In particular, $\tUpsilon_{\mu}^{\alpha} \colon \R \to \tD_{\hsf}$ is uniformly bounded and $q$-localized, while $\tPhi_{\mu}^{\alpha} \colon \R \to \tD_{\hsf}$ is uniformly bounded.
Both maps are $\S$-reversible.

\begin{equation}\label{eqn: chi0-star on tPsf-mu-alpha}
\chi_0^*[\tPsf_{\mu}^{\alpha}(X)]
= \chi_0^*[\tUpsilon_{\mu}^{\alpha}(X)]
= 0.
\end{equation}

All of the functions defined above are valid for $\mu$ and $\alpha$ in the ranges

\begin{equation}\label{eqn: Lombardi intervals}
0 < \mu < \tmu_*
\quadword{and}
\tAlpha_0\exp\left(-\frac{b\omega}{\Lfrak_0^{1/2}\mu^{1/2}}\right) \le \alpha \le \tAlpha_1,
\end{equation}

where $\tAlpha_1 > 0$ is independent of $b$ and $q$, while $\tmu_*$ and $\tAlpha_0$ may depend on $b$ and $q$.
We may assume that $\tAlpha_0$, $\tAlpha_1$, and $\tmu_*$ are small enough that $\T_{\mu}(\Wsf_{\mu}^{\alpha}(x))$ and $\T_{\mu}(\tPsf_{\mu}^{\alpha}(x))$ are defined for all $x \in \R$ and $\mu$ and $\alpha$  in the intervals (<ref>).

It will also be convenient to abbreviate

\begin{equation}\label{eqn: tSigma-mu-alpha}
\tSigma_{\mu}^{\alpha}(X)
:= -\frac{3\Lfrak_0}{2\Qfrak_0}\sech^2\left(\frac{\Lfrak_0^{1/2}X}{2}\right)\chi_1
+ \mu^{1/2}\tUpsilon_{\mu}^{\alpha}(X),
\end{equation}

so that $\tSigma_{\mu}^{\alpha}$ is uniformly bounded and $q$-localized, while the solution $\Wsf_{\mu}^{\alpha}$ has the “localized + (asymptotically) periodic structure”

\begin{equation}\label{eqn: W nano loc + per}
\Wsf_{\mu}^{\alpha}(x)
= \mu\tSigma_{\mu}^{\alpha}(\mu^{1/2}x) + \alpha\mu\tPhi_{\mu}^{\alpha}(\Tup_{\mu}^{\alpha}(\mu^{1/2}x)).
\end{equation}

In the following sections we will convert these solutions to the reduced problem (<ref>) into solutions to the original problem (<ref>) by undoing the changes of variables above.
The subsequent work is not particularly difficult, but we need to keep careful track of various powers of $\mu$ and $\alpha$.

§.§.§ Construction of the “periodic + growing” solutions $\Psf_{\mu}^{\alpha}$

Per (<ref>), since the map $\tPsf_{\mu}^{\alpha}(s) = \alpha\mu\tPhi_{\mu}^{\alpha}(\mu^{1/2}s)$ solves the reduced problem (<ref>), the map

\begin{multline}\label{eqn: Psf-mu-alpha origin}
\Psf_{\mu}^{\alpha}(x)
:= \left(\int_0^x \Gamma_{\mu}\big(\T_{\mu}(\alpha\mu\tPhi_{\mu}^{\alpha}(\mu^{1/2}s))\chi_3
+ \T_{\mu}(\alpha\mu\tPhi_{\mu}^{\alpha}(\mu^{1/2}x))\chi_3 \\
+ \alpha\mu\tPhi_{\mu}^{\alpha}(\mu^{1/2}x)
\end{multline}

solves the original problem (<ref>).
We will rewrite this expression in the form (<ref>).

We first put

\begin{equation}\label{eqn: tPhi-T}
\tPhi_{\mu}^{\alpha,\T}(X)
:= \alpha^{-1}\mu^{-2}\T_{\mu}(\alpha\mu\tPhi_{\mu}^{\alpha}(X)).
\end{equation}

Then $\tPhi_{\mu}^{\alpha,\T}$ is periodic, and it is uniformly bounded in the sense of Definition <ref> by part <ref> of Lemma <ref>.
It is also even, since the $\S$-reversibility of $\tPhi_{\mu}^{\alpha}$ and part <ref> of Lemma <ref> imply

\begin{equation}\label{eqn: tPhi-mu-alpha-T even}
\alpha\mu^2\tPhi_{\mu}^{\alpha,\T}(-X)
= \T_{\mu}(\alpha\mu\tPhi_{\mu}^{\alpha}(-X))
= \T_{\mu}(\alpha\mu\S\tPhi_{\mu}^{\alpha}(X))
= \T_{\mu}(\alpha\mu\tPhi_{\mu}^{\alpha}(X)).
\end{equation}

Next, we define

\begin{equation}\label{eqn: Phi-mu-alpha nano defn}
\Phi_{\mu}^{\alpha}(X)
:= \mu\tPhi_{\mu}^{\alpha,\T}(X)\chi_3 + \tPhi_{\mu}^{\alpha}(X)
\end{equation}

to see that $\Phi_{\mu}^{\alpha}$ is uniformly bounded and

\begin{equation}\label{eqn: TPchi3 + P id}
\T_{\mu}(\alpha\mu\tPhi_{\mu}^{\alpha}(X))\chi_3
+ \alpha\mu\tPhi_{\mu}^{\alpha}(X)
= \alpha\mu\Phi_{\mu}^{\alpha}(X).
\end{equation}

It remains for us to treat the integral in (<ref>).
With $\Gamma_{\mu}$ and $\Gamma_{\mu}^*$ defined in (<ref>), we have

\[
\Gamma_{\mu}(\T_{\mu}\big(\alpha\mu\tPhi_{\mu}^{\alpha}(X))\chi_3
= \alpha\mu\chi_1^*[\tPhi_{\mu}^{\alpha}(X)]
+ \Gamma_{\mu}^*\big(\T(\alpha\mu\tPhi_{\mu}^{\alpha}(X),\mu,0)\chi_3
\]

\begin{equation}\label{eqn: tPhi-mu-alpha-chi}
\tPhi_{\mu}^{\alpha,\chi}(X)
:= \alpha^{-1}\chi_1^*[\tPhi_{\mu}^{\alpha}(X)].
\end{equation}

This map is periodic and uniformly bounded by the expansions (<ref>) and (<ref>).
It is also even, by a calculation like (<ref>).

Next, let

\[
\tPhi_{\mu}^{\alpha,\Gamma}(X)
:= (\alpha\mu^2)^{-1}\Gamma_{\mu}^*\big(\T_{\mu}(\alpha\mu\tPhi_{\mu}^{\alpha}(X))\chi_3
\]

Part <ref> of Lemma <ref> guarantees that $\tPhi_{\mu}^{\alpha,\Gamma}$ is  uniformly bounded, and it is also periodic.
Finally, it is even, since

\[
\Gamma_{\mu}\big(\T_{\mu}\big(\alpha\mu\tPhi_{\mu}^{\alpha}(-X))\chi_3 + \alpha\mu\tPhi_{\mu}^{\alpha}(-X)\big)
= \Gamma_{\mu}\big(\S(\T_{\mu}(\alpha\mu\tPhi_{\mu}^{\alpha}(X))\chi_3+\alpha\tPhi_{\mu}^{\alpha}(X)\big)
\]

by the $\S$-reversibility of $\tPhi_{\mu}^{\alpha}$, part <ref> of Lemma <ref>, part <ref> of Lemma <ref>, and part <ref> of Lemma <ref>.

We now abbreviate

\begin{equation}\label{eqn: Phi-mu-alpha-int}
\Phi_{\mu}^{\alpha,\int}(X)
:= \big(\alpha\mu(\alpha+\mu)\big)^{-1}\big(\alpha^2\mu\tPhi_{\mu}^{\alpha,\chi}(X) + \alpha\mu^2\tPhi_{\mu}^{\alpha,\Gamma}(X)\big),
\end{equation}

so that $\Phi_{\mu}^{\alpha,\int}$ is, once again, periodic, uniformly bounded, and even.
We integrate and find

\begin{equation}\label{eqn: Mu-T-Phi id}
\int_0^x \Gamma_{\mu}\big(\T_{\mu}(\alpha\mu\tPhi_{\mu}^{\alpha}(\mu^{1/2}s))\chi_3
= \alpha\mu^{1/2}(\alpha+\mu)\int_0^{\mu^{1/2}x} \Phi_{\mu}^{\alpha,\int}(s) \ds.
\end{equation}

The estimate (<ref>) follows from all the uniform bounds mentioned above.
The Lipschitz estimate (<ref>) follows from (<ref>) and (<ref>) and the various uniform bounds above.
Finally, the formula (<ref>) follows from (<ref>) and (<ref>).

For the map $\Psf_{\mu}^{\alpha}$ to be periodic, it is necessary and sufficient that the integral term in (<ref>) be periodic.
Since the integrand is periodic and real analytic, the periodicity of the integral is equivalent to the vanishing of the zeroth Fourier coefficient of the integrand.
However, the integrand here is quite complicated due to the three terms in $\Gamma_{\mu}$ from (<ref>), where now each of these terms is a composition involving both $\T_{\mu}$ and $\tPhi_{\mu}^{\alpha}$.
We know a great deal about these maps from Lemma <ref> and part <ref> of Theorem <ref>, but our present store of knowledge is not sufficient to help us divine whether or not the zeroth Fourier mode vanishes.

§.§.§ Construction of the full “nanopteron + growing” solutions $\Usf_{\mu}^{\alpha}$

We may use (<ref>) to see that, with $\Wsf_{\mu}^{\alpha}$ defined in (<ref>), the map

\begin{equation}\label{eqn: UsF-mu-alpha origin}
\Usf_{\mu}^{\alpha}(x)
:= \left(\int_0^x \Gamma_{\mu}\big(\T_{\mu}(\Wsf_{\mu}^{\alpha}(s))\chi_3 + \Wsf_{\mu}^{\alpha}(s)\big) \ds\right)\chi_0
+ \T_{\mu}(\Wsf_{\mu}^{\alpha}(x))\chi_3
+ \Wsf_{\mu}^{\alpha}(x)
\end{equation}

solves the original problem (<ref>).
As in Section <ref>, we rewrite a number of the terms above to highlight certain powers of $\mu$ and $\alpha$, and now also with an eye toward separating exponentially localized terms from (asymptotically) periodic ones.

We begin this time with the integral in (<ref>).
First, from the definitions of $\Gamma$ and $\Gamma_1$ in (<ref>) and the definition of $\tPhi_{\mu}^{\alpha,\chi}$ in (<ref>), we calculate

\begin{multline}\label{eqn: Mu(T(W))1}
\Gamma_{\mu}\big(\T_{\mu}(\Wsf_{\mu}^{\alpha}(s))\chi_3 + \Wsf_{\mu}^{\alpha}(s)\big)
= -\frac{3\Lfrak_0}{2\Qfrak_0}\mu\sech^2\left(\frac{\Lfrak_0^{1/2}\mu^{1/2}s}{2}\right)
+ \mu^{3/2}\chi_1^*[\tUpsilon_{\mu}^{\alpha}(\mu^{1/2}s)] \\
+ \alpha^2\mu\tPhi_{\mu}^{\alpha,\chi}(\Tup_{\mu}^{\alpha}(\mu^{1/2}s))
+ \Gamma_{\mu}^*\big(\T_{\mu}(\Wsf_{\mu}^{\alpha}(s))\chi_3 + \Wsf_{\mu}^{\alpha}(s)\big).
\end{multline}

\[
\tSigma_{\mu}^{\alpha,\T}(X)
:= \mu^{-2}\big(\T_{\mu}(\mu\tSigma_{\mu}^{\alpha}(X) + \alpha\mu\tPhi_{\mu}^{\alpha}(X))-\T_{\mu}(\alpha\mu\tPhi_{\mu}^{\alpha}(X))\big),
\]

where $\tSigma_{\mu}^{\alpha}$ was defined in (<ref>).
Part <ref> of Lemma <ref> guarantees that $\tSigma_{\mu}^{\alpha,\T}$ is uniformly bounded.
The methods of the preceding section also show that $\tSigma_{\mu}^{\alpha,\T}$ is even.
Finally, the expansion of $\Wsf_{\mu}^{\alpha}$ in (<ref>) gives

\[
\T_{\mu}(\Wsf_{\mu}^{\alpha}(x))
= \mu^2\tSigma_{\mu}^{\alpha,\T}(\mu^{1/2}x) + \alpha\mu^2\tPhi_{\mu}^{\alpha,\T}(\Tup_{\mu}^{\alpha}(\mu^{1/2}x)),
\]

where $\tPhi_{\mu}^{\alpha,\T}$ was defined in (<ref>).

Now set

\begin{multline*}
\tSigma_{\mu}^{\alpha,\Gamma}(X)
:= \mu^{-2}\Gamma_{\mu}^*\big([\alpha\mu^2\tPhi_{\mu}^{\alpha,\T}(\Tup_{\mu}^{\alpha}(X))\chi_3+\alpha\mu\tPhi_{\mu}^{\alpha}(\Tup_{\mu}^{\alpha}(X))] + [\mu^2\tSigma_{\mu}^{\alpha,\T}(X)\chi_3 + \mu\tSigma_{\mu}^{\alpha}(X)]\big) \\
-  \mu^{-2}\Gamma_{\mu}^*\big(\alpha\mu^2\tPhi_{\mu}^{\alpha,\T}(\Tup_{\mu}^{\alpha}(X))\chi_3+\alpha\mu\tPhi_{\mu}^{\alpha}(\Tup_{\mu}^{\alpha}(X))\big).
\end{multline*}

Part <ref> of Lemma <ref> implies that $\tSigma_{\mu}^{\alpha,\Gamma}$ is uniformly bounded and $q$-localized in the sense of Definition <ref>, and the methods of Section <ref> show that it is even.
Last, abbreviate

\[
\tUpsilon_{\mu}^{\alpha,\int}(X)
:= \chi_1^*[\tUpsilon_{\mu}^{\alpha}(X)] + \mu^{1/2}\tSigma_{\mu}^{\alpha,\Gamma}(X),
\]

so $\tUpsilon_{\mu}^{\alpha,\int}$ is uniformly bounded and $q$-localized and even.

We may now use the work above and the definition of $\Phi_{\mu}^{\alpha,\int}$ in (<ref>) to rearrange (<ref>) into

\begin{multline*}
\Gamma_{\mu}\big(\T_{\mu}(\Wsf_{\mu}^{\alpha}(s))\chi_3 + \Wsf_{\mu}^{\alpha}(s)\big)
= -\frac{3\Lfrak_0}{2\Qfrak_0}\mu\sech^2\left(\frac{\Lfrak_0^{1/2}\mu^{1/2}s}{2}\right)
+ \mu^{3/2}\Upsilon_{\mu}^{\alpha,\int}(\mu^{1/2}s) \\
+ \alpha\mu(\alpha+\mu)\Phi_{\mu}^{\alpha,\int}(\Tup_{\mu}^{\alpha}(\mu^{1/2}s)),
\end{multline*}

and so

\begin{multline}\label{eqn: Mu int W}
\int_0^x \Gamma_{\mu}\big(\T_{\mu}(\Wsf_{\mu}^{\alpha}(s))\chi_3 + \Wsf_{\mu}^{\alpha}(s)\big) \ds
= -\frac{3\Lfrak_0^{1/2}}{\Qfrak_0}\mu^{1/2}\tanh\left(\frac{\Lfrak_0^{1/2}\mu^{1/2}x}{2}\right)
+ \mu\int_0^{\mu^{1/2}x}\tUpsilon_{\mu}^{\alpha,\int}(s) \ds \\
+ \alpha\mu^{1/2}(\alpha+\mu)\int_0^{\mu^{1/2}x}\Phi_{\mu}^{\alpha,\int}(\Tup_{\mu}^{\alpha}(s)) \ds.
\end{multline}

We rewrite the remaining two integrals in (<ref>) to isolate their leading-order behavior.
Part <ref> of Lemma <ref> (with $\nu = \mu^{1/2}$ and $q_* = \Lfrak_0^{1/2}$) allows us to write

\begin{equation}\label{eqn: Upsilon1}
\int_0^X \tUpsilon_{\mu}^{\alpha,\int}(s) \ds
= \Lup_{\mu,1}^{\alpha}\tanh\left(\frac{\Lfrak_0^{1/2}X}{2}\right)
+ \Upsilon_{\mu}^{\alpha,1}(X),
\end{equation}

where $\Lup_{\mu}^{\alpha,\int}$ is uniformly bounded and $\Upsilon_{\mu}^{\alpha,1}$ is uniformly bounded and $q$-localized.
Part <ref> of Lemma <ref> (now with $q_* = \Lfrak_0^{1/2}/2$) allows us to write

\begin{equation}\label{eqn: Upsilon2}
\int_0^X \Phi_{\mu}^{\alpha,\int}(\Tup_{\mu}^{\alpha}(s)) \ds
= \int_0^{\Tup_{\mu}^{\alpha}(X)} \Phi_{\mu}^{\alpha,\int}(s) \ds
+ \mu^{1/2}\Lup_{\mu,2}^{\alpha,2}\tanh\left(\frac{\Lfrak_0^{1/2}X}{2}\right)
+ \mu^{1/2}\Upsilon_{\mu}^{\alpha,2}(X),
\end{equation}

where $\Lup_{\mu}^{\alpha,\infty}$ is uniformly bounded and $\Upsilon_{\mu}^{\alpha,2}$ is uniformly bounded and $\Lfrak_0^{1/2}$-localized, and therefore $q$-localized, since $0 < q < \Lfrak_0^{1/2}$.

\[
\Lup_{\mu}^{\alpha}
:= \Lup_{\mu}^{\alpha,1} + \alpha(\alpha+\mu)\Lup_{\mu}^{\alpha,2}
\]

\begin{equation}\label{eqn: Upsilon-mu-alpha-0}
\Upsilon_{\mu}^{\alpha,0}(X)
:= \Upsilon_{\mu}^{\alpha,1}(X) + \alpha(\alpha+\mu)\Upsilon_{\mu}^{\alpha,2}(X)
\end{equation}

Then $\Lup_{\mu}^{\alpha}$ is uniformly bounded and $\Upsilon_{\mu}^{\alpha,0}$ is uniformly bounded and $q$-localized.
By inspection of (<ref>) and (<ref>), we see that $\partial_X[\Upsilon_{\mu}^{\alpha,0}]$ is also uniformly bounded and $q$-localized.

Last, define

\begin{equation}\label{eqn: Upsilon-mu-alpha-star}
\Upsilon_{\mu}^{\alpha,*}(X)
:= \mu^{1/2}\tSigma_{\mu}^{\alpha,\T}(X)\chi_3 + \tUpsilon_{\mu}^{\alpha}(X).
\end{equation}

By (<ref>), we have $\chi_0^*[\Upsilon_{\mu}^{\alpha,*}(X)] = 0$.
Then the definition of $\Wsf_{\mu}^{\alpha}$ in (<ref>), the definition of $\Phi_{\mu}^{\alpha}$ in (<ref>), and the integral identity (<ref>) allow us to rewrite the solution $\Usf_{\mu}^{\alpha}$ from (<ref>) into the version (<ref>) foretold in the statement of Theorem <ref>.

§.§.§ Reversibility proofs

We show that each of the terms in the definition of $\Usf_{\mu}^{\alpha}$ in (<ref>) is $\S$-reversible.
We will need the following lemma, whose proof is a direct calculation using Definition <ref> and Hypothesis <ref>–<ref>.

Let $f \colon \R \to \R$ be a function and define $g_k(X) := f_k(X)\chi_k$, for $k=1,\ldots,4$, where $\chi_k$ is one of the generalized eigenvectors from Hypothesis <ref>.
Then $g_k$ is reversible if $f$ is an odd function and $k$ is an even integer, or if $f$ is an even function and $k$ is an odd integer.

It follows from (<ref>) and (<ref>) and the evenness of $\tUpsilon_{\mu}^{\alpha,\int}$ and $\Phi_{\mu}^{\alpha,\int}$ that the maps $\Upsilon_{\mu,1}^{\alpha,0}$ and $\Upsilon_{\mu}^{\alpha,1}$ are odd.
Then from its definition in (<ref>), the map $\Upsilon_{\mu}^{\alpha,0}$ is also odd.
Consequently, the prefactor functions on $\chi_0$ in (<ref>) are odd, and so Lemma <ref> shows that the $\chi_0$-term in (<ref>) is $\S$-reversible.

Next, the $\sech^2$-prefactor on $\chi_1$ is even, so that term in (<ref>) is $\S$-reversible.
Also, the map $\tSigma_{\mu}^{\alpha,\T}$ is even, so the $\chi_3$-term in the definition of $\Upsilon_{\mu}^{\alpha,*}$ in (<ref>) is $\S$-reversible.
Last, the map $\tUpsilon_{\mu}^{\alpha}$ is $\S$-reversible by Theorem <ref>.
Thus $\Upsilon_{\mu}^{\alpha,*}$ is $\S$-reversible.

Since $\Tup_{\mu}^{\alpha}$ is odd, the last two terms in (<ref>) will be $\S$-reversible if we know that $\Psf_{\mu}^{\alpha}$, defined in (<ref>), is $\S$-reversible.
From its definition in (<ref>), the map $\Phi_{\mu}^{\alpha}$ is $\S$-reversible by the same arguments as for $\Upsilon_{\mu}^{\alpha,*}$.
The $\chi_0$-term in (<ref>) is $\S$-reversible by the evenness of the integrand $\Phi_{\mu}^{\alpha,\int}$ and Lemma <ref>.
Thus $\Psf_{\mu}^{\alpha}$ is $\S$-reversible.

§ THE POSITION TRAVELING WAVE PROBLEM

We will reformulate the position traveling wave problem (<ref>) for the general dimer into a first-order differential equation posed on an infinite-dimensional Banach space.
After choosing a small parameter that controls the distance of the wave speed from the speed of sound, we will show that this equation satisfies the hypotheses of Section <ref>.
We are able to verify all the hypothesis except the symmetry for the general dimer; to obtain a symmetry, we must specialize to either a mass or a spring dimer.

We develop this first-order equation in Section <ref> and introduce the near-sonic small parameter in Section <ref>.
In between and afterward, we uncover a host of properties of this equation that are needed to verify the hypotheses.
Table <ref> outlines precisely where the different hypotheses are considered for the concrete lattice problems.
Table <ref> connects the notation of various operators and quantities in the abstract problem of Section <ref> to the concrete lattice situation of this section.

\[
\begin{tabular}{l|l}
\hline
{\bf{Hypothesis}} &{\bf{Discussion and Verification}} \\
\hline
\ref{hypo: F structure}: structural properties &Section \ref{sec: near-sonic} \\
\ref{hypo: symmetry}: symmetry &Section \ref{sec: symmetries} \\
\ref{hypo: spectrum}: center spectrum &Propositions \ref{prop: eigenprops} and \ref{prop: Lambda props}  \\
\ref{hypo: eigenproj}: zero eigenprojection &Sections \ref{sec: gen ev}, \ref{sec: eigenproj}, and \ref{sec: omega ev}  \\
\ref{hypo: first int}: first integral &Section \ref{sec: first int} \\
\ref{hypo: nondegen}: nondegeneracies &Section \ref{sec: nondegen} \\
\ref{hypo: opt reg}: optimal regularity &Section \ref{sec: opt reg} \\
\hline
\end{tabular}
\]

Location of the proofs of the various hypotheses

\[
\begin{tabular}{Sc|Sc|Sl}
\hline
{\bf{Section \ref{sec: abstract}}} &{\bf{Section \ref{sec: tw prob redux}}} &{\bf{Definition in Section \ref{sec: tw prob redux}}} \\
\hline
$\X$ &$\X$ & \eqref{eqn: X defn} \\
\hline
$\D$ &$\D$ & \eqref{eqn: D defn} \\
\hline
$U$ &$\Ub$ &\eqref{eqn: Ub} \\
\hline
$\F(U,\mu)$ &$\F(\Ub,\mu;\kappa,\beta,\Vscr_1,\Vscr_2,w)$ &\eqref{eqn: F-ep-kappa-beta-w} \\
\hline
$\L_0$ &$\L_0(\kappa,w)$ &\eqref{eqn: star simplicity} \\
\hline
$\L_1(\mu)$ &$\L_1(\kappa,w)$ &\eqref{eqn: Lsuper1} \\
\hline
$\nl(U,\mu)$ &$\nl(\Ub,\mu;\kappa,\beta,\Vscr_1,\Vscr_2,w)$ &\eqref{eqn: concrete nl} \\
\hline
$\nl_0(U,\grave{U})$ &$\nl_0(\Ub,\grave{\Ub};\kappa,\beta,w)$ &\eqref{eqn: concrete nl0} \\
\hline
$\S$ &$\S_{\Mb}$ or $\S_{\Kb}$ &\eqref{eqn: md symm}, \eqref{eqn: sd symm} \\
\hline
$\omega$ &$\omega_*(\kappa,w)$ &\eqref{eqn: star simplicity}; Proposition \ref{prop: Lambda props}, parts \ref{part: Lambda-1}, \ref{part: Lambda-3}  \\
\hline
$\chi_k$ &$\chib_k(\kappa,w)$ &\eqref{eqn: chib0}, \eqref{eqn: chib1}, \eqref{eqn: chib2}, \eqref{eqn: chib3} \\
\hline
$\chi_k^*[U]$ &$\chib_k^*[\Ub;\kappa,w]$ &\eqref{eqn: chib0-star}, \eqref{eqn: chib1-star}, \eqref{eqn: chib2-star}, \eqref{eqn: chib3-star} \\
\hline
$\chi_{\pm}$ &$\chib_{\pm}(\kappa,w)$ &\eqref{eqn: chib-pm-omega} \\
\hline
$\chi_{\pm}^*[U]$ &$\chib_{\pm}^*[\Ub;\kappa,w]$ &Section \ref{sec: omega ev} \\
\hline
$\J_{\mu}(U)$ &$\J_{\mu}(\Ub;\V_1,\V_2,\kappa,w)$ &\eqref{eqn: concrete first int} \\
\hline
$\Jscr_*(\mu)$ &$\Jscr_*(\mu;\kappa,w)$ &\eqref{eqn: Jscr concrete} \\
\hline
$\Lfrak_0$ &$\Lfrak_0(\kappa,w)$ &\eqref{eqn: lin nondegen concrete} \\
\hline
$\Qfrak_0$ &$\Qfrak_0(\kappa,\beta,w)$ &\eqref{eqn: quad nondegen concrete} \\
\hline
\end{tabular}
\]

Correspondence of notation between Sections <ref> and <ref>

§.§ The Iooss–Kirchgässner change of variables

Recall that $p_1$ and $p_2$ are the traveling wave profiles in position coordinates, and $p_1$ and $p_2$ must solve (<ref>).
Following Iooss and Kirchgässner <cit.>, we set $\xi_j = p_j'$ and then define functions $P_j$ on $\R \times [-1,1]$ by

\begin{equation}\label{eqn: Pj}
:= p_j(x+v).
\end{equation}

Observe that

\[
P_j(x,0) = p_j(x+0) = p_j(x)
\quadword{and}
\partial_x[P_j](x,v) = p_j(x+v) = \partial_v[P_j](x,v).
\]

We collect these new coordinates into one vector as

\begin{equation}\label{eqn: Ub}
\Ub(x)
:= \big(p_1(x),p_2(x),\xi_1(x),\xi_2(x),P_1(x,\cdot),P_2(x,\cdot)\big),
\end{equation}

so that $\Ub(x) \in \X$, where

\begin{equation}\label{eqn: X defn}
\X := \set{(p_1,p_2,\xi_1,\xi_2,P_1,P_2) \in \R^4 \times \Cal([-1,1]) \times \Cal([-1,1])}{P_1(0) = p_1, \ P_2(0) = p_2}
\end{equation}

We will be rather cavalier about whether we write elements of $\X$ as “row” or “column” vectors.
We will also work on the space

\begin{equation}\label{eqn: D defn}
\D
:= \X \cap \big(\R^4 \times \Cal^1([-1,1]) \times \Cal^1([-1,1])\big).
\end{equation}

The spaces $\D$ and $\X$ are Banach spaces with the usual maximum norms.
For $\Ub \in \X$ we will denote sometimes its components by $(\Ub)_j$ for $j=1,\ldots,6$, so that if $\Ub = (p_1,p_2,\xi_1,\xi_2,P_1,P_2) \in \X$, then

\begin{equation}\label{eqn: component convention}
(\Ub)_1 := p_1,
\quad
(\Ub)_2 := p_2,
\quad
(\Ub)_3 := \xi_1,
\quad
(\Ub)_4 := \xi_2,
\quad
(\Ub)_5 := P_1,
\quad\text{and}\quad
(\Ub)_6 := P_2.
\end{equation}

We are almost ready to express the traveling wave problem (<ref>) as a first-order equation in $\X$.
Define the evaluation operators $\delta^{\pm1}$ by

\begin{equation}\label{eqn: delta defn}
:= P_j(x,\pm1)
= p_j(x\pm1).
\end{equation}

If $f = f(v)$ is a function of a single variable $v$, we will still write $\delta^{\pm1}f := f(\pm1)$.

The traveling wave problem (<ref>) is then equivalent to

\begin{equation}\label{eqn: IK system}
\Ub'(x)
= \F(\Ub(x);\V_1,\V_2,w,c),
\end{equation}

where, for $\Ub = (p_1,p_2,\xi_1,\xi_2,P_1,P_2) \in \X$, we define

\begin{equation}\label{eqn: Fc}
\F(\Ub;\V_1,\V_2,w,c)
:= \begin{pmatrix*}
\xi_1 \\
\xi_2 \\
c^{-2}\V_1'(\delta^1P_2-p_1)-c^{-2}\V_2'(p_1-\delta^{-1}P_2) \\
c^{-2}w\V_2'(\delta^1P_1-p_2)-c^{-2}w\V_1'(p_2-\delta^{-1}P_1) \\
\partial_v[P_1] \\
\partial_v[P_2]
\end{pmatrix*}.
\end{equation}

The parameter $w > 0$ is the (reciprocal of) the mass ratio, per (<ref>), and the functions $\V_1$ and $\V_2$ are the spring potentials from (<ref>).

We can expand $\F$ as the superposition of linear, quadratic, and superquadratic terms:

\begin{equation}\label{eqn: Fc L+Q0+Q1}
\F(\Ub;c)
= \L(\kappa,w,c)\Ub + c^{-2}\nl_0(\Ub,\Ub;\beta,w) + c^{-2}\nl_1(\Ub;\Vscr_1,\Vscr_2,w).
\end{equation}

We define these terms as follows.
First, the linear operator $\L(\kappa,w,c)$ is the block diagonal matrix

\begin{equation}\label{eqn: Lc defn}
\L(\kappa,w,c)
:= \begin{bmatrix*}
0 &\ind &0 \\[10pt]
-c^{-2}(1+\kappa)\diag(1,w) &0 &c^{-2}\Delta(\kappa,w) \\
0 &0 &\partial_v\ind
\end{bmatrix*},
\qquad
\ind := \begin{bmatrix*}
1 &0 \\
0 &1
\end{bmatrix*},
\end{equation}

and the component operator $\Delta(\kappa,w)$ is

\begin{equation}\label{eqn: Delta defn}
\Delta(\kappa,w)
:= \begin{bmatrix*}
0 &\big(\delta^1+\kappa\delta^{-1}\big) \\
w\big(\kappa\delta^1+\delta^{-1}\big) &0
\end{bmatrix*}.
\end{equation}

Next, the quadratic term $\nl_0$ is

\begin{equation}\label{eqn: nl0 defn}
\nl_0(\Ub,\grave{\Ub};\kappa,\beta,w)
:= \begin{pmatrix*}
0 \\
0 \\
(\delta^1P_2-p_1)(\delta^1\grave{P}_2-\grave{p}_1) - \beta(p_1-\delta^{-1}P_2)(\grave{p}_1-\delta^{-1}\grave{P}_2) \\
w\beta(\delta^1P_1-p_2)(\delta^1\grave{P}_1-\grave{p}_2) - w(p_2-\delta^{-1}P_1)(\grave{p}_2-\delta^{-1}\grave{P}_1) \\
0 \\
\end{pmatrix*}.
\end{equation}

Here we are writing $\grave{\Ub} = (\grave{p}_1,\grave{p}_2,\grave{\xi}_1,\grave{\xi}_2,\grave{P}_1,\grave{P}_2)$.
Finally, the superquadratic term is

\begin{equation}\label{eqn: nl1 defn}
\nl_1(\Ub;\Vscr_1,\Vscr_2,w)
= \begin{pmatrix*}
0 \\
0 \\
\Vscr_1(\delta^1P_2-p_1)-\Vscr_2(p_1-\delta^{-1}P_2) \\
w\Vscr_2(\delta^1P_1-p_2)-w\Vscr_1(p_2-\delta^{-1}P_1) \\
\partial_v[P_1] \\
\partial_v[P_2]
\end{pmatrix*}.
\end{equation}

The functions $\Vscr_1$ and $\Vscr_2$ represent the superquadratic terms in the spring forces, per (<ref>).

The problem (<ref>) is not the one to which we will apply our abstract theory from Section <ref>.
In particular, we have not yet introduced a small parameter into (<ref>); we will do so in Section <ref> by taking the wave speed $c$ to be sufficiently close to the speed of sound.
Before that, however, we develop some general properties of (<ref>) that are valid for arbitrary $c$.

§.§ Lattice symmetries

We first observe a symmetry at the level of the original position coordinates.
For a mass dimer ($m_{j+2} = m_j$ and $\V_j = \V_0$ for all $j$), one can check that if $\{u_j\}_{j \in \Z}$ is a solution set for (<ref>), then so is $\{-u_{-j}\}_{j \in \Z}$.
For a spring dimer ($\V_{j+2} = \V_j$ and $m_j = m_0$ for all $j$), the set $\{-u_{-j+1}\}_{j \in \Z}$ is a solution whenever $\{u_j\}_{j \in \Z}$ is.
This mismatch in symmetries between the mass and spring dimers appears to be a perennial leitmotif of the respective traveling wave problems.
For example, at the level of relative displacement traveling waves, with the traveling wave profiles denoted by $\varrho_1$ and $\varrho_2$ as in (<ref>), the articles [21, 36, 19] on the mass dimer assume that $\varrho_1$ is even and $\varrho_2$ is odd, while for the spring dimer [16] takes both $\varrho_1$ and $\varrho_2$ to be even.

We now define operators $\S_{\Mb}$, $\S_{\Kb} \in \b(\X)$ such that $\S_{\Mb}^2 = \S_{\Kb}^2 = \ind_{\X}$,  $\S_{\Mb}$ anticommutes with the mass dimer's system, and $\S_{\Kb}$ anticommutes with the spring dimer's system.
That is, for the mass dimer, we need

\begin{equation}\label{eqn: mass dimer anticommute}
\begin{cases}
\L(1,w,c)\S_{\Mb}\Ub = -\S_{\Mb}\L(1,w,c)\Ub \\
\nl_0(\S_{\Mb}\Ub,\S_{\Mb}\Ub;1,w) = -\S_{\Mb}\nl_0(\Ub,\Ub;1,w) \\
\nl_1(\S_{\Mb}\Ub;\Vscr,\Vscr,w) = -S_{\Mb}\nl_1(\Ub,\Vscr,\Vscr,w).
\end{cases}
\end{equation}

For the spring dimer, we need

\begin{equation}\label{eqn: spring dimer anticommute}
\begin{cases}
\L(\kappa,1,c)\S_{\Kb}\Ub = -\S_{\Kb}\L(\kappa,1,c)\Ub \\
\nl_0(\S_{\Kb}\Ub,\S_{\Kb}\Ub;\beta,1) = -\S_{\Kb}\nl_0(\Ub,\Ub;\beta,1) \\
\nl_1(\S_{\Kb}\Ub;\Vscr_1,\Vscr_2,1) = -\S_{\Kb}\nl_1(\Ub;\Vscr_1,\Vscr_2,1).
\end{cases}
\end{equation}

To construct these symmetries, first define a “reflection” operator by

\begin{equation}\label{eqn: R refl}
:= f(-v),
\end{equation}

where $f$ is any function defined on $[-1,1]$.
The chain rule tells us that $R$ and $\partial_v$ anticommute.
With $(\delta^{\pm1}f)(v) := f(\pm1)$ as in (<ref>), we have $\delta^{\pm}R = \delta^{\mp}$.
Then the operators

\begin{equation}\label{eqn: md symm}
\S_{\Mb}
:= \begin{bmatrix*}
-\ind &0 &0 \\
0 &\ind &0 \\
0 &0 &-R\ind
\end{bmatrix*}
\end{equation}

\begin{equation}\label{eqn: sd symm}
\S_{\Kb}
:= \begin{bmatrix*}
-\Jbb &0 &0 \\
0 &\Jbb &0 \\
0 &0 &-R\Jbb
\end{bmatrix*},
\qquad \Jbb
:= \begin{bmatrix*}
0 &1 \\
1 &0
\end{bmatrix*}
\end{equation}

satisfy (<ref>) and (<ref>), respectively.
It is straightforward to check that the operator norms of $\S_{\Mb}$ and $\S_{\Kb}$ are both 1.

The structures of these symmetries are related to our earlier observation that if $\{u_j\}_{j \in \Z}$ solves (<ref>), then $\{-u_{-j}\}_{j \in \Z}$ is also a solution set for the mass dimer, while $\{-u_{-j-1}\}_{j \in \Z}$ is a solution set for the spring dimer.
We view the reflection operator $R$ in both cases as arising from the switching of $j$ to $-j$, while the various factors of $-1$ on the diagonal in both $\S_{\Mb}$ and $S_{\Kb}$ come from the prefactor of $-1$ on $u_{-j}$ in the mass dimer and $u_{-j-1}$ in the spring dimer.
Last, the “flip” operator $\Jbb$ for the spring dimer arises from the offset index $-j-1$.

We note with considerable interest, and frustration, that there does not appear to be a general symmetry that anticommutes with the terms of (<ref>) when we are not specifically in the situation of a mass dimer or a spring dimer.
To be clear, we have not proved nonexistence of such a symmetry.
However, in light of the different even-odd and even-even symmetries for the prior relative displacement problems, and, indeed, the asymmetric appearance of the general dimer in Figure <ref>, we think it is unlikely that such a symmetry exists.
This absence of symmetry is the chief reason why we report results separately in Theorems <ref> and <ref> for mass and spring dimers.
Nonetheless, we will be able to verify all of the non-symmetry hypotheses for the general dimer, which suggests that if a substitute could be found for the role of the symmetry in Lombardi's methods, our methods would readily extend to the general dimer.

§.§ Spectral analysis of the operator $\L(\kappa,w,c)$

We exhaustively analyze the operator $\L(\kappa,w,c)$, defined in (<ref>) as an operator in the space $\X$ from (<ref>) with domain $\D$ from (<ref>).
In particular, we calculate its spectrum and a family of generalized eigenvectors associated with the eigenvalue 0.
We summarize all of our spectral-theoretic conventions in Appendix <ref>.

§.§.§ A characterization of the spectrum of $\L(\kappa,w,c)$

Let $z \in \C$ and suppose that $\Ub = (p_1,p_2,\xi_1,\xi_2,P_1,P_2) \in \D$ and $\grave{\Ub} = (\grave{p}_1,\grave{p}_2,\grave{\xi}_1,\grave{\xi}_2,\grave{P}_1,\grave{P}_2) \in \X$

\[
z\Ub-\L(\kappa,w,c)\Ub = \grave{\Ub}.
\]

We will find a formula for $\Ub$ in terms of $z$ and $\grave{\Ub}$; that is, we calculate the resolvent of $\L(\kappa,w,c)$.

Using the definition of $\L(\kappa,w,c)$ in (<ref>), we first see that $P_j$, $\grave{P}_j$, and $p_j$ must satisfy the initial value problems

\[
\begin{cases}
\partial_v[P_j] -z{P}_j = -\grave{P}_j \\
P_j(0) = p_j.
\end{cases}
\]

We solve them with Duhamel's formula:

\begin{equation}\label{eqn: P-resolvent}
= e^{z{v}}p_j + \I_{z}[\grave{P}_j](v),
\end{equation}

where, for $P \in \Cal([-1,1])$, we put

\begin{equation}\label{eqn: Ilambda}
\I_{z}[P](v)
:= -\int_0^v e^{z(v-s)}P(s) \ds, \qquad P \in \Cal([-1,1]).
\end{equation}

Thus to find $P_j$ it suffices to discover $p_j$.

Again using the definition of $\L(\kappa,w,c)$ in (<ref>), we see that $p_j$, $\xi_j$, and $\grave{p}_j$ must satisfy

\[
= \grave{p}_j,
\]

and so

\begin{equation}\label{eqn: xi-resolvent}
\xi_j
= z{p}_j - \grave{p}_j.
\end{equation}

Then $p_j$, $P_j$, and $\grave{\xi}_j$ must satisfy

\begin{equation}\label{eqn: xi P p}
\begin{cases}
z\xi_1 + \left(\frac{1+\kappa}{c^2}\right)p_1 -\left(\frac{\delta^1+\kappa\delta^{-1}}{c^2}\right)P_2 = \grave{\xi}_1 \\
\\
z\xi_2 + \left(\frac{w(1+\kappa)}{c^2}\right)p_2 - \left(\frac{w(\kappa\delta^1+\delta^{-1})}{c^2}\right)P_1 = \grave{\xi}_2.
\end{cases}
\end{equation}

We calculate

\[
\delta^{\pm1}P_j
= P_j(\pm1)
= e^{\pm{z}}p_j(x) + \I_{z}[\grave{P}_j](\pm1)
\]

and replace $P_j$ and $\xi_j$ in (<ref>) with their $p_j$-dependent formulas to find

\begin{equation}\label{eqn: spectrum pre prod}
\begin{cases}
z(z{p}_1 - \grave{p}_1)
+ \left(\frac{1+\kappa}{c^2}\right)p_1
- \frac{\delta^1\I_{z}[\grave{P}_2]+\kappa\delta^{-1}\I_{z}[\grave{P}_2]}{c^2} = \grave{\xi}_1 \\
\\
z(z{p}_2 - \grave{p}_2)
+ \left(\frac{w(1+\kappa)}{c^2}\right)p_2
- \left(\frac{\kappa{w}{e}^{z}+we^{-z}}{c^2}\right)p_1
- \frac{\kappa{w}\delta^{1}\I_{z}[\grave{P}_1]+w\delta^{-1}\I_{z}[\grave{P}_1]}{c^2} = \grave{\xi}_2.
\end{cases}
\end{equation}

After some rearrangements, this will be equivalent to a matrix-vector equation in $\C^2$.
We define a matrix $\M(z;\kappa,w,c) \in \C^{2 \times 2}$ by

\begin{equation}\label{eqn: Mc defn}
\M(z;\kappa,w,c)
:= \begin{bmatrix*}
\big(c^2z^2+1+\kappa\big) &-\big(e^{z}+\kappa{e}^{-z}\big) \\[5pt]
-w\big(\kappa{e}^{z}+e^{-z}\big) &\big(c^2z^2+w(1+\kappa)\big)
\end{bmatrix*}
\end{equation}

and an operator $\B(z;\kappa,w,c) \colon \X \to \C^2$ by

\begin{equation}\label{eqn: Bc defn}
\B(z;\kappa,w,c)
\grave{\Ub}
:= \begin{pmatrix*}
c^2z\grave{p}_1 + c^2\grave{\xi}_1 + \delta^1\I_{z}[\grave{P}_2]+\kappa\delta^{-1}\I_{z}[\grave{P}_2] \\[5pt]
c^2z\grave{p}_2 + c^2\grave{\xi}_2 + \kappa{w}\delta^{1}\I_{z}[\grave{P}_1]+w\delta^{-1}\I_{z}[\grave{P}_1]
\end{pmatrix*}.
\end{equation}

Then with $\pb = (p_1,p_2) \in \C^2$, the system (<ref>) becomes

\begin{equation}\label{eqn: matrix-vector product}
\M(z;\kappa,w,c)\pb
= \B(z;\kappa,w,c)\grave{\Ub}.
\end{equation}

We can solve (<ref>) uniquely for $\pb$ given $\grave{\Ub}$ if and only if

\begin{equation}\label{eqn: Mc det}
\det\big(\M(z;\kappa,w,c)\big)
= c^4z^4+c^2(1+\kappa)(1+w)z^2+2\kappa{w}(1-\cosh(2z))
\ne 0.
\end{equation}

Since we can always solve for $P_j$ and $\xi_j$ via (<ref>) and (<ref>), we can solve $z\Ub - \L(\kappa,w,c)\Ub = \grave{\Ub}$ uniquely for $\Ub \in \D$ given $\grave{\Ub} \in \X$ if and only if we can solve (<ref>) for $\pb \in \C^2$ given $\grave{\Ub} \in \X$.
And so we have the following characterization of the spectrum.

The spectrum of $\L(\kappa,w,c) \colon \D \to \X$ is the set

\begin{equation}\label{eqn: spectrum}
\sigma(\L(\kappa,w,c))
= \set{z \in \C}{\det\big(\M(z;\kappa,w,c)\big) = 0}.
\end{equation}

Suppose $|c| > 1$.
Then the following hold.

* The spectrum consists entirely of eigenvalues.

* Each eigenvalue is an isolated point of the spectrum, and the algebraic multiplicity of an eigenvalue equals its multiplicity as a root of the map $\det\big(\M(\cdot;\kappa,w,c)\big)$.

Each eigenvalue is geometrically simple.

* The spectrum is symmetric with respect to the real and complex axes in $\C$: if $z \in \sigma(\L(\kappa,w,c))$, then $\overline{z}, -z \in \sigma(\L(\kappa,w,c))$, too.

Let $\Ub = (p_1,p_2,\xi_1,\xi_2,P_1,P_2) \in \X$ and $z \in \rho(\L(\kappa,w,c))$.
The six components of the resolvent operator $\rhs(z;\kappa,w,c)\Ub := \big(z\ind_{\D}-\L(\kappa,w,c)\big)^{-1}\Ub$ are

\begin{align}
\begin{split}
\big(\rhs(z;\kappa,w,c)\Ub\big)_1
&= \frac{c^2z^2+w(1+\kappa)}{\det\big(\M(z;\kappa,w,c)\big)}\big(c^2\xi_1+c^2z{p}_1 + \delta^1\I_{z}[P_2] + \kappa\delta^{-1}\I_{z}[P_2]\big) \\
&+\frac{e^{z}+\kappa{e}^{-z}}{\det\big(\M(z;\kappa,w,c)\big)}\big(c^2\xi_2 + c^2z{p}_2 + \kappa{w}\delta^1\I_{z}[P_1] + w\delta^{-1}\I_{z}[P_1]\big)
\end{split} \label{eqn: resolv comp1} \\
\nonumber \\
\begin{split}
\big(\rhs(z;\kappa,w,c)\Ub\big)_2
&= \frac{w(\kappa{e}^{z}+e^{-z})}{\det\big(\M(z;\kappa,w,c)\big)}\big(c^2\xi_1+c^2z{p}_1 + \delta^1\I_{z}[P_2] + \kappa\delta^{-1}\I_{z}[P_2]\big) \\
&+\frac{c^2z^2+1+\kappa}{\det\big(\M(z;\kappa,w,c)\big)}\big(c^2\xi_2 + c^2z{p}_2 + \kappa{w}\delta^1\I_{z}[P_1] + w\delta^{-1}\I_{z}[P_1]\big)
\end{split} \label{eqn: resolv comp2} \\
\nonumber \\
\big(\rhs(z;\kappa,w,c)\Ub\big)_3
&= z\big(\rhs(z;\kappa,w,c)\Ub\big)_1-p_1 \\
\nonumber \\
\big(\rhs(z;\kappa,w,c)\Ub\big)_4
&= z\big(\rhs(z;\kappa,w,c)\Ub\big)_2-p_2 \\
\nonumber \\
\big(\rhs(z;\kappa,w,c)\Ub\big)_5(v)
&= e^{zv}\big(\rhs(z;\kappa,w,c)\Ub\big)_1+\I_z[P_1], \ -1 \le v \le 1 \\
\nonumber \\
\big(\rhs(z;\kappa,w,c)\Ub\big)_6(v)
&= e^{zv}\big(\rhs(z;\kappa,w,c)\Ub\big)_2+\I_z[P_2], \ -1 \le v \le 1.
\end{align}

For simplicity of notation in this proof, we suppress the dependence of (almost) all operators and functions on $c$, $\kappa$, and $w$; in particular, we just write $\M(z;c)$ for the matrix in (<ref>) and $\L(c)$ instead of $\L(\kappa,w,c)$.
The set equality (<ref>) was established by the remarks preceding the statement of this proposition.

* Let $z \in \sigma(\L)$.
We first claim that if $|c| > 1$ and $z \in \sigma(\L)$, then

\begin{equation}\label{eqn: spectrum claim}
c^2z^2 + 1+ \kappa
\ne 0.
\end{equation}

Otherwise, if $c^2z^2+1+\kappa = 0$, then we can solve for $z$ as

\[
= \pm{i}\frac{\sqrt{1+\kappa}}{|c|}.
\]

Since $z \in \sigma(\L)$, we have

\[
\det\left(\M\left(\frac{\sqrt{1+\kappa}}{|c|}\right);c\right)
= 0.
\]

Some straightforward algebraic rearrangements then reveal that

\begin{equation}\label{eqn: spectrum claim aux}
\cos\left(\frac{2\sqrt{1+\kappa}}{|c|}\right)
= -\left(\frac{\kappa^2+1}{2}\right)
< -1,
\end{equation}

which is impossible.

So, we may put

\begin{equation}\label{eqn: Eb-c}
= E(z;\kappa,c)
:= \frac{e^{z}+\kappa{e}^{-z}}{c^2z^2+1+\kappa}
\quadword{and}
\pb := (E(z;\kappa,c),1).
\end{equation}

Then $\M(z;c)\pb = 0$, and so with

\begin{equation}\label{eqn: Ub-kappa-lambda}
\Eb
= \Eb(z;\kappa,c)
:= (E(z;\kappa,c), 1, zE(z;\kappa,c),z, E(z;\kappa,c)e^{z\cdot}, e^{z\cdot}),
\end{equation}

it is a direct calculation that $\L\Eb=z\Eb$.
That is, $z$ is an eigenvalue of $\L$.

* Suppose that $z \in \C\setminus\sigma(\L) = \rho(\L)$.
Using (<ref>), (<ref>), and (<ref>) we can write the resolvent operator of $\L$ at $z$ in the form

\begin{equation}\label{eqn: resolvent0}
\rhs(z)
= \frac{1}{\det(\M(z;c))}\rhs_1(z) + \rhs_2(z),
\end{equation}

where $\rhs_1$, $\rhs_2 \colon \rho(\L) \to \b(\X)$ are analytic.
From its definition in (<ref>), we see that the mapping $\M(\cdot,z) \colon \C \to \C^{2 \times 2}$ is entire, and so $\det(\M(\cdot;c))$ is also entire.
Lemma <ref> then guarantees that the eigenvalues of $\L$ are isolated and that the algebraic multiplicity of an eigenvalue equals its multiplicity as a root of $\det(\M(\cdot;c))$.

* Fix $z \in \sigma(\L)$ and suppose $\L\Ub = z\Ub$ for some $\Ub = (p_1,p_2,\xi_1,\xi_2,P_1,P_2) \in \X$.
We use (<ref>) with $\grave{\Ub} = 0$ to obtain $p_1 = E(z;\kappa,c)p_2$, with $E$ defined in (<ref>).
Then we use (<ref>) and (<ref>), again with $\grave{\Ub} = 0$, to solve for the other components of $\Ub$ as

\[
\xi_1 = zE(z;\kappa,c)p_2,
\quad
\xi_2 = z{p}_2,
\quad
P_1(v) = zE(z;\kappa,c)p_2e^{z{v}},
\quad\text{and}\quad
P_2(v) = p_2e^{z{v}}.
\]

That is,

\begin{equation}\label{eqn: eigenvector}
\Ub
= p_2\big(E(z;\kappa,c), 1, zE(z;\kappa,c), z, E(z;\kappa,c)e^{z\cdot}, e^{z\cdot}\big)
= p_2\Eb(z;\kappa,c)
\end{equation}

as in (<ref>), and so the eigenspace corresponding to $z$ is 1-dimensional.

* The determinant (<ref>) is even in $z$, so if $z \in \sigma(\L)$, then $-z \in \sigma(\L)$.
The coefficients in (<ref>) are real and $\overline{\cosh(2z)} = \cosh(2\overline{z})$, so if $z \in \sigma(\L)$, then $\overline{z} \in \sigma(\L)$.

* This follows by unraveling the calculations in (<ref>), (<ref>), and (<ref>).

§.§.§ The center spectrum of $\L(\kappa,w,c)$

We study $\sigma(\L(\kappa,w,c)) \cap i\R$.
Given $k \in \R$, we have $ik \in \sigma(\L(\kappa,w,c))$ if and only if $k$ satisfies the “dispersion relation”

\begin{equation}\label{eqn: Lambdac defn}
= \det(\M(ik;\kappa,w,c))
= \bunderbrace{c^4k^4-c^2(1+w)(1+\kappa)k^2+2\kappa{w}(1-\cos(2k))}{\Lambda(k;\kappa,w,c)}.
\end{equation}

Clearly $\Lambda(0;\kappa,w,c) = 0$; we are interested in the multiplicity of the root 0, and in the behavior of other roots of $\Lambda(\cdot;\kappa,w,c)$.

Let $\kappa$, $w > 0$ with $\max\{\kappa,w\} > 1$.

There exists $c_-(\kappa,w) \in (0,1)$ such that if $|c| \ge c_-(\kappa,w)$, then there exists a unique $\omega_c = \omega_c(\kappa,w) > 0$ such that $\Lambda(k;\kappa,w,c) = 0$ if and only if $k =0$ or $k = \pm\omega_c(\kappa,w)$.

The numbers $\pm\omega_c(\kappa,w)$ from part <ref> are both simple roots of $\Lambda(\cdot;\kappa,w,c)$.

\begin{equation}\label{eqn: cs defn}
\cs(\kappa,w)
:= \sqrt{\frac{4\kappa{w}}{(1+\kappa)(1+w)}}.
\end{equation}

Then 0 has multiplicity 4 as a root of $\Lambda(\cdot;\kappa,w,\cs(\kappa,w))$ and multiplicity 2 as a root of $\Lambda(\cdot;\kappa,w,c)$ when $c \ne \cs(\kappa,w)$.