# Breather Solutions for a Quasilinear $(1+1)$-dimensional Wave Equation

Simon Kohler and Wolfgang Reichel  Institute for Analysis, Karlsruhe Institute
of Technology (KIT), D-76128 Karlsruhe, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: September 3, 2024)

###### Abstract.

We consider the $(1+1)$-dimensional quasilinear wave equation
$g(x)w_{tt}-w_{xx}+h(x)(w_{t}^{3})_{t}=0$ on $\mathbb{R}\times\mathbb{R}$
which arises in the study of localized electromagnetic waves modeled by Kerr-
nonlinear Maxwell equations. We are interested in time-periodic, spatially
localized solutions. Here $g\in{L^{\infty}(\mathbb{R})}$ is even with
$g\not\equiv 0$ and $h(x)=\gamma\,\delta_{0}(x)$ with
$\gamma\in\mathbb{R}\backslash\\{0\\}$ and $\delta_{0}$ the delta-distribution
supported in $0$. We assume that $0$ lies in a spectral gap of the operators
$L_{k}=-\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}-k^{2}\omega^{2}g$ on
${L^{2}(\mathbb{R})}$ for all $k\in 2\mathbb{Z}+1$ together with additional
properties of the fundamental set of solutions of $L_{k}$. By expanding $w$
into a Fourier series in time we transfer the problem of finding a suitably
defined weak solution to finding a minimizer of a functional on a sequence
space. The solutions that we have found are exponentially localized in space.
Moreover, we show that they can be well approximated by truncating the Fourier
series in time. The guiding examples, where all assumptions are fulfilled, are
explicitely given step potentials and periodic step potentials $g$. In these
examples we even find infinitely many distinct breathers.

##  1\. Introduction and Main Results

We study the $(1+1)$-dimensional quasilinear wave equation

(1) $\displaystyle g(x)w_{tt}-w_{xx}+h(x)(w_{t}^{3})_{t}=0\quad\text{for
}(x,t)\in\mathbb{R}\times\mathbb{R},$

and we look for real-valued, time-periodic and spatially localized solutions
$w(x,t)$. At the end of this introduction we give a motivation for this
equation arising in the study of localized electromagnetic waves modeled by
Kerr-nonlinear Maxwell equations. We also cite some relevant papers. To the
best of our knowledge for (1) in its general form no rigorous existence
results are available. A first result is given in this paper by taking an
extreme case where $h(x)$ is a spatial delta distribution at $x=0$. Our basic
assumption on the coefficients $g$ and $h$ is the following:

(C0) $\displaystyle g\in{L^{\infty}(\mathbb{R})}\text{ even, }g\not\equiv
0\text{ and }h(x)=\gamma\delta_{0}(x)\mbox{ with }\gamma\neq 0$

where $\delta_{0}$ denotes the delta-distribution supported in $0$. We have
two prototypical examples for the potential $g$: a step potential (Theorem
1.1) and a periodic step potential (Theorem 1.2). The general version is given
in Theorem 1.5 below.

######  Theorem 1.1.

For $a,b,c>0$ let

$g(x)\coloneqq\left\\{\begin{array}[]{ll}-a,&\mbox{ if
}\,\absolutevalue{x}>c,\\\ b,&\mbox{ if
}\,\absolutevalue{x}<c.\end{array}\right.$

For every frequency $\omega$ such that $\sqrt{b}\omega
c\frac{2}{\pi}\in\frac{2\mathbb{N}+1}{2\mathbb{N}+1}$ and $\gamma<0$ there
exist infinitely many nontrivial, real-valued, spatially localized and time-
periodic weak solutions of (1) with period $T=\frac{2\pi}{\omega}$. For each
solution $w$ there are constants $C,\rho>0$ such that
$\absolutevalue{w(x,t)}\leq C\mathrm{e}^{-\rho\absolutevalue{x}}$.

######  Theorem 1.2.

For $a,b>0$, $a\not=b$ and $\Theta\in(0,1)$ let

$g(x)\coloneqq\left\\{\begin{array}[]{ll}a,&\mbox{ if
}\,\absolutevalue{x}<\pi\Theta,\\\ b,&\mbox{ if
}\,\pi\Theta<\absolutevalue{x}<\pi\end{array}\right.$

and extend $g$ as a $2\pi$-periodic function to $\mathbb{R}$. Assume in
addition

$\displaystyle\sqrt{\frac{b}{a}}\,\frac{1-\Theta}{\Theta}\in\frac{2\mathbb{N}+1}{2\mathbb{N}+1}.$

For every frequency $\omega$ such that
$4\sqrt{a}\theta\omega\in\frac{2\mathbb{N}+1}{2\mathbb{N}+1}$ there exist
infinitely many nontrivial, real-valued, spatially localized and time-periodic
weak solutions of (1) with period $T=\frac{2\pi}{\omega}$. For each solution
$w$ there are constants $C,\rho>0$ such that $\absolutevalue{w(x,t)}\leq
C\mathrm{e}^{-\rho\absolutevalue{x}}$.

Weak solutions of (1) are understood in the following sense. We write
$D\coloneqq{\mathbb{R}\times\mathbb{T}_{T}}$ and denote by $\mathbb{T}_{T}$
the one-dimensional torus with period $T$.

######  Definition 1.3.

Under the assumption (C0) a function
$w\in{H^{1}({\mathbb{R}\times\mathbb{T}_{T}})}$ with
$\partial_{t}w(0,\cdot)\in{L^{3}(\mathbb{T}_{T})}$ is called a weak solution
of (1) if for every
$\psi\in{C^{\infty}_{c}({\mathbb{R}\times\mathbb{T}_{T}})}$

(2)
$\displaystyle\int_{D}-g(x)\partial_{t}w\,\partial_{t}\psi+\partial_{x}w\,\partial_{x}\psi\differential{(x,t)}-\gamma\int_{0}^{T}(\partial_{t}w(0,t))^{3}\partial_{t}\psi(0,t)\differential{t}=0.$

Theorem 1.1 and Theorem 1.2 are special cases of Theorem 1.5, which applies to
much more general potentials $g$. In Section 6.1 and Section 6.2 of the
Appendix we will show that the special potentials $g$ from these two theorems
satisfy the conditions (C1) and (C2) of Theorem 1.5. The basic preparations
and assumptions for Theorem 1.5 will be given next.

Since we are looking for time-periodic solutions, it is appropriate to make
the Fourier ansatz
$w(x,t)=\sum_{k\in{\mathbb{Z}_{odd}}}w_{k}(x)\mathrm{e}^{\mathrm{i}k\omega t}$
with ${\mathbb{Z}_{odd}}\coloneqq 2\mathbb{Z}+1$. The reason for dropping even
Fourier modes is that the $0$-mode belongs to the kernel of the wave operator
$L=g(x)\partial_{t}^{2}-\partial_{x}^{2}$. The restriction to odd Fourier
modes generates $T/2=\pi/\omega$-antiperiodic functions $w$, is therefore
compatible with the structure of (1) and in particular the cubic nonlinearity.
Notice the decomposition
$(Lw)(x,t)=\sum_{k\in{\mathbb{Z}_{odd}}}L_{k}w_{k}(x)\mathrm{e}^{\mathrm{i}k\omega
t}$ with self-adjoint operators

$\displaystyle
L_{k}=-\frac{d^{2}}{dx^{2}}-k^{2}\omega^{2}g(x):H^{2}(\mathbb{R})\subset
L^{2}(\mathbb{R})\rightarrow L^{2}(\mathbb{R}).$

Clearly $L_{k}=L_{-k}$ so that is suffices to study $L_{k}$ for
$k\in{\mathbb{N}_{odd}}$. Our first assumption is concerned with the spectrum
$\sigma(L_{k})$:

(C1)
$\displaystyle\forall\,k\in{\mathbb{N}_{odd}},0\not\in\sigma_{\mathrm{ess}}(L_{k})\cup\sigma_{\mathrm{D}}(L_{k}),$

where by $\sigma_{\mathrm{D}}(L_{k})$ we denote the spectrum of $L_{k}$ with
an extra Dirichlet condition at $0$, i.e., the spectrum of $L_{k}$ restricted
to $\\{\varphi\in{H^{2}(\mathbb{R})}\leavevmode\nobreak\ |\leavevmode\nobreak\
\varphi(0)=0\\}$. This is the same as the spectrum of $L_{k}$ restricted to
functions which are odd around $x=0$.

######  Lemma 1.4.

Under the assumption (C0) and (C1) there exists for every
$k\in{\mathbb{N}_{odd}}$ a function $\Phi_{k}\in H^{2}(0,\infty)$ with
$L_{k}\Phi_{k}=0$ on $(0,\infty)$ and $\Phi_{k}(0)=1$.

###### Proof.

We have either that $0$ is in the point spectrum (but not the Dirichlet
spectrum) or that $0$ is in the resolvent set of $L_{k}$. In the first case
there is an eigenfunction $\Phi_{k}\in H^{2}(\mathbb{R})$ with
$L_{k}\Phi_{k}=0$ and w.l.o.g. $\Phi_{k}(0)=1$. In the second case
$0\in\rho(L_{k})$ so that there exists a unique solution $\tilde{\Phi}_{k}$ of
$L_{k}\tilde{\Phi}_{k}=1_{[-2,-1]}$ on $\mathbb{R}$. Clearly, if restricted to
$(0,\infty)$, the function $\tilde{\Phi}_{k}$ solves $L_{k}\tilde{\Phi}_{k}=$
on $(0,\infty)$. Moreover, $\tilde{\Phi}_{k}(0)\not=0$ since otherwise
$\tilde{\Phi}_{k}$ would be an odd eigenfunction of $L_{k}$ which is excluded
due to $0\in\rho(L_{k})$. Thus a suitably rescaled version of
$\tilde{\Phi}_{k}$ satisfies the claim of the lemma. ∎

Our second set of assumptions concerns the structure of the decaying
fundamental solution according to Lemma 1.4.

(C2) $\displaystyle\mbox{There exist $\rho,M>0$ such that for all
}k\in{\mathbb{N}_{odd}}\colon|\Phi_{k}(x)|\leq Me^{-\rho x}\mbox{ on
}[0,\infty).$

Now we can formulate our third main theorem as a generalization of Theorem 1.1
and Theorem 1.2. The fact that the solutions which we find, can be well
approximated by truncation of the Fourier series in time, is explained in
Lemma 5.2 below. Moreover, a further extension yielding infinitely many
different solutions is given in Theorem 4.1 in Section 4.

######  Theorem 1.5.

Assume (C0), (C1) and (C2) for a potential $g$ and a frequency $\omega>0$.
Then (1) has a nontrivial, $T$-periodic weak solution $w$ in the sense of
Definition 1.3 with $T=\frac{2\pi}{\omega}$ provided

  * (i)

$\gamma<0$ and the sequence
$\left(\Phi^{\prime}_{k}(0)\right)_{k\in{\mathbb{N}_{odd}}}$ has at least one
positive element,

  * (ii)

$\gamma>0$ and the sequence
$\left(\Phi^{\prime}_{k}(0)\right)_{k\in{\mathbb{N}_{odd}}}$ has at least one
negative element.

Moreover, there is a constant $C>0$ such that $\absolutevalue{w(x,t)}\leq
C\mathrm{e}^{-\rho\absolutevalue{x}}$ for all $(x,t)\in\mathbb{R}^{2}$ with
$\rho$ as in (C2).

######  Remark 1.6.

(a) It turns out that the above assumptions can be weakened as follows: it
suffices to verify (C1) and (C2) and (i), (ii) for all integers $k\in
r\cdot{\mathbb{Z}_{odd}}$ for some $r\in{\mathbb{N}_{odd}}$. We will prove
this observation in Section 4.

(b) Our variational approach also works if we consider (1) with Dirichlet
boundary conditions on a bounded interval $(-l,l)$ instead of the real line.
There are many possible results. For illustration purposes we just formulate
the simplest one. E.g., if we assume that $\frac{\omega
l}{\pi}\in\frac{{\mathbb{Z}_{odd}}}{4\mathbb{Z}}$ then

$\displaystyle w_{tt}-w_{xx}+\gamma\delta_{0}(x)(w_{t}^{3})_{t}=0\mbox{ on
}(-l,l)\times\mathbb{R}\mbox{ with }w(\pm l,t)=0\mbox{ for all }t$

has a nontrivial, real-valued time-periodic weak solution with period
$T=\frac{2\pi}{\omega}$ both for $\gamma>0$ and $\gamma<0$. The operator
$L_{k}=-\frac{d^{2}}{dx^{2}}-\omega^{2}k^{2}$ is now a self-adjoint operator
on $H^{2}(-l,l)\cap H_{0}^{1}(-l,l)$. The assumption $\frac{\omega
l}{\pi}\in\frac{{\mathbb{Z}_{odd}}}{4\mathbb{Z}}$ guarantees (C1) for all
$k\in{\mathbb{Z}_{odd}}$. The functions $\Phi_{k}$ are given by
$\Phi_{k}(x)=\frac{\sin(\omega k(l-x))}{\sin(\omega kk)}$ so that
$\Phi_{k}^{\prime}(0)=-\omega k\cot(\omega kl)$. The assumption $\frac{\omega
l}{\pi}\in\frac{{\mathbb{Z}_{odd}}}{4\mathbb{Z}}$ now guarantees that the
sequence $\\{\cot(\omega kl)\leavevmode\nobreak\ |\leavevmode\nobreak\
k\in{\mathbb{Z}_{odd}}\\}$ is finite and does not contain $0$ or $\pm\infty$.
Moreover $\frac{\omega l}{\pi}=\frac{2p+1}{4q}$ yields
$\Phi_{k}^{\prime}(0)\Phi_{k+2q}^{\prime}(0)<0$, i.e., we also have the
required sign-change which allows for both signs of $\gamma$.

We observe that the growth of
$\left(\Phi^{\prime}_{k}(0)\right)_{k\in{\mathbb{Z}_{odd}}}$ is connected to
regularity properties of our solutions.

######  Theorem 1.7.

Assume (C0), (C1) and (C2) and additionally $\Phi^{\prime}_{k}(0)=O(k)$. Then
the weak solution $w$ from Theorem 1.5 belongs to
${H^{1+\nu}_{per}(\mathbb{T}_{T},{L^{2}(\mathbb{R})})}\cap{H^{\nu}_{per}(\mathbb{T}_{T},{H^{1}(\mathbb{R})})}$
for any $\nu\in(0,\frac{1}{4})$.

Here, for $\nu\in\mathbb{R}$ the fractional Sobolev spaces of time-periodic
functions are defined by

$\displaystyle{H^{\nu}_{per}(\mathbb{T}_{T},{L^{2}(\mathbb{R})})}$
$\displaystyle\coloneqq\left\\{u(x,t)=\sum_{k\in\mathbb{Z}}\hat{u}_{k}(x)\mathrm{e}^{\mathrm{i}\omega
kt}\leavevmode\nobreak\ \bigg{|}\leavevmode\nobreak\
\sum_{k\in\mathbb{Z}}\left(1+\absolutevalue{k}^{2}\right)^{\nu}\norm{\hat{u}_{k}}^{2}_{L^{2}(\mathbb{R})}<\infty\right\\},$
$\displaystyle{H^{\nu}_{per}(\mathbb{T}_{T},{H^{1}(\mathbb{R})})}$
$\displaystyle\coloneqq\left\\{u(x,t)=\sum_{k\in\mathbb{Z}}\hat{u}_{k}(x)\mathrm{e}^{\mathrm{i}\omega
kt}\leavevmode\nobreak\ \bigg{|}\leavevmode\nobreak\
\sum_{k\in\mathbb{Z}}\left(1+\absolutevalue{k}^{2}\right)^{\nu}\norm{\hat{u}_{k}}^{2}_{H^{1}(\mathbb{R})}<\infty\right\\}.$

We shortly motivate (1) and give some references to the literature. Consider
Maxwell’s equations in the absence of charges and currents

$\displaystyle\nabla\cdot\mathbf{D}$ $\displaystyle=0,$
$\displaystyle\nabla\times\mathbf{E}\,=$
$\displaystyle-\partial_{t}\mathbf{B},$ $\displaystyle\mathbf{D}=$
$\displaystyle\varepsilon_{0}\mathbf{E}+\mathbf{P}(\mathbf{E}),$
$\displaystyle\nabla\cdot\mathbf{B}$ $\displaystyle=0,$
$\displaystyle\nabla\times\mathbf{H}=$
$\displaystyle\,\partial_{t}\mathbf{D},$ $\displaystyle\mathbf{B}=$
$\displaystyle\mu_{0}\mathbf{H}.$

We assume that the dependence of the polarization $\mathbf{P}$ on the electric
field $\mathbf{E}$ is instantaneous and it is the sum of a linear and a cubic
term given by
$\mathbf{P}(\mathbf{E})=\varepsilon_{0}\chi_{1}(\mathbf{x})\mathbf{E}+\varepsilon_{0}\chi_{3}(\mathbf{x})\absolutevalue{\mathbf{E}}^{2}\mathbf{E}$,
cf. [1], Section 2.3 (for simplicity, more general cases where instead of a
factor multiplying $\absolutevalue{\mathbf{E}}^{2}\mathbf{E}$ one can take
$\chi_{3}$ as an $\mathbf{x}$-dependent tensor of type $(1,3)$ are not
considered here). Here $\varepsilon_{0},\mu_{0}$ are constants such that
$c^{2}=(\varepsilon_{0}\mu_{0})^{-1}$ with $c$ being the speed of light in
vacuum and $\chi_{1},\chi_{3}$ are given material functions. By direct
calculations one obtains the quasilinear curl-curl-equation

(3) $\displaystyle
0=\nabla\times\nabla\times\mathbf{E}+\partial_{t}^{2}\left(V(\mathbf{x})\mathbf{E}+\Gamma(\mathbf{x})\absolutevalue{\mathbf{E}}^{2}\mathbf{E}\right),$

where
$V(\mathbf{x})=\mu_{0}\varepsilon_{0}\left(1+\chi_{1}(\mathbf{x})\right)$ and
$\Gamma(\mathbf{x})=\mu_{0}\varepsilon_{0}\chi_{3}(\mathbf{x})$. Once (3) is
solved for the electric field $\mathbf{E}$, the magnetic induction
$\mathbf{B}$ is obtained by time-integration from
$\nabla\times\mathbf{E}=-\partial_{t}\mathbf{B}$ and it will satisfy
$\nabla\cdot\mathbf{B}=0$ provided it does so at time $t=0$. By construction,
the magnetic field $\mathbf{H}=\frac{1}{\mu_{0}}\mathbf{B}$ satisfies
$\nabla\times\mathbf{H}=\partial_{t}\mathbf{D}$. In order to complete the full
set of nonlinear Maxwell’s equations one only needs to check Gauss’s law
$\nabla\cdot\mathbf{D}=0$ in the absence of external charges. This will follow
directly from the constitutive equation
$\mathbf{D}=\varepsilon_{0}(1+\chi_{1}(\mathbf{x}))\mathbf{E}+\varepsilon_{0}\chi_{3}(\mathbf{x})\absolutevalue{\mathbf{E}}^{2}\mathbf{E}$
and the two different specific forms of $\mathbf{E}$ given next:

$\displaystyle\mathbf{E}(\mathbf{x},t)$ $\displaystyle=(0,u(x_{1}-\kappa
t,x_{3}),0)^{T}$ polarized wave traveling in $x_{1}$-direction
$\displaystyle\mathbf{E}(\mathbf{x},t)$ $\displaystyle=(0,u(x_{1},t),0)^{T}$
polarized standing wave

In the first case $\mathbf{E}$ is a polarized wave independent of $x_{2}$
traveling with speed $\kappa$ in the $x_{1}$ direction and with profile $u$.
If additionally $V(\mathbf{x})=V(x_{3})$ and
$\Gamma(\mathbf{x})=\Gamma(x_{3})$ then the quasilinear curl-curl-equation (3)
turns into the following equation for $u=u(\tau,x_{3})$ with the moving
coordinate $\tau=x_{1}-\kappa t$:

$\displaystyle-
u_{x_{3}x_{3}}+(\kappa^{2}V(x_{3})-1)u_{\tau\tau}+\kappa^{2}\Gamma(x_{3})(u^{3})_{\tau\tau}=0.$

Setting $u=w_{\tau}$ and integrating once w.r.t. $\tau$ we obtain (1).

In the second case $\mathbf{E}$ is a polarized standing wave which is
independent of $x_{2},x_{3}$. If we assume furthermore that
$V(\mathbf{x})=V(x_{1})$ and $\Gamma(\mathbf{x})=\Gamma(x_{1})$ then this time
the quasilinear curl-curl-equation (3) for $u=w_{t}$ turns (after one time-
integration) directly into (1).

In the literature, (3) has mostly been studied by considering time-harmonic
waves $\mathbf{E}(\mathbf{x},t)=\mathbf{U}(\mathbf{x})e^{\mathrm{i}\kappa t}$.
This reduces the problem to the stationary elliptic equation

(4)
$0=\nabla\times\nabla\times\mathbf{U}-\kappa^{2}\left(V(\mathbf{x})\mathbf{U}+\Gamma(\mathbf{x})\absolutevalue{\mathbf{U}}^{2}\mathbf{U}\right)\mbox{
in }\mathbb{R}^{3}.$

Here case $\mathbf{E}$ is no longer real-valued. This may be justified by
extending the ansatz to
$\mathbf{E}(\mathbf{x},t)=\mathbf{U}(\mathbf{x})e^{\mathrm{i}\kappa t}+c.c.$
and by either neglecting higher harmonics generated from the cubic
nonlinearity or by assuming the time-averaged constitutive relation
$\mathbf{P}(\mathbf{E})=\varepsilon_{0}\chi_{1}(\mathbf{x})\mathbf{E}+\varepsilon_{0}\chi_{3}(\mathbf{x})\frac{1}{T}\int_{0}^{T}\absolutevalue{\mathbf{E}}^{2}\,dt\mathbf{E}$
with $T=2\pi/\kappa$, cf. [11], [13]. For results on (4) we refer to [2], [7]
and in particular to the survey [3]. Time-harmonic traveling waves have been
found in a series of papers [10, 11, 12]. The number of results for
monochromatic standing polarized wave profiles $U(\mathbf{x})=(0,u(x_{1}),0)$
with $u$ satisfying
$0=-u^{\prime\prime}-\kappa^{2}\left(V(x_{1})u+\Gamma(x_{1})|u|^{2}u\right)$
on $\mathbb{R}$ is too large to cite so we restrict ourselves to Cazenave’s
book [4].

Our approach differs substantially from the approaches by monochromatic waves
described above. Our ansatz
$w(x,t)=\sum_{k\in{\mathbb{Z}_{odd}}}w_{k}(x)\mathrm{e}^{\mathrm{i}k\omega t}$
with ${\mathbb{Z}_{odd}}\coloneqq 2\mathbb{Z}+1$ is automatically
polychromatic since it couples all integer multiples of the frequency
$\omega$. A similar polychromatic approach is considered in [8]. The authors
seek spatially localized traveling wave solutions of the 1+1-dimensional
quasilinear Maxwell model, where in the direction of propagation $\chi_{1}$ is
a periodic arrangement of delta functions. Based on a multiple scale
approximation ansatz, the field profile is expanded into infinitely many modes
which are time-periodic in both the fast and slow time variables. Since the
periodicities in the fast and slow time variables differ, the field becomes
quasiperiodic in time. To a certain extent the authors of [8] analytically
deal with the resulting system for these infinitely many coupled modes through
bifurcation methods, with a rigorous existence proof still missing. However,
numerical results from [8] indicate that spatially localized traveling waves
could exist.

With our case of allowing $\chi_{1}$ to be a bounded function but taking
$\chi_{3}$ to be a delta function at $x=0$ we consider an extreme case. On the
other hand our existence results (possibly for the first time) rigorously
establish localized solutions of the full nonlinear Maxwell problem (3)
without making the assumption of either neglecting higher harmonics or of
assuming a time-averaged nonlinear constitutive law.

The existence of localized breathers of the quasilinear problem (1) with
bounded coefficients $g,h$ remains generally open. We can, however, provide
specific functions $g$, $h$ for which (1) has a breather-type solution that
decays to $0$ as $|x|\to\infty$. Let

$\displaystyle b(x)\coloneqq(1+x^{2})^{-1/2},\quad
h(x)\coloneqq\frac{1-2x^{2}}{1+x^{2}},\quad
g(x)\coloneqq\frac{2+x^{4}}{(1+x^{2})^{2}}$

and consider a time-periodic solution $a$ of the ODE

$\displaystyle-a^{\prime\prime}-(a^{\prime 3})^{\prime}=a$

with minimal prescribed period $T\in(0,2\pi)$. Then $w(x,t)\coloneqq a(t)b(x)$
satisfies (1). Note that $h$ is sign-changing and $w$ is not exponentially
localized. We found this solution by inserting the ansatz for $w$ with
separated variables into (1). We then defined $b(x)\coloneqq(1+x^{2})^{-1/2}$
and set $g(x)\coloneqq-b^{\prime\prime}(x)/b(x)$ and
$h(x)\coloneqq-b^{\prime\prime}(x)/b(x)^{3}$. The remaining equation for $a$
then turned out to be the above one.

The paper is structured as follows: In Section 2 we develop the variational
setting and give the proof of Theorem 1.5. The proof of the additional
regularity results of Theorem 1.7 is given in Section 3. In Section 4 we give
the proof of Theorem 4.1 on the existence of infinitely many different
breathers. In Section 5 we show that our breathers can be well approximated by
truncation of the Fourier series in time. Finally, in the Appendix we give
details on the background and proof of Theorem 1.1 (Section 6.1) and Theorem
1.2 (Section 6.2) as well as a technical detail on a particular embedding of
Hölder spaces into Sobolev spaces (Section 6.3).

##  2\. Variational Approach and Proof of Theorem 1.5

The main result of our paper is Theorem 1.5 which will be proved in this
section. It is a consequence of Lemma 2.4 and Theorem 2.6 below.

Formally (1) is the Euler-Lagrange-equation of the functional

(5)
$I(w)\coloneqq\int_{D}-\frac{1}{2}g(x)\absolutevalue{\partial_{t}w}^{2}+\frac{1}{2}\absolutevalue{\partial_{x}w}^{2}\differential{(x,t)}-\frac{1}{4}\gamma\int_{0}^{T}\absolutevalue{\partial_{t}w(0,t)}^{4}\differential{t}$

defined on a suitable space of $T$-periodic functions. Instead of directly
searching for a critical point of this functional we first rewrite the problem
into a nonlinear Neumann boundary value problem under the assumption that $w$
is even in $x$. In this case (1) amounts to the following linear wave equation
on the half-axis with nonlinear Neumann boundary conditions:

(6) $\displaystyle\begin{cases}g(x)w_{tt}-w_{xx}=0&\text{for
}(x,t)\in(0,\infty)\times\mathbb{R},\\\
2w_{x}(0_{+},t)=\gamma\left(w_{t}(0,t)^{3}\right)_{t}&\text{for
}t\in\mathbb{R}\end{cases}$

where solutions $w\in{H^{1}([0,\infty)\times\mathbb{T}_{T})}$ with
$\partial_{t}w(0,\cdot)\in{L^{3}(\mathbb{T}_{T})}$ of (6) are understood in
the sense that

(7) $\displaystyle
2\int_{D_{+}}-g(x)\partial_{t}w\,\partial_{t}\psi+\partial_{x}w\,\partial_{x}\psi\differential{(x,t)}-\gamma\int_{0}^{T}(\partial_{t}w(0,t))^{3}\partial_{t}\psi(0,t)\differential{t}=0$

for all $\psi\in{C^{\infty}_{c}([0,\infty)\times\mathbb{T}_{T})}$ with
$D_{+}=(0,\infty)\times\mathbb{T}_{T}$. It is clear that evenly extended
solutions $w$ of (7) also satisfy (2). To see this note that every
$\psi\in{C^{\infty}_{c}(\mathbb{R}\times\mathbb{T}_{T})}$ can be split into an
even and an odd part $\psi=\psi_{e}+\psi_{o}$ both belonging to
${C^{\infty}_{c}(\mathbb{R}\times\mathbb{T}_{T})}$. Testing with $\psi_{o}$ in
(2) produces zeroes in all spatial integrals due to the evenness of $w$ and
also in the temporal integral since $\psi_{o}(0,\cdot)\equiv 0$ due to
oddness. Testing with $\psi_{e}$ in (2) produces twice the spatial integrals
appearing in (7). In the following we concentrate on finding solutions of (6)
for the linear wave equation with nonlinear Neumann boundary conditions.

Motivated by the linear wave equation in (6) we make the ansatz that

(8)
$w(x,t)=\sum_{k\in{\mathbb{Z}_{odd}}}\frac{\hat{\alpha}_{k}}{k}\Phi_{k}(\absolutevalue{x})e_{k}(t),$

where $e_{k}(t)\coloneqq\frac{1}{\sqrt{T}}\mathrm{e}^{\mathrm{i}\omega kt}$
denotes the ${L^{2}(\mathbb{T}_{T})}$-orthonormal Fourier base of
$\mathbb{T}_{T}$, and where $\Phi_{k}$ are the decaying fundamental solutions
$\Phi_{k}$ of $L_{k}$, cf. Lemma 1.4. Such a function $w$ will always solve
the linear wave equation in (6) and we will determine real sequences
$\hat{\alpha}=(\hat{\alpha}_{k})_{k\in{\mathbb{Z}_{odd}}}$ such that the
nonlinear Neumann condition is satisfied as well. The additional factor
$\frac{1}{k}$ is only for convenience, since $\partial_{t}$ generates a
multiplicative factor $\mathrm{i}\omega k$.

The convolution between two sequences
$\hat{z},\hat{y}\in\mathbb{R}^{\mathbb{Z}}$ is defined pointwise (whenever it
converges) by
$(\hat{z}*\hat{y})_{k}\coloneqq\sum_{l\in\mathbb{Z}}\hat{z}_{l}\hat{y}_{k-l}$.

In order to obtain real-valued functions $w$ by the ansatz (8) we require the
sequence $\hat{\alpha}$ to be real and odd in $k$, i.e.,
$\hat{\alpha}_{k}\in\mathbb{R}$ and $\hat{\alpha}_{k}=-\hat{\alpha}_{-k}$.
Since (8) already solves the wave equation in (6), it remains to find
$\hat{\alpha}$ such that

$\displaystyle
2w_{x}(0_{+},t)=2\sum_{k\in{\mathbb{Z}_{odd}}}\frac{\hat{\alpha}_{k}}{k}\Phi_{k}^{\prime}(0)e_{k}(t)\stackrel{{\scriptstyle!}}{{\leavevmode\nobreak\
=\leavevmode\nobreak\
}}\frac{1}{T}\sum_{k\in{\mathbb{Z}_{odd}}}\gamma\omega^{4}k(\hat{\alpha}*\hat{\alpha}*\hat{\alpha})_{k}e_{k}(t)=\gamma(w_{t}(0,t)^{3})_{t},$

where we have used $\Phi_{k}(0)=1$. As the above identity needs to hold for
all $t\in\mathbb{R}$ we find

(9)
$(\hat{\alpha}*\hat{\alpha}*\hat{\alpha})_{k}=\frac{2T\Phi_{k}^{\prime}(0)}{\gamma\omega^{4}k^{2}}\hat{\alpha}_{k}\quad\mbox{
for all }k\in{\mathbb{Z}_{odd}}.$

This will be accomplished by searching for critical points $\hat{\alpha}$ of
the functional

$\displaystyle
J(\hat{z})\coloneqq\frac{1}{4}(\hat{z}*\hat{z}*\hat{z}*\hat{z})_{0}+\frac{T}{\gamma\omega^{4}}\sum_{k}\frac{\Phi^{\prime}_{k}(0)}{k^{2}}\hat{z}_{k}^{2}.$

defined on a suitable Banach space of real sequences $\hat{z}$ with
$\hat{z}_{k}=-\hat{z}_{-k}$. Indeed, computing (formally) the Fréchet
derivative of $J$ at $\hat{\alpha}$ we find

(10)
$J^{\prime}(\hat{\alpha})[\hat{y}]=\left(\hat{\alpha}*\hat{\alpha}*\hat{\alpha}*\hat{y}\right)_{0}+\frac{2T}{\gamma\omega^{4}}\sum_{k}\frac{\Phi^{\prime}_{k}(0)}{k^{2}}\hat{\alpha}_{k}\hat{y}_{k}.$

Let us indicate how (10) amounts to (9). For fixed
$k_{0}\in{\mathbb{Z}_{odd}}$ we define the test sequence
$\hat{y}\coloneqq(\delta_{k,k_{0}}-\delta_{k,-k_{0}})_{k\in{\mathbb{Z}_{odd}}}$
which has exactly two non-vanishing entries at $k_{0}$ and at $-k_{0}$. Thus,
$\hat{y}$ belongs to the same space of odd, real sequences as $\hat{\alpha}$
and can therefore be used as a test sequence in
$J^{\prime}(\hat{\alpha})[\hat{y}]=0$. After a short calculation using
$\hat{\alpha}_{k}=-\hat{\alpha}_{-k}$, $\Phi_{k}^{\prime}=\Phi_{-k}^{\prime}$
we obtain (9) for $k_{0}$.

It turns out that a real Banach space of real-valued sequences which is
suitable for $J$ can be given by

$\displaystyle{\mathcal{D}(J)}\coloneqq\left\\{\hat{z}\in\mathbb{R}^{\mathbb{Z}_{odd}}\leavevmode\nobreak\
\big{|}\leavevmode\nobreak\
{\interleave\hat{z}\interleave}<\infty,\leavevmode\nobreak\
\hat{z}_{k}=-\hat{z}_{-k}\right\\}\mbox{ where
}{\interleave\hat{z}\interleave}\coloneqq\norm{\hat{z}*\hat{z}}_{l^{2}}^{\frac{1}{2}}.$

The relation between the function $I$ defined in (5) and the new functional
$J$ is formally given by

$\displaystyle
I\left(\sum_{k\in{\mathbb{Z}_{odd}}}\frac{\hat{z}_{k}}{k}\Phi_{k}(\absolutevalue{x})e_{k}(t)\right)=-\frac{\gamma\omega^{4}}{T}J\left(\hat{z}\right).$

######  Lemma 2.1.

The space $({\mathcal{D}(J)},{\interleave\cdot\interleave})$ is a separable,
reflexive, real Banach space and isometrically embedded into the real Banach
space ${L^{4}(\mathbb{T}_{T},\mathrm{i}\mathbb{R})}$ of purely imaginary-
valued measurable functions. Moreover for
$\hat{u},\hat{v},\hat{w},\hat{z}\in{\mathcal{D}(J)}$ we have

(11) $\displaystyle(\hat{u}*\hat{u}*\hat{u}*\hat{u})_{0}$
$\displaystyle={\interleave\hat{u}\interleave}^{4},$ (12)
$\displaystyle\absolutevalue{(\hat{u}*\hat{v}*\hat{w}*\hat{z})_{0}}$
$\displaystyle\leq{\interleave\hat{u}\interleave}\,{\interleave\hat{v}\interleave}\,{\interleave\hat{w}\interleave}\,{\interleave\hat{z}\interleave},$
(13) $\displaystyle\norm{\hat{z}}_{{l^{2}}}$
$\displaystyle\leq{\interleave\hat{z}\interleave}.$

###### Proof.

We first recall the correspondence between real-valued sequences $\hat{z}\in
l_{2}$ with $\hat{z}_{k}=-\hat{z}_{-k}$ and purely imaginary-valued functions
$z\in{L^{2}(\mathbb{T}_{T},\mathrm{i}\mathbb{R})}$ by setting

$\displaystyle\hat{z}_{k}\coloneqq\left<z,e_{k}\right>_{L^{2}(\mathbb{T}_{T})}\mbox{
and }z(t)\coloneqq\sum_{k\in\mathbb{Z}}\hat{z}_{k}e_{k}(t)$

Parseval’s identity provides the isomorphism
$\norm{z}_{{L^{2}(\mathbb{T}_{T})}}=\|\hat{z}\|_{l^{2}}$. The following
identity

$\displaystyle
T\norm{z}_{{L^{4}(\mathbb{T}_{T})}}^{4}=T\int_{0}^{T}z(t)^{4}\,dt=(\hat{z}*\hat{z}*\hat{z}*\hat{z})_{0}=\|\hat{z}*\hat{z}\|_{l^{2}}^{2}={\interleave\hat{z}\interleave}^{4}$

shows that ${\interleave\cdot\interleave}$ is indeed a norm on
${\mathcal{D}(J)}$ and its provides the isometric embedding of
${\mathcal{D}(J)}$ into a subspace of
$L^{4}(\mathbb{T}_{T},\mathrm{i}\mathbb{R})$. By Parseval’s equality and
Hölder’s inequality we see that

$\displaystyle\norm{\hat{z}}_{{l^{2}}}=\norm{z}_{{L^{2}(\mathbb{T}_{T})}}\leq
T^{\frac{1}{4}}\norm{z}_{{L^{4}(\mathbb{T}_{T})}}={\interleave\hat{z}\interleave}$

so that ${\mathcal{D}(J)}$ is indeed a subspace of $l^{2}$. Finally, for any
$\hat{u},\hat{v},\hat{w},\hat{z}\in{\mathcal{D}(J)}$ we see that

$\displaystyle\absolutevalue{(\hat{u}*\hat{v}*\hat{w}*\hat{z})_{0}}=T\absolutevalue{\int_{0}^{T}u(t)v(t)w(t)z(t)\,dt}\leq
T\norm{u}_{L^{4}}\norm{v}_{L^{4}}\norm{w}_{L^{4}}\norm{z}_{L^{4}}={\interleave\hat{u}\interleave}\,{\interleave\hat{v}\interleave}\,{\interleave\hat{w}\interleave}\,{\interleave\hat{z}\interleave}.$

This finishes the proof of the lemma. ∎

For $\frac{T}{2}$-anti-periodic functions $\psi\colon D\to\mathbb{R}$ of the
space-time variable $(x,t)\in D$ we use the notation

(14)
$\psi(x,t)=\sum_{k\in{\mathbb{Z}_{odd}}}\hat{\psi}_{k}(x)e_{k}(t)=\sum_{k\in{\mathbb{Z}_{odd}}}\frac{1}{k}\Psi_{k}(x)e_{k}(t)$

with
$\frac{1}{k}\Psi_{k}(x)=\hat{\psi}_{k}(x)\coloneqq\left<\psi(x,\cdot),e_{k}\right>_{L^{2}(\mathbb{T}_{T})}$.
The Parseval identity and the definition of ${\interleave\cdot\interleave}$
immediately lead to the following lemma.

######  Lemma 2.2.

For $\psi:D\to\mathbb{R}$ as in (14) the following holds:

  * (i)

$\|\psi_{x}\|_{L^{2}(D)}^{2}=\sum_{k}\frac{1}{k^{2}}\|\Psi_{k}^{\prime}\|_{{L^{2}(\mathbb{R})}}^{2}$,

  * (ii)

$\|\psi_{t}\|_{L^{2}(D)}^{2}=\omega^{2}\sum_{k}\|\Psi_{k}\|_{{L^{2}(\mathbb{R})}}^{2}$,

  * (iii)

$T\|\psi_{t}(0,\cdot)\|_{L^{4}(\mathbb{T}_{T})}^{4}=\omega^{4}{\interleave\hat{y}\interleave}^{4}$
where $\hat{y}_{k}=\Psi_{k}(0)$ for $k\in{\mathbb{Z}_{odd}}$.

The next result give some estimates on the growth of norms of $\Phi_{k}$. It
serves as a preparation for the proof of regularity properties for functions
$w$ as in (8) stated in Lemma 2.4.

######  Lemma 2.3.

Assume (C0), (C1) and (C2). Then

(15)
$\|\Phi_{k}\|_{L^{2}(0,\infty)}=O(1),\quad\|\Phi_{k}^{\prime}\|_{L^{2}(0,\infty)}=O(k),\quad\|\Phi_{k}^{\prime}\|_{L^{\infty}(0,\infty)}=O(k^{\frac{3}{2}}).$

In particular $|\Phi_{k}^{\prime}(0)|=O(k^{\frac{3}{2}})$.

###### Proof.

The first part of (15) is a direct consequence of (C2).

Multiplying $L_{k}\Phi_{k}=0$ with $\Phi_{k}$, $\Phi_{k}^{\prime}$ and
integrating from $a\geq 0$ to $\infty$ we get

(16)
$\displaystyle\int_{a}^{\infty}-\omega^{2}k^{2}g(x)\Phi_{k}(x)^{2}+\Phi_{k}^{\prime}(x)^{2}\,dx$
$\displaystyle=-\Phi_{k}(a)\Phi_{k}^{\prime}(a),$ (17)
$\displaystyle\int_{a}^{\infty}-2\omega^{2}k^{2}g(x)\Phi_{k}(x)\Phi_{k}^{\prime}(x)\,dx$
$\displaystyle=-\Phi_{k}^{\prime}(a)^{2},$

respectively. Applying the Cauchy-Schwarz inequality to (17) and using the
first part of (15) we find

(18) $\|\Phi_{k}^{\prime}\|_{L^{\infty}(0,\infty)}^{2}\leq
O(k^{2})\|\Phi_{k}^{\prime}\|_{L^{2}(0,\infty)}$

and from (16), (18) we get

$\displaystyle\|\Phi_{k}^{\prime}\|_{L^{2}(0,\infty)}^{2}$ $\displaystyle\leq
O(k^{2})+\|\Phi_{k}\|_{L^{\infty}(0,\infty)}\|\Phi_{k}^{\prime}\|_{L^{\infty}(0,\infty)}$
$\displaystyle\leq
O(k^{2})+\|\Phi_{k}\|_{L^{\infty}(0,\infty)}O(k)\|\Phi_{k}^{\prime}\|_{L^{2}(0,\infty)}^{\frac{1}{2}}.$

The $L^{\infty}$-assumption in (C2) leads to

$\displaystyle\|\Phi_{k}^{\prime}\|_{L^{2}(0,\infty)}^{2}\leq
O(k^{2})+O(k)\|\Phi_{k}^{\prime}\|_{L^{2}(0,\infty)}^{\frac{1}{2}}\leq
O(k^{2})+C_{\epsilon}O(k^{\frac{4}{3}})+\epsilon\|\Phi_{k}^{\prime}\|_{L^{2}(0,\infty)}^{2},$

where we have used Young’s inequality with exponents $4/3$ and $4$. This
implies the second inequality in (15). Inserting this into (18) we obtain the
third inequality in (15). ∎

######  Lemma 2.4.

Assume (C0), (C1) and (C2). For $\hat{\alpha}\in{\mathcal{D}(J)}$ and
$w:D\to\mathbb{R}$ as in (8) we have $w_{x},w_{t}\in L^{2}(D)$,
$w_{t}(0,\cdot)\in{L^{4}(\mathbb{T}_{T})}$ and there are values $C>0$ and
$\rho>0$ such that $\absolutevalue{w(x,t)}\leq
C\mathrm{e}^{-\rho\absolutevalue{x}}$.

######  Remark 2.5.

The lemma does not require $\hat{\alpha}$ to be a critical point of $J$. The
smoothness and decay of $w$ as in (8) is simply a consequence of
$\hat{\alpha}\in{\mathcal{D}(J)}$ and (C2).

###### Proof.

We use the characterization from Lemma 2.2. Let us begin with the estimate for
$\norm{\partial_{t}w}_{{L^{2}(D)}}$. By Lemma 2.3 we have
$\sup_{k}\|\Phi_{k}\|_{L^{2}(0,\infty)}<\infty$ so that

$\displaystyle\norm{\partial_{t}w}_{{L^{2}(D)}}^{2}$
$\displaystyle=2\omega^{2}\sum_{k}\hat{\alpha}_{k}^{2}\norm{\Phi_{k}}_{L^{2}(0,\infty)}^{2}\leq
2\omega^{2}\Bigl{(}\sup_{k}\norm{\Phi_{k}}_{L^{2}(0,\infty)}\Bigr{)}^{2}\norm{\hat{\alpha}}_{l^{2}}^{2}$
$\displaystyle\leq
2\omega^{2}\Bigl{(}\sup_{k}\norm{\Phi_{k}}_{L^{2}(0,\infty)}\Bigr{)}^{2}{\interleave\hat{\alpha}\interleave}^{2}<\infty,$

which finishes our first goal. Next we estimate
$\norm{\partial_{x}w}_{{L^{2}(D)}}$. Here we use again Lemma 2.3 to find

$\displaystyle\norm{\partial_{x}w}_{{L^{2}(D)}}^{2}=2\sum_{k}\frac{\hat{\alpha}_{k}^{2}}{k^{2}}\norm{\Phi_{k}^{\prime}}_{L^{2}(0,\infty)}^{2}\leq
C\|\hat{\alpha}\|_{l^{2}}^{2}\leq
C{\interleave\hat{\alpha}\interleave}^{2}<\infty$

which finishes our second goal. Next we show that
$w_{t}(0,\cdot)\in{L^{4}(\mathbb{T}_{T})}$. Using $\Phi_{k}(0)=1$ we observe
that

$\displaystyle
T\norm{w_{t}(0,\cdot)}_{L^{4}(\mathbb{T}_{T})}^{4}=T\int_{0}^{T}\Bigl{(}\sum_{k\in{\mathbb{Z}_{odd}}}\mathrm{i}\omega\hat{\alpha}_{k}\Phi_{k}(0)e_{k}(t)\Bigr{)}^{4}\differential{t}$
$\displaystyle=\omega^{4}{\interleave\hat{\alpha}\interleave}^{4}<\infty.$

Finally we show the uniform-in-time exponential decay of $w$. By construction
$w$ is even in $x$, hence we only consider $x>0$. By (C2) we see that

$\displaystyle\absolutevalue{w(x,t)}\leq\sum_{k}\frac{\absolutevalue{\hat{\alpha}_{k}}}{\absolutevalue{k}}\absolutevalue{\Phi_{k}(x)}=\sum_{k}\frac{\absolutevalue{\hat{\alpha}_{k}}}{\absolutevalue{k}}Ce^{-\alpha
x}\leq\|\hat{\alpha}\|_{l^{2}}\left(\sum_{k}\frac{1}{k^{2}}\right)^{1/2}Ce^{-\alpha
x}\leq\tilde{C}e^{-\alpha x}$

which finishes the proof of the lemma. ∎

In the following result we will show that minimizers of $J$ on
${\mathcal{D}(J)}$ exist, are solutions of (9) and indeed correspond to weak
solutions of (1).

######  Theorem 2.6.

Assume (C0), (C1) and (C2). Then the functional $J$ is well defined on its
domain ${\mathcal{D}(J)}$, Fréchet-differentiable, bounded from below and
attains its negative minimum provided

  * (i)

$\gamma<0$ and the sequence
$\left(\Phi^{\prime}_{k}(0)\right)_{k\in{\mathbb{N}_{odd}}}$ has at least one
positive element, or

  * (ii)

$\gamma>0$ and the sequence
$\left(\Phi^{\prime}_{k}(0)\right)_{k\in{\mathbb{N}_{odd}}}$ has at least one
negative element.

For every critical point $\hat{\alpha}\in{\mathcal{D}(J)}$ the corresponding
function
$w(x,t)\coloneqq\sum_{k\in{\mathbb{Z}_{odd}}}\frac{\hat{\alpha}_{k}}{k}\Phi_{k}(\absolutevalue{x})e_{k}(t)$
is a nontrivial weak solution of (1).

###### Proof.

Note that
$J(\hat{z})=\frac{1}{4}{\interleave\hat{z}\interleave}^{4}+J_{1}(\hat{z})$,
where $J_{1}(\hat{z})=\sum_{k}a_{k}\hat{z}_{k}^{2}$ with
$a_{k}=\frac{T\Phi_{k}^{\prime}(0)}{\gamma\omega^{4}k^{2}}$. By Lemma 2.3 the
sequence $(a_{k})_{k}$ is converging to $0$ as $|k|\to\infty$, so in
particular it is bounded. Due to (13) one finds that $J$ is well defined and
continuous on ${\mathcal{D}(J)}$, and moreover, that for
$\hat{z}\in{\mathcal{D}(J)}$

$\displaystyle
J(\hat{z})\geq\frac{1}{4}{\interleave\hat{z}\interleave}^{4}-\sup_{k}|a_{k}|\sum_{k}\hat{z}_{k}^{2}\geq\frac{1}{4}{\interleave\hat{z}\interleave}^{4}-\sup_{k}|a_{k}|{\interleave\hat{z}\interleave}^{2}.$

This implies that $J$ is coercive and bounded from below. The weak lower semi-
continuity of $J$ follows from the convexity and continuity of the map
$\hat{z}\mapsto{\interleave\hat{z}\interleave}^{4}$ and the weak continuity of
$J_{1}$. To see the latter take an arbitrary $\epsilon>0$. Then there is
$k_{0}\in\mathbb{N}$ such that $|a_{k}|\leq\epsilon$ for $|k|>k_{0}$ and this
implies the inequality

(19)
$\displaystyle|J_{1}(\hat{z})-J_{1}(\hat{y})|\leq\sup_{k}|a_{k}|\sum_{|k|\leq
k_{0}}|\hat{z}_{k}^{2}-\hat{y_{k}}^{2}|+\epsilon(\|\hat{z}\|_{l^{2}}^{2}+\|\hat{y}\|_{l^{2}}^{2})\quad\forall\,\hat{z},\hat{y}\in{\mathcal{D}(J)}.$

Since $({\mathcal{D}(J)},{\interleave\cdot\interleave})$ continuously embeds
into $l^{2}$ any weakly convergent sequence in
$({\mathcal{D}(J)},{\interleave\cdot\interleave})$ also weakly converges in
$l^{2}$ and in particular pointwise. This pointwise convergence together with
the boundedness of the sequence and (19) yields the weak continuity of $J_{1}$
and thus the weak lower semi-continuity of $J$. As a consequence, cf. Theorem
1.2 in [9], we get the existence of a minimizer.

In order to check that the minimizer is nontrivial is suffices to verify that
$J$ attains negative values. Here we distinguish between case (i) and (ii) in
the assumptions of the theorem. In case (i) when $\gamma<0$ we find an index
$k_{0}$ such that $\Phi_{k_{0}}^{\prime}(0)>0$. In case (ii) when $\gamma>0$
we choose $k_{0}$ such that $\Phi_{k_{0}}^{\prime}(0)<0$. In both cases we
obtain that $\Phi_{k_{0}}^{\prime}(0)/\gamma<0$. If we set
$\hat{y}\coloneqq(\delta_{k,k_{0}}-\delta_{k,-k_{0}})_{k\in{\mathbb{Z}_{odd}}}$
then $\hat{y}$ has exactly two non-vanishing entries, namely $+1$ at $k_{0}$
and $-1$ at $-k_{0}$. Hence $\hat{y}\in{\mathcal{D}(J)}$. Using the property
$\Phi_{k_{0}}^{\prime}=\Phi_{-k_{0}}^{\prime}$ we find for $t\in\mathbb{R}$

$\displaystyle
J(t\hat{y})=t^{4}\frac{1}{4}{\interleave\hat{y}\interleave}^{4}+2t^{2}\frac{T\Phi^{\prime}_{k_{0}}(0)}{\gamma\omega^{4}k_{0}^{2}}$

which is negative by the choice of $k_{0}$ provided $t>0$ is sufficiently
small. Thus, $\inf_{{\mathcal{D}(J)}}J<0$ and every minimizer $\hat{\alpha}$
is nontrivial.

Next we show for every critical point $\hat{\alpha}$ of $J$ that
$w(x,t)\coloneqq\sum_{k\in{\mathbb{Z}_{odd}}}\frac{\hat{\alpha}_{k}}{k}\Phi_{k}(\absolutevalue{x})e_{k}(t)$
is a weak solution of (1). The regularity properties
$w\in{H^{1}({\mathbb{R}\times\mathbb{T}_{T}})}$,
$\partial_{t}w(0,\cdot)\in{L^{4}(\mathbb{T}_{T})}$ and the exponential decay
have already been shown in Lemma 2.4. We skip the standard proof that
$J\in{C^{1}({\mathcal{D}(J)},\mathbb{R})}$ and that its Fréchet-derivative is
given by (10). We will show that (2) holds for any $\psi$ as in (14) with even
functions $\Psi_{k}\in H^{1}(\mathbb{R})$, $\Psi_{k}=-\Psi_{-k}$ such that
$\psi_{x},\psi_{t}\in L^{2}(D)$ and $\psi(0,\cdot)\in L^{4}(\mathbb{T}_{T})$
as described in Lemma 2.2. We begin by deriving expressions and estimates for
the functionals

$\displaystyle H_{1}(\psi)=\int_{D}g(x)w_{t}\psi_{t}\,d(x,t),\quad
H_{2}(\psi)=\int_{D}w_{x}\psi_{x}\,d(x,t),\quad
H_{3}(\psi)=\int_{0}^{T}w_{t}(0,t)^{3}\psi_{t}(0,t)\,dt.$

In a first step we assume that the sum in (14) is finite in order to justify
the exchange of summation and integration in the following. Then, starting
with $H_{1}$ we find

$\displaystyle H_{1}(\psi)$
$\displaystyle=-\omega^{2}\int_{D}g(x)\sum_{k,l}\hat{\alpha}_{k}\Phi_{k}(\absolutevalue{x})\Psi_{l}(|x|)e_{k}(t)e_{l}(t)\differential{(x,t)}$
$\displaystyle=-2\omega^{2}\sum_{k}\hat{\alpha}_{k}\int_{0}^{\infty}g(x)\Phi_{k}(x)\Psi_{-k}(x)\differential{x}$
$\displaystyle=2\omega^{2}\sum_{k}\hat{\alpha}_{k}\int_{0}^{\infty}g(x)\Phi_{k}(x)\Psi_{k}(x)\differential{x},$
$\displaystyle|H_{1}(\psi)|$ $\displaystyle\leq
2\omega^{2}\|g\|_{L^{\infty}(\mathbb{R})}\Bigl{(}\sum_{k}\hat{\alpha}_{k}^{2}\|\Phi_{k}\|^{2}_{L^{2}(0,\infty)}\Bigr{)}^{\frac{1}{2}}\Bigl{(}\sum_{k}\|\Psi_{k}\|_{L^{2}(0,\infty)}^{2}\Bigr{)}^{\frac{1}{2}}=\|g\|_{L^{\infty}(\mathbb{R})}\|w_{t}\|_{L^{2}(D)}\|\psi_{t}\|_{L^{2}(D)}$

and similarly for $H_{2}$ we find using (24)

$\displaystyle H_{2}(\psi)$
$\displaystyle=\int_{D}\sum_{k,l}\frac{\hat{\alpha}_{k}}{k}\Phi_{k}^{\prime}(\absolutevalue{x})\frac{1}{l}\Psi_{l}^{\prime}(|x|)e_{k}(t)e_{l}(t)\differential{(x,t)}$
$\displaystyle=2\sum_{k}\frac{\hat{\alpha}_{k}}{-k^{2}}\int_{0}^{\infty}\Phi_{k}^{\prime}(x)\Psi_{-k}^{\prime}(x)\differential{x}$
$\displaystyle=2\sum_{k}\frac{\hat{\alpha}_{k}}{k^{2}}\int_{0}^{\infty}\Phi_{k}^{\prime}(x)\Psi_{k}^{\prime}(x)\differential{x}$
$\displaystyle=2\omega^{2}\sum_{k}\hat{\alpha}_{k}\int_{0}^{\infty}g(x)\Phi_{k}(x)\Psi_{k}(x)\differential{x}-2\sum_{k}\frac{\hat{\alpha}_{k}}{k^{2}}\Phi_{k}^{\prime}(0)\Psi_{k}(0),$
$\displaystyle|H_{2}(\psi)|$ $\displaystyle\leq
2\Bigl{(}\sum_{k}\frac{\hat{\alpha}_{k}^{2}}{k^{2}}\|\Phi_{k}^{\prime}\|_{L^{2}(0,\infty)}^{2}\Bigr{)}^{\frac{1}{2}}\Bigl{(}\sum_{k}\frac{1}{k^{2}}\|\Psi_{k}^{\prime}\|_{L^{2}(0,\infty)}^{2}\Bigr{)}^{\frac{1}{2}}=\|w_{x}\|_{L^{2}(D)}\|\psi_{x}\|_{L^{2}(D)}.$

Moreover, considering $H_{3}$ and setting $\hat{y}_{k}\coloneqq\Psi_{k}(0)$
for $k\in{\mathbb{Z}_{odd}}$ one sees

$\displaystyle H_{3}(\psi)$
$\displaystyle=\omega^{4}\int_{0}^{T}\Bigl{(}\sum_{k}\hat{\alpha}_{k}e_{k}(t)\Bigr{)}^{3}\Bigl{(}\sum_{l}\Psi_{l}(0)e_{l}(t)\Bigr{)}\differential{t}$
$\displaystyle=\frac{\omega^{4}}{T}(\hat{\alpha}*\hat{\alpha}*\hat{\alpha}*\hat{y})_{0},$
$\displaystyle\absolutevalue{H_{3}(\psi)}$
$\displaystyle\leq\frac{\omega^{4}}{T}{\interleave\hat{\alpha}\interleave}^{3}{\interleave\hat{y}\interleave}=\norm{w_{t}(0,\cdot)}_{L^{4}(\mathbb{T}_{T})}^{3}\norm{\psi_{t}(0,\cdot)}_{L^{4}(\mathbb{T}_{T})}.$

Hence $H_{1},H_{2}$ and $H_{3}$ are bounded linear functionals of the variable
$\psi$ as in (14) with $\psi_{x},\psi_{t}\in L^{2}(D)$ and
$\psi_{t}(0,\cdot)\in{L^{4}(\mathbb{T}_{T})}$. For such $\psi$ we use the
above formulae for $H_{1},H_{2},H_{3}$ and compute the linear combination

$\displaystyle-H_{1}(\psi)+H_{2}(\psi)-\gamma
H_{3}(\psi)=-2\sum_{k}\frac{\hat{\alpha}_{k}}{k^{2}}\Phi_{k}^{\prime}(0)\Psi_{k}(0)-\frac{\gamma\omega^{4}}{T}(\hat{\alpha}*\hat{\alpha}*\hat{\alpha}*\hat{y})_{0}=0$

due to the Euler-Lagrange equation for the functional $J$, i.e., the vanishing
of $J^{\prime}(\hat{\alpha})[\hat{y}]$ in (10) for all
$\hat{y}\in{\mathcal{D}(J)}$. The last equality means that $w$ is a weak
solution of (1). ∎

##  3\. Further Regularity

Here we prove Theorem 1.7. We observe first that in the example of a periodic
step-potential in Theorem 1.2 we find that not only
$\Phi^{\prime}_{k}(0)=O(k^{\frac{3}{2}})$ holds (as Lemma 2.3 shows) but even
$\Phi^{\prime}_{k}(0)=O(k)$ is satisfied. It is exactly this weaker growth
that we can exploit in order to prove additional smoothness of the solutions
of (1). We begin by defining for $\nu>0$ the Banach space of sequences

$\displaystyle{h^{\nu}}\coloneqq\Bigl{\\{}\hat{z}\in{l^{2}}\mbox{ s.t.
}\|\hat{z}\|_{h^{\nu}}^{2}\coloneqq\sum_{k}(1+k^{2})^{\nu}\absolutevalue{\hat{z}_{k}}^{2}<\infty\Bigr{\\}}.$

Moreover, we use the isometric isomorphism between $h^{\nu}$ and

$\displaystyle
H^{\nu}(\mathbb{T}_{T})=\Bigl{\\{}z(t)=\sum_{k}\hat{z}_{k}e_{k}(t)\text{ s.t.
}\hat{z}\in{h^{\nu}}\Bigr{\\}}$

by setting $\|z\|_{H^{\nu}}\coloneqq\|\hat{z}\|_{h^{\nu}}$. We also use the
Morrey embedding
${H^{1+\nu}(\mathbb{T}_{T})}\to{C^{0,\frac{1}{2}+\nu}(\mathbb{T}_{T})}$ for
$\nu\in(0,1/2)$ and the following embedding:
${C^{0,\nu}(\mathbb{T}_{T})}\to{H^{\tilde{\nu}}(\mathbb{T}_{T})}$ for
$0<\tilde{\nu}<\nu\leq 1$, cf. Lemma 6.1 in the Appendix. The latter embedding
means that $\hat{z}\in h^{\tilde{\nu}}$ provided
$z\in{C^{0,\nu}(\mathbb{T}_{T})}$ and $0<\tilde{\nu}<\nu\leq 1$.

######  Theorem 3.1.

Assume (C0), (C1), (C2) and in addition $\Phi^{\prime}_{k}(0)=O(k)$. For every
$\hat{\alpha}\in{\mathcal{D}(J)}$ with $J^{\prime}(\hat{\alpha})=0$ we have
$\hat{\alpha}\in h^{\nu}$ for every $\nu\in(0,1/4)$.

###### Proof.

Let $\hat{\alpha}\in{\mathcal{D}(J)}$ with $J^{\prime}(\hat{\alpha})=0$.
Recall from (9) that

(20)
$(\hat{\alpha}*\hat{\alpha}*\hat{\alpha})_{k}=\hat{\eta}_{k}\hat{\alpha}_{k}\quad\mbox{
where
}\quad\hat{\eta}_{k}\coloneqq\frac{2T\Phi^{\prime}_{k}(0)}{\gamma\omega^{4}k^{2}}\mbox{
for }k\in{\mathbb{Z}_{odd}}$

so that $|\hat{\eta}_{k}|\leq C/k$. If we define the convolution of two
$T$-periodic functions $f,g\in{L^{2}(\mathbb{T}_{T})}$ on the torus
$\mathbb{T}_{T}$ as

$\displaystyle\left(f*g\right)(t)\coloneqq\frac{1}{\sqrt{T}}\int_{0}^{T}f(s)g(t-s)\differential{s}$

and if we set

$\displaystyle\alpha(t)\coloneqq\sum_{k}\hat{\alpha}_{k}e_{k}(t),\quad\eta(t)\coloneqq\sum_{k}\hat{\eta}_{k}e_{k}(t)$

then the equation

(21) $\alpha^{3}=\alpha*\eta$

for the $T$-periodic function $\alpha\in{L^{4}(\mathbb{T}_{T})}$ is equivalent
to the equation (20) for the sequence $\hat{\alpha}\in{\mathcal{D}(J)}$. We
will analyze (21) with a bootstrap argument.

_Step 1:_ We show that $\alpha\in{C^{0,\frac{1}{6}}(\mathbb{T}_{T})}$. The
right hand side of (21) is an ${H^{1}(\mathbb{T}_{T})}$-function since

$\displaystyle\norm{\alpha*\eta}_{H^{1}(\mathbb{T}_{T})}^{2}=\norm{\hat{\alpha}\hat{\eta}}_{h^{1}}^{2}\leq\sum_{k\in{\mathbb{Z}_{odd}}}(1+k^{2})\hat{\alpha}_{k}^{2}\frac{C^{2}}{k^{2}}\leq
2C^{2}\|\hat{\alpha}\|_{l^{2}}^{2}<\infty.$

Therefore, using (21) we see that $\alpha^{3}\in H^{1}(\mathbb{T}_{T})$ and by
the Morrey embedding that $\alpha^{3}\in{C^{0,\frac{1}{2}}(\mathbb{T}_{T})}$.
Since the inverse of the mapping $x\mapsto x^{3}$ is given by
$x\mapsto|x|^{-\frac{2}{3}}x$, which is a
${C^{0,\frac{1}{3}}}(\mathbb{R})$-function, we obtain
$\alpha\in{C^{0,\frac{1}{6}}(\mathbb{T}_{T})}$.

_Step 2:_ We fix $q\in(0,1)$ and show that if
$\alpha\in{C^{0,\nu_{n}}(\mathbb{T}_{T})}$ for some $\nu_{n}\in(0,1/2)$ solves
(21) then $\alpha\in{C^{0,\nu_{n+1}}(\mathbb{T}_{T})}$ with
$\nu_{n+1}=\frac{q\nu_{n}}{3}+\frac{1}{6}$. For the proof we iterate the
process from Step 1 and we start with
$\alpha\in{C^{0,\nu_{n}}(\mathbb{T}_{T})}$. Then, according to Lemma 6.1 of
the Appendix, $\alpha\in{H^{q\nu_{n}}(\mathbb{T}_{T})}$ and hence
$\hat{\alpha}\in{h^{q\nu_{n}}}$. Then as before the convolution of $\alpha$
with $\eta$ generates one more weak derivative, namely

$\displaystyle\norm{\alpha*\eta}_{H^{1+q\nu_{n}}(\mathbb{T}_{T})}^{2}=\norm{\hat{\alpha}\hat{\eta}}_{h^{1+q\nu_{n}}}^{2}\leq\sum_{k}(1+k^{2})^{1+q\nu_{n}}\hat{\alpha}_{k}^{2}\frac{C^{2}}{k^{2}}\leq
C^{2}\|\hat{\alpha}\|_{h^{q\nu_{n}}}<\infty.$

Hence by (21) we conclude $\alpha^{3}\in{H^{1+q\nu_{n}}(\mathbb{T}_{T})}$ and
by the Morrey embedding
$\alpha^{3}\in{C^{0,\frac{1}{2}+q\nu_{n}}(\mathbb{T}_{T})}$ provided
$q\nu_{n}\in(0,1/2)$. As in Step 1 this implies
$\alpha\in{C^{0,\nu_{n+1}}(\mathbb{T}_{T})}$ with
$\nu_{n+1}=\frac{1}{6}+\frac{q\nu_{n}}{3}$.

Starting with $\nu_{1}=1/6$ from Step 1 we see by Step 2 that
$\nu_{n}\nearrow\frac{1}{2(3-q)}$. Since $q\in(0,1)$ can be chosen arbitrarily
close to $1$ this finishes the proof. ∎

With this preparation the proof of Theorem 1.7 is now immediate.

###### Proof of Theorem 1.7.

Let
$w(x,t)=\sum_{k\in{\mathbb{Z}_{odd}}}\frac{\hat{\alpha}_{k}}{k}\Phi_{k}(|x|)e_{k}(t)$
with $\hat{\alpha}\in{\mathcal{D}(J)}$ such that $J^{\prime}(\hat{\alpha})=0$.
Recall from assumption (C2) that
$C\coloneqq\sup_{k}\norm{\Phi_{k}}_{L^{2}(0,\infty)}^{2}<\infty$. Likewise,
from Lemma 2.3 we have
$\norm{\Phi_{k}^{\prime}}_{L^{2}(0,\infty)}^{2}\leq\tilde{C}k^{2}$ for all
$k\in{\mathbb{Z}_{odd}}$ and some $\tilde{C}>0$. Therefore, using Theorem 3.1
we find for all $\nu<\frac{1}{4}$

$\displaystyle\norm{\partial_{t}^{1+\nu}w}_{L^{2}(D)}^{2}=2\omega^{2+2\nu}\sum_{k}\hat{\alpha}_{k}^{2}|k|^{2\nu}\norm{\Phi_{k}}_{L^{2}(0,\infty)}^{2}\leq
2\omega^{2+2\nu}C\|\hat{\alpha}\|_{h^{\nu}}^{2}<\infty$

and likewise

$\displaystyle\norm{\partial_{t}^{\nu}w_{x}}_{L^{2}(D)}^{2}=2\omega^{2\nu}\sum_{k}\hat{\alpha}_{k}^{2}|k|^{2\nu-2}\norm{\Phi_{k}^{\prime}}_{L^{2}(0,\infty)}^{2}\leq
2\omega^{2\nu}\tilde{C}\|\hat{\alpha}\|_{h^{\nu}}^{2}<\infty.$

This establishes the claim. ∎

##  4\. Existence of Infinitely Many Breathers

In this section we extend Theorem 1.5 by the following multiplicity result.

######  Theorem 4.1.

Assume (C0), (C1) and (C2). Then (1) has infinitely many nontrivial,
$T$-periodic weak solution $w$ in the sense of Definition 1.3 with
$T=\frac{2\pi}{\omega}$ provided

  * (i)

$\gamma<0$ and there exists an integer $l_{-}\in{\mathbb{N}_{odd}}$ such that
for infinitely many $j\in\mathbb{N}$ the sequence
$\Bigl{(}\Phi^{\prime}_{m\cdot l_{-}^{j}}(0)\Bigr{)}_{m\in{\mathbb{N}_{odd}}}$
has at least one positive element,

  * (ii)

$\gamma>0$ and there exists an integer $l_{+}\in{\mathbb{N}_{odd}}$ such that
for infinitely many $j\in\mathbb{N}$ the sequence
$\Bigl{(}\Phi^{\prime}_{m\cdot l_{+}^{j}}(0)\Bigr{)}_{m\in{\mathbb{N}_{odd}}}$
has at least one negative element.

######  Remark 4.2.

In the above Theorem, conditions (C1) and (C2) can be weakened: instead of
requiring them for all $k\in{\mathbb{N}_{odd}}$ it suffices to require them
for $k\in l_{-}^{j}{\mathbb{N}_{odd}}$, $k\in l_{+}^{j}{\mathbb{N}_{odd}}$
respectively. We prove this observation together with the one in Remark 1.6 at
the end of this section.

We start with an investigation about the types of symmetries which are
compatible with our equation. The Euler-Lagrange equation (9) for critical
points $\hat{\alpha}\in{\mathcal{D}(J)}$ of $J$ takes the form
$(\hat{\alpha}*\hat{\alpha}*\hat{\alpha})_{k}=\hat{\eta}_{k}\hat{\alpha}_{k}$
with
$\hat{\eta}_{k}\coloneqq\frac{2T\Phi^{\prime}_{k}(0)}{\gamma\omega^{4}k^{2}}$
for $k\in{\mathbb{Z}_{odd}}$. Next we describe subspaces of ${\mathcal{D}(J)}$
which are invariant under triple convolution and pointwise multiplication with
$(\hat{\eta}_{k})_{k\in{\mathbb{Z}_{odd}}}$. It turns out that these subspaces
are made of sequences $\hat{z}$ where only the $r^{th}$ entry modulus $2r$ is
occupied.

######  Definition 4.3.

For $r\in{\mathbb{N}_{odd}},p\in{\mathbb{N}_{even}}$ with $r<p$ let

$\displaystyle{\mathcal{D}(J)}_{r,p}=\\{\hat{z}\in{\mathcal{D}(J)}:\forall\,k\in\mathbb{Z},k\neq
r\leavevmode\nobreak\ \mathrm{mod}\leavevmode\nobreak\
p\colon\hat{z}_{k}=0\\}.$

######  Lemma 4.4.

For $r\in{\mathbb{N}_{odd}},p\in{\mathbb{N}_{even}}$ with $r<p$ and $p\not=2r$
we have ${\mathcal{D}(J)}_{r,p}=\\{0\\}$.

###### Proof.

Let $\hat{z}\in{\mathcal{D}(J)}_{r,p}$. For all $k\not\in r+p\mathbb{Z}$ we
have $\hat{z}_{k}=0$ by definition of ${\mathcal{D}(J)}_{r,p}$. Let therefore
$k=r+pl_{1}$ for some $l_{1}\in\mathbb{Z}$. Then $-k=-r-pl_{1}\not\in
r+p\mathbb{Z}$ because otherwise $2r=-p(l_{1}+l_{2})=p|l_{1}+l_{2}|$ for some
$l_{2}\in\mathbb{Z}$. Since by assumption $p>r$ we get $|l_{1}+l_{2}|<2$. But
clearly $|l_{1}+l_{2}|\not\in\\{0,1\\}$ since $r\not=0$ and $p\not=2r$ by
assumption. By this contradiction we have shown $-k\not\in r+p\mathbb{Z}$ so
that necessarily $0=\hat{z}_{-k}=-\hat{z}_{k}$. This shows $\hat{z}=0$. ∎

In the following we continue by only considering
$\mathcal{D}_{r}\coloneqq{\mathcal{D}(J)}_{r,2r}$ for
$r\in{\mathbb{N}_{odd}}$.

######  Proposition 4.5.

Let $r\in{\mathbb{N}_{odd}}$.

  * (i)

The elements $\hat{z}\in\mathcal{D}_{r}$ are exactly those elements of
${\mathcal{D}(J)}$ which generate $\frac{T}{2r}$-antiperiodic functions
$\sum_{k\in{\mathbb{Z}_{odd}}}\frac{\hat{z}_{k}}{k}\Phi_{k}(x)e_{k}(t)$.

  * (ii)

If $\hat{z}\in\mathcal{D}_{r}$ then $(\hat{z}*\hat{z}*\hat{z})_{k}=0$ for all
$k\not\in r+2r\mathbb{Z}$.

###### Proof.

(i) An element $\hat{z}\in{\mathcal{D}(J)}$ generates a
$\frac{T}{2r}$-antiperiodic function
$z(x,t)=\sum_{k\in{\mathbb{Z}_{odd}}}\frac{\hat{z}_{k}}{k}\Phi_{k}(x)e_{k}(t)$
if and only if $z(x,t+\frac{T}{2r})=-z(x,t)$. Comparing the Fourier
coefficients we see that this is the case if for all $k\in{\mathbb{Z}_{odd}}$
we have $\hat{z}_{k}\bigl{(}\exp(\frac{\mathrm{i}\omega kT}{2r})+1\bigr{)}=0$,
i.e., either $k\in r+2r\mathbb{Z}$ or $\hat{z}_{k}=0$. This is exactly the
condition that $\hat{z}\in\mathcal{D}_{r}$.
(ii) Let $\hat{z}\in\mathcal{D}_{r}$ and assume that there is $k\in\mathbb{Z}$
such that
$0\not=(\hat{z}*\hat{z}*\hat{z})_{k}=\sum_{l,m}\hat{z}_{l}\hat{z}_{m-l}\hat{z}_{k-m}$.
So there is $l_{0},m_{0}\in{\mathbb{Z}_{odd}}$ such that
$\hat{z}_{l_{0}},\hat{z}_{m_{0}-l_{0}},\hat{z}_{k-m_{0}}\not=0$ which means by
the definition of $\mathcal{D}_{r}$ that $l_{0},m_{0}-l_{0},k-m_{0}\in
r+2r\mathbb{Z}$. Thus $k=l_{0}+m_{0}-l_{0}+k-m_{0}\in
3r+2r\mathbb{Z}=r+2r\mathbb{Z}$. ∎

###### Proof of Theorem 4.1.

We give the proof in case (i); for case (ii) the proof only needs a trivial
modification. Let $r=l^{j}$ where $j$ is an index such that the sequence
$\Bigl{(}\Phi^{\prime}_{k\cdot l^{j}}(0)\Bigr{)}_{k\in{\mathbb{N}_{odd}}}$ has
a positive element (we have changed the notation from $l_{-}$ to $l$ for the
sake of readability). Since $\mathcal{D}_{r}$ is a closed subspace of
${\mathcal{D}(J)}$ we have as before in Theorem 2.6 the existence of a
minimizer $\hat{\alpha}^{(r)}\in\mathcal{D}_{r}$, i.e.,
$J(\hat{\alpha}^{(r)})=\min_{\mathcal{D}_{r}}J<0$. Moreover,
$\hat{\alpha}^{(r)}$ satisfies the restricted Euler-Lagrange-equation

(22)
$0=J^{\prime}\left(\hat{\alpha}^{(r)}\right)\left[\hat{x}\right]=\left(\hat{\alpha}^{(r)}*\hat{\alpha}^{(r)}*\hat{\alpha}^{(r)}*\hat{x}\right)_{0}+\frac{2T}{\gamma\omega^{4}}\sum_{k}\frac{\Phi^{\prime}_{k}(0)}{k^{2}}\hat{\alpha}^{(r)}_{k}\hat{x}_{k}\qquad\forall\,\hat{x}\in\mathcal{D}_{r}.$

We need to show that (22) holds for every $\hat{z}\in{\mathcal{D}(J)}$. If for
an arbitrary $\hat{z}\in{\mathcal{D}(J)}$ we define
$\hat{x}_{k}\coloneqq\hat{z}_{k}$ for $k\in r+2r\mathbb{Z}$ and
$\hat{x}_{k}\coloneqq 0$ else then $\hat{x}\in\mathcal{D}_{r}$. If we
furthermore define $\hat{y}\coloneqq\hat{z}-\hat{x}$ then $\hat{y}_{k}=0$ for
all $k\in r+2r\mathbb{Z}$. This implies in particular that

$\displaystyle\sum_{k}\frac{\Phi^{\prime}_{k}(0)}{k^{2}}\hat{\alpha}^{(r)}_{k}\hat{y}_{k}=0$

and by using (ii) of Proposition 4.5 also

$\displaystyle(\hat{\alpha}*\hat{\alpha}*\hat{\alpha}*\hat{y})_{0}=\sum_{k}\left(\hat{\alpha}^{(r)}*\hat{\alpha}^{(r)}*\hat{\alpha}^{(r)}\right)_{k}\hat{y}_{-k}=0.$

This implies $J^{\prime}(\hat{\alpha}^{(r)})[\hat{y}]=0$ and since by (22)
also $J^{\prime}(\hat{\alpha}^{(r)})[\hat{x}]=0$ we have succeeded in proving
that $J^{\prime}(\hat{\alpha}^{(r)})=0$.

It remains to show the multiplicity result. For this purpose we only consider
$r=l^{j_{m}}$ for $j_{m}\to\infty$ as $m\to\infty$ where $j_{m}$ is an index
such that the sequence
$\Bigl{(}\Phi^{\prime}_{l^{j_{m}}k}(0)\Bigr{)}_{k\in{\mathbb{N}_{odd}}}$ has a
positive element. First we observe that
$\mathcal{D}_{l^{j_{m}}}\supsetneq\mathcal{D}_{l^{j_{m+1}}}$. Assume for
contradiction that the set $\\{\hat{\alpha}^{(l^{j_{m}})}\\}$ is finite. Then
we have a subsequence $(j_{m_{n}})_{n\in\mathbb{N}}$ such that
$\hat{\alpha}=\hat{\alpha}^{(l^{j_{m_{n}}})}$ is constant. But then

$\displaystyle\hat{\alpha}\in\bigcap_{n\in\mathbb{N}}\mathcal{D}_{l^{j_{m_{n}}}}=\bigcap_{j\in\mathbb{N}}\mathcal{D}_{l^{j}}=\\{0\\}.$

This contradiction shows the existence of infinitely many distinct critical
points of the function $J$ and finishes the proof of the theorem. ∎

###### Proof of Remark 1.6 and Remark 4.2.

The proof of Theorem 4.1 works on the basis that it suffices to minimize the
functional $J$ on $\mathcal{D}_{r}$. In this way a $\frac{T}{2r}$-antiperiodic
breather is obtained. For $\hat{z}\in\mathcal{D}_{r}$ only the entries
$\hat{z}_{k}$ with $k\in r{\mathbb{Z}_{odd}}$ are nontrivial while all other
entries vanish. Therefore, (C1) and (C2) and the values of
$\Phi_{k}^{\prime}(0)$ are only relevant for $k\in r{\mathbb{Z}_{odd}}$. In
the special case of Remark 4.2 we take $r=l_{\pm}^{j}$. ∎

##  5\. Approximation by Finitely Many Harmonics

Here we give some analytical results on finite dimensional approximation of
the breathers obtained in Theorem 1.5. The finite dimensional approximation is
obtained by cutting-off the ansatz (8) and only considering harmonics of order
$|k|\leq N$. Here a summand in the series (8) of the form
$\Phi_{k}(|x|)e_{k}(t)$ is a called a harmonic since it satisfies the linear
wave equation in (6). We will prove that $J$ restricted to spaces
${\mathcal{D}(J^{(N)})}$ of cut-off ansatz functions still attains its minimum
and that the sequence of the corresponding minimizers converges up to a
subsequence to a minimizer of $J$ on ${\mathcal{D}(J)}$.

######  Definition 5.1.

Let $N\in{\mathbb{N}_{odd}}$. Define

$\displaystyle J^{(N)}\coloneqq
J|_{\mathcal{D}(J^{(N)})},\qquad{\mathcal{D}(J^{(N)})}\coloneqq\left\\{\hat{z}\in{\mathcal{D}(J)}\leavevmode\nobreak\
\big{|}\leavevmode\nobreak\
\forall\,\absolutevalue{k}>N\colon\hat{z}_{k}=0\right\\}$

######  Lemma 5.2.

Under the assumptions of Theorem 1.5 the following holds:

  1. (i)

For every $N\in{\mathbb{N}_{odd}}$ sufficiently large there exists
$\hat{\alpha}^{(N)}\in{\mathcal{D}(J^{(N)})}$ such that
$J(\hat{\alpha}^{(N)})=\inf J^{(N)}<0$ and
$\lim_{N\to\infty}J(\hat{\alpha}^{(N)})=\inf J$.

  2. (ii)

There is $\hat{\alpha}\in{\mathcal{D}(J)}$ such that up to a subsequence
(again denoted by $(\hat{\alpha}^{(N)})_{N}$) we have

$\displaystyle\hat{\alpha}^{(N)}\to\hat{\alpha}\qquad\text{ in
}\leavevmode\nobreak\ {\mathcal{D}(J)}$

and $J(\hat{\alpha})=\inf J$.

######  Remark 5.3.

The Euler-Lagrange-equation for $\hat{\alpha}^{(N)}$ reads:

$\displaystyle
0=J^{\prime}\left(\hat{\alpha}^{(N)}\right)[\hat{y}]=\left(\hat{\alpha}^{(N)}*\hat{\alpha}^{(N)}*\hat{\alpha}^{(N)}*\hat{y}\right)_{0}+\frac{2T}{\gamma\omega^{4}}\sum_{k}\frac{\Phi^{\prime}_{k}(0)}{k^{2}}\hat{\alpha}^{(N)}_{k}\hat{y}_{k}\qquad\forall\,\hat{y}\in{\mathcal{D}(J^{(N)})}.$

This amounts to satisfying (2) in Definition 1.3 for functions
$\psi(x,t)=\sum_{k\in{\mathbb{Z}_{odd}},|k|\leq N}\hat{\psi}_{k}(x)e_{k}(t)$
with $\hat{\psi}_{k}\in H^{1}(\mathbb{R})$. Clearly, in general
$\hat{\alpha}^{(N)}$ is not a critical point of $J$.

###### Proof.

(i) We choose $N\in{\mathbb{N}_{odd}}$ so large, such that we have the assumed
sign of the the one element in
$\left(\Phi_{k}^{\prime}(0)\right)_{\absolutevalue{k}\leq N}$. The restriction
of $J$ to the $\frac{N+1}{2}$-dimensional space ${\mathcal{D}(J^{(N)})}$
preserves coercivity. The continuity of $J^{(N)}$ therefore guarantees the
existence of a minimizer $\hat{\alpha}^{(N)}\in{\mathcal{D}(J^{(N)})}$. As
before we see that $J(\hat{\alpha}^{(N)})=\inf J^{(N)}<0$, so in particular
$\hat{\alpha}^{(N)}\neq 0$. Next we observe that
${\mathcal{D}(J^{(N)})}\subset{\mathcal{D}(J)}$, i.e.,
$J(\hat{\alpha}^{(N)})\geq\inf J=J(\hat{\beta})$ for a minimizer
$\hat{\beta}\in{\mathcal{D}(J)}$ of $J$. Let us define
$\hat{\beta}^{(N)}_{k}=\hat{\beta}_{k}$ for $\absolutevalue{k}\leq N$ and
$\hat{\beta}^{(N)}_{k}=0$. Since the Fourier-series
$\beta(t)=\sum_{k}\hat{\beta}_{k}e_{k}(t)$ converges in $L^{4}(\mathbb{T})$,
cf. Theorem 4.1.8 in [6], we see that
$\hat{\beta}^{(N)}\rightarrow\hat{\beta}$ in ${\mathcal{D}(J)}$. By the
minimality of $\hat{\alpha}^{(N)}\in{\mathcal{D}(J^{(N)})}$ and continuity of
$J$ we conclude

$\displaystyle\inf_{{\mathcal{D}(J)}}J\leq J(\hat{\alpha}^{(N)})\leq
J(\hat{\beta}^{(N)})\longrightarrow J(\hat{\beta})=\inf_{{\mathcal{D}(J)}}J.$

Hence $\lim_{N\to\infty}J(\hat{\alpha}^{(N)})=\inf J$ as claimed.

(ii) Since
${\mathcal{D}(J^{(N)})}\subset{\mathcal{D}(J^{(N+1)})}\subset{\mathcal{D}(J)}$
we see that $J(\hat{\alpha}^{(N)})\geq J(\hat{\alpha}^{(N+1)})\geq\inf J$ so
that in particular the sequence $(J(\hat{\alpha}^{(N)}))_{N}$ is bounded. By
coercivity of $J$ we conclude that $(\hat{\alpha}^{(N)})_{N}$ is bounded in
${\mathcal{D}(J)}$ so that there is $\hat{\alpha}\in{\mathcal{D}(J)}$ and a
subsequence (again denoted by $(\hat{\alpha}^{(N)})_{N}$) such that

$\displaystyle\hat{\alpha}^{(N)}\rightharpoonup\hat{\alpha}\qquad\text{ in
}\leavevmode\nobreak\ {\mathcal{D}(J)}.$

By part (i) and weak lower semi-continuity of $J$ we obtain

$\displaystyle\inf J=\lim_{N\to\infty}J(\hat{\alpha}^{(N)})\geq
J(\hat{\alpha}),$

i.e., $\hat{\alpha}$ is a minimizer of $J$. Recall that
$J(\cdot)=\frac{1}{4}{\interleave\cdot\interleave}^{4}+J_{1}(\cdot)$ where
$J_{1}$ is weakly continuous, cf. proof of Theorem 2.6. Therefore, since
$\hat{\alpha}^{(N)}\rightharpoonup\hat{\alpha}$ and $J(\hat{\alpha}^{(N)})\to
J(\hat{\alpha})$ we see that
${\interleave\hat{\alpha}^{(N)}\interleave}\to{\interleave\hat{\alpha}\interleave}$
as $N\to\infty$. Since ${\mathcal{D}(J)}$ is strictly uniformly convex, we
obtain the norm-convergence of $(\hat{\alpha}^{(N)})_{N}$ to $\hat{\alpha}$. ∎

##  6\. Appendix

###  6.1. Details on exponentially decreasing fundamental solutions for step
potentials

Here we consider a second-order ordinary differential operator

$\displaystyle L_{k}\coloneqq-\frac{d^{2}}{dx^{2}}-k^{2}\omega^{2}g(x)$

with $g$ as in Theorem 1.1. Clearly, $L_{k}$ is a self-adjoint operator on
$L^{2}(\mathbb{R})$ with domain $H^{2}(\mathbb{R})$. Moreover,
$\sigma_{ess}(L_{k})=[k^{2}\omega^{2}a,\infty)$. By the assumption on $\omega$
we have

$\displaystyle\sqrt{b}\omega c\frac{2}{\pi}=\frac{p}{q}\mbox{ with
}p,q\in{\mathbb{N}_{odd}}.$

Hence, with $k\in q{\mathbb{N}_{odd}}$, $k\sqrt{b}\omega c$ is an odd multiple
of $\pi/2$. In the following we shall see that $0$ is not an eigenvalue of
$L_{k}$ for $k\in q{\mathbb{N}_{odd}}$ so that (C1) as in Remark 4.2 is
fulfilled. A potential eigenfunction $\phi_{k}$ for the eigenvalue $0$ would
have to look like

(23)
$\phi_{k}(x)=\begin{cases}-A\sin(k\omega\sqrt{b}c)e^{k\omega\sqrt{a}(x+c)},&\leavevmode\nobreak\
\phantom{-c<}x<-c,\\\
A\sin(k\omega\sqrt{b}x)+B\cos(k\omega\sqrt{b}x),&-c<x<c,\\\
A\sin(k\omega\sqrt{b}c)e^{-k\omega\sqrt{a}(x-c)},&\phantom{-}c<x.\end{cases}$

with $A,B\in\mathbb{R}$ to be determined. Note that we have used
$\cos(k\omega\sqrt{b}c)=0$. The $C^{1}$-matching of $\phi_{k}$ at $x=\pm c$
lead to the two equations

$\displaystyle-Bk\omega\sqrt{b}\sin(k\omega\sqrt{b}c)$
$\displaystyle=-Ak\omega\sqrt{a}\sin(k\omega\sqrt{b}c),$ $\displaystyle
Bk\omega\sqrt{b}\sin(k\omega\sqrt{b}c)$
$\displaystyle=-Ak\omega\sqrt{a}\sin(k\omega\sqrt{b}c)$

and since $\sin(k\omega\sqrt{b}c)=\pm 1$ this implies $A=B=0$ so that there is
no eigenvalue $0$ of $L_{k}$. Next we need to find the fundamental solution
$\phi_{k}$ of $L_{k}$ that decays to zero at $+\infty$ and is normalized by
$\phi_{k}(0)=1$. Here we can use the same ansatz as in (23) and just ignore
the part of $\phi_{k}$ on $(-\infty,0)$. Now the normalization $\phi_{k}(0)=1$
leads to $B=1$ and the $C^{1}$-matching at $x=c$ leads to
$A=\sqrt{\frac{b}{a}}B=\sqrt{\frac{b}{a}}$ so that the decaying fundamental
solution is completely determined. We find that

$\displaystyle\absolutevalue{\phi_{k}(x)}\leq\left\\{\begin{array}[]{ll}A+B,&0\leq
x\leq c\vspace{\jot}\\\ A,&c<x\leq 2c\vspace{\jot}\\\
Ae^{-\frac{1}{2}k\omega\sqrt{a}x},&x>2c\end{array}\right.$

so that $|\phi_{k}(x)|\leq(A+B)e^{-\rho_{k}x}\leq Me^{-\rho x}$ on
$[0,\infty)$ with $\rho_{k}=\frac{1}{2}k\omega\sqrt{a}$,
$\rho=\frac{1}{2}\omega\sqrt{a}$ and $M=A+B$. This shows that also (C2) holds.
Finally, since $\phi_{k}^{\prime}(0)=\frac{bk\omega}{\sqrt{a}}>0$ the
existence of infinitely many breathers can only be shown for $\gamma<0$. At
the same time, due to $|\phi_{k}(0)|=O(k)$, Theorem 1.7 applies.

###  6.2. Details on Bloch Modes for periodic step potentials

Here we consider a second-order periodic ordinary differential operator

$\displaystyle L\coloneqq-\frac{d^{2}}{dx^{2}}+V(x)$

with $V\in L^{\infty}(\mathbb{R})$ which we assume to be even and
$2\pi$-periodic. Moreover, we assume that $0$ does not belong to the spectrum
of $L:H^{2}(\mathbb{R})\subset L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$. We
first describe what Bloch modes are and why they exists. Later we show that
this is the situation which occurs in Theorem 1.5 and we verify conditions
(C1) and (C2).

A function $\Phi\in{C^{1}(\mathbb{R})}$ which is twice almost everywhere
differentiable such that

(24) $L\Phi=0\quad\text{ a.e. in
}\mathbb{R},\qquad\Phi(\cdot+2\pi)=\rho\Phi(\cdot).$

with $\rho\in(-1,1)\setminus\\{0\\}$ is called the (exponentially decreasing
for $x\to\infty$) Bloch mode of $L$ and $\rho$ is called the Floquet
multiplier. The existence of $\Phi$ is guaranteed by the assumption that
$0\notin\sigma(L)$. This is essentially Hill’s theorem, cf. [5]. Note that
$\Psi(x)\coloneqq\Phi(-x)$ is a second Bloch mode of $L$, which is
exponentially increasing for $x\to\infty$. The functions $\Phi$ and $\Psi$
form a fundamental system of solutions for operator $L$ on $\mathbb{R}$. Next
we explain how $\Phi$ is constructed, why it can be taken real-valued and why
it does not vanish at $x=0$ so that we can assume w.l.o.g $\Phi(0)=1$.

According to [5], Theorem 1.1.1 there are linearly independent functions
$\Psi_{1},\Psi_{2}\colon\mathbb{R}\rightarrow\mathbb{C}$ and Floquet-
multipliers $\rho_{1},\rho_{2}\in\mathbb{C}$ such that $L\Psi_{j}=0$ a.e. on
$\mathbb{R}$ and $\Psi_{j}(x+2\pi)=\rho_{j}\Psi_{j}(x)$ for $j=1,2$. We define
$\phi_{j}$, $j=1,2$ as the solutions to the initial value problems

$\displaystyle\begin{cases}L\phi_{1}=0,\\\
\phi_{1}(0)=1,\quad\phi_{1}^{\prime}(0)=0,\end{cases}\quad\text{and}\qquad\begin{cases}L\phi_{2}=0,\\\
\phi_{2}(0)=0,\quad\phi_{2}^{\prime}(0)=1\end{cases}$

and consider the Wronskian

(25) $\displaystyle W(x)\coloneqq\begin{pmatrix}\phi_{1}(x)&\phi_{2}(x)\\\
\phi^{\prime}_{1}(x)&\phi^{\prime}_{2}(x)\end{pmatrix}$

and the monodromy matrix

(26) $\displaystyle A\coloneqq
W(2\pi)=\begin{pmatrix}\phi_{1}(2\pi)&\phi_{2}(2\pi)\\\
\phi^{\prime}_{1}(2\pi)&\phi^{\prime}_{2}(2\pi)\end{pmatrix}.$

Then $\det A=1$ is the Wronskian determinant of the fundamental system
$\phi_{1},\phi_{2}$ and the Floquet multipliers
$\rho_{1,2}=\frac{1}{2}\left(\tr(A)\pm\sqrt{\tr(A)^{2}-4}\right)$ are the
eigenvalues of $A$ with corresponding eigenvectors
$v_{1}=(v_{1,1},v_{1,2})\in\mathbb{C}^{2}$ and
$v_{2}=(v_{2,1},v_{2,2})\in\mathbb{C}^{2}$. Thus,
$\Psi_{j}(x)=v_{j,1}\phi_{1}(x)+v_{j,2}\phi_{2}(x)$. By Hill’s theorem (see
[5]) we know that

$\displaystyle
0\in\sigma(L)\qquad\Leftrightarrow\qquad\absolutevalue{\tr(A)}\leq 2.$

Due to the assumption that $0\not\in\sigma(L)$ we see that $\rho_{1},\rho_{2}$
are real with $\rho_{1},\rho_{2}\in\mathbb{R}\setminus\\{-1,0,1\\}$ and
$\rho_{1}\rho_{2}=1$, i.e., one of the two Floquet multipliers has modulus
smaller then one and other one has modulus bigger than one. W.l.o.g. we assume
$0<|\rho_{2}|<1<|\rho_{1}|$. Furthermore, since $\rho_{1},\rho_{2}$ are real
and $A$ has real entries we can choose $v_{1},v_{2}$ to be real and so
$\Psi_{1},\Psi_{2}$ are both real valued. As a result we have found a real-
valued Bloch mode $\Psi_{2}(x)$ which is exponentially decreasing as
$x\to\infty$ due to $|\rho_{2}|<1$. Let us finally verify that
$\Psi_{2}(0)\not=0$ so that we may assume by rescaling that $\Psi_{2}(0)=1$.
Assume for contradiction that $\Psi_{2}(0)=0$. Since the potential $V(x)$ is
even in $x$ this implies that $\Psi_{2}$ is odd and hence (due to the
exponential decay at $+\infty$) in $L^{2}(\mathbb{R})$. But this contradicts
that $0\not\in\sigma(L)$.

Now we explain how the precise choice of the data $a,b>0,\Theta\in(0,1)$ and
$\omega$ for the step-potential $g$ in Theorem 1.2 allows to fulfill the
conditions (C1) and (C2). Let us define

$\displaystyle\tilde{g}(x)\coloneqq\begin{cases}a,&x\in[0,2\Theta\pi),\\\
b,&x\in(2\Theta\pi,2\pi).\end{cases}$

and extend $\tilde{g}$ as a $2\pi$-periodic function to $\mathbb{R}$. Then
$\tilde{g}(x)=g(x-\Theta\pi)$, and the corresponding exponentially decaying
Bloch modes $\tilde{\phi}_{k}$ and $\phi_{k}$ are similarly related by
$\tilde{\phi}_{k}(x)=\phi_{k}(x-\Theta\pi)$. For the computation of the
exponentially decaying Bloch modes, it is, however, more convenient to use the
definition $\tilde{g}$ instead of $g$.

Now we will calculate the monodromy matrix $A_{k}$ from (26) for the operator
$L_{k}$. For a constant value $c>0$ the solution of the initial value problem

$\displaystyle-\phi^{\prime\prime}(x)-k^{2}\omega^{2}c\phi(x)=0,\quad\phi(x_{0})=\alpha,\quad\phi^{\prime}(x_{0})=\beta$

is given by

$\displaystyle\begin{pmatrix}\phi(x)\\\
\phi^{\prime}(x)\end{pmatrix}=T_{k}(x-x_{0},c)\begin{pmatrix}\alpha\\\
\beta\end{pmatrix}$

with the propagation matrix

$\displaystyle
T_{k}(s,c)\coloneqq\begin{pmatrix}\cos(k\omega\sqrt{c}s)&\frac{1}{k\omega\sqrt{c}}\sin(k\omega\sqrt{c}s)\\\
-k\omega\sqrt{c}\sin(k\omega\sqrt{c}s)&\cos(k\omega\sqrt{c}s)\end{pmatrix}.$

Therefore we can write the Wronskian as follows

$\displaystyle W_{k}(x)$
$\displaystyle=\begin{cases}T_{k}(x,a)&x\in[0,2\Theta\pi]\\\
T_{k}(x-2\Theta\pi,b)T_{k}(2\Theta\pi,a)&x\in[2\Theta\pi,2\pi]\end{cases}$

and the monodromy matrix as

$\displaystyle A_{k}=W_{k}(2\pi)=T_{k}(2\pi(1-\Theta),b)T_{k}(2\Theta\pi,a).$

To get the exact form of $A_{k}$ let us use the notation

$\displaystyle l\coloneqq\sqrt{\frac{b}{a}}\,\frac{1-\Theta}{\Theta},\qquad
m\coloneqq 2\sqrt{a}\Theta\omega.$

Hence

$\displaystyle A_{k}=$ $\displaystyle\sin(kml\pi)\sin(km\pi)$
$\displaystyle\cdot\begin{pmatrix}\cot(kml\pi)\cot(km\pi)-\sqrt{\frac{a}{b}}&\frac{1}{k\omega\sqrt{a}}\cot(kml\pi)+\frac{1}{k\omega\sqrt{b}}\cot(km\pi)\\\
-k\omega\sqrt{b}\cot(km\pi)-k\omega\sqrt{a}\cot(kml\pi)&-\sqrt{\frac{b}{a}}+\cot(kml\pi)\cot(km\pi)\end{pmatrix}$

and

$\displaystyle\tr(A_{k})=2\cos(kml\pi)\cos(km\pi)-\Bigl{(}\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}\Bigr{)}\sin(kml\pi)\sin(km\pi).$

In order to verify (C1) we aim for $\absolutevalue{\tr(A_{k})}>2$. However,
instead of showing $\absolutevalue{\tr(A_{k})}>2$ for all
$k\in{\mathbb{Z}_{odd}}$ we may restrict to $k\in r\cdot{\mathbb{Z}_{odd}}$
for fixed $r\in{\mathbb{N}_{odd}}$ according to Remark 1.6. Next we will
choose $r\in{\mathbb{Z}_{odd}}$. Due to the assumptions from Theorem 1.2 we
have

(27) $l=\frac{\tilde{p}}{\tilde{q}},\leavevmode\nobreak\ \leavevmode\nobreak\
2m=\frac{p}{q}\in\frac{{\mathbb{N}_{odd}}}{{\mathbb{N}_{odd}}}.$

Therefore, by setting $r=\tilde{q}q$111Instead of $r=\tilde{q}q$ we may have
chosen any odd multiple of $\tilde{q}q$, e.g. $r=(\tilde{q}q)^{j}$ for any
$j\in\mathbb{N}$. This is important for the applicability of Theorem 4.1 to
obtain infinitely many breathers. we obtain $\cos(km\pi)=\cos(kml\pi)=0$ and
$\sin(km\pi),\sin(kml\pi)\in\\{\pm 1\\}$ for all $k\in
r\cdot{\mathbb{Z}_{odd}}$. Together with $a\not=b$ this implies
$|\tr(A_{k})|=\sqrt{\frac{a}{b}}+\sqrt{\frac{b}{a}}>2$ so that (C1) holds and
$A_{k}$ takes the simple diagonal form

$\displaystyle
A_{k}=\begin{pmatrix}-\sqrt{\frac{a}{b}}\sin(kml\pi)\sin(km\pi)&\\\
0&-\sqrt{\frac{b}{a}}\sin(kml\pi)\sin(km\pi).\end{pmatrix}$

In the following we assume w.l.o.g $0<a<b$, i.e., the Floquet exponent with
modulus less than $1$ is
$\rho_{k}=-\sqrt{\frac{a}{b}}\sin(kml\pi)\sin(km\pi)$. Note that
$|\rho_{k}|=\sqrt{a/b}$ is independent of $k$. Furthermore the Bloch mode
$\tilde{\phi}_{k}$ that is decaying to $0$ at $+\infty$ and normalized by
$\tilde{\phi}_{k}(\Theta\pi)=1$ is deduced from the upper left element of the
Wronskian, i.e.,

$\displaystyle\tilde{\phi}_{k}(x)=\frac{1}{\cos(k\omega\sqrt{a}\Theta\pi)}\left\\{\begin{array}[]{ll}\cos(k\omega\sqrt{a}x),&x\in(0,2\Theta\pi),\vspace{\jot}\\\
\cos(k\omega\sqrt{b}(x-2\Theta\pi))\cos(k\omega\sqrt{a}2\Theta\pi)&\\\
-\sqrt{\frac{a}{b}}\sin(k\omega\sqrt{b}(x-2\Theta\pi))\sin(k\omega\sqrt{a}2\Theta\pi),&x\in(2\Theta\pi,2\pi)\end{array}\right.$

and on shifted intervals of lengths $2\pi$ one has
$\tilde{\phi}_{k}(x+2m\pi)=\rho_{k}^{m}\tilde{\phi}_{k}(x)$. Notice that by
(27) the expression $k\omega\sqrt{a}\Theta\pi=k\frac{p}{q}\frac{\pi}{4}$ is an
odd multiple of $\pi/4$ since $k\in q\tilde{q}{\mathbb{Z}_{odd}}$ and hence
$|\cos(k\omega\sqrt{a}\Theta\pi)|=1/\sqrt{2}$. Therefore
$\|\phi_{k}\|_{L^{\infty}(0,\infty)}=\|\tilde{\phi}_{k}\|_{L^{\infty}(\Theta\pi,\infty)}\leq\|\tilde{\phi}_{k}\|_{L^{\infty}(0,2\pi)}\leq\sqrt{2}(1+\sqrt{a/b})$.
Thus we have shown that $|\phi_{k}(x)|\leq Me^{-\rho x}$ for $x\in[0,\infty)$
with $M>0$ and $\rho=\frac{1}{4\pi}(\ln b-\ln a)>0$. Finally, let us compute

$\displaystyle\phi_{k}^{\prime}(0)=\tilde{\phi}_{k}^{\prime}(\Theta\pi)=-k\omega\sqrt{a}\tan(k\omega\sqrt{a}\Theta\pi)\in\\{\pm
k\omega\sqrt{a}\\}.$

This shows that $|\phi_{k}^{\prime}(0)|=O(k)$ holds which allows to apply
Theorem 1.7. It also shows that the estimate
$|\phi_{k}(0)|=O(k^{\frac{3}{2}})$ from Lemma 2.3 can be improved in special
cases. To see that $\phi_{k}^{\prime}(0)$ is alternating in $k$, observe that
moving from $k\in r{\mathbb{Z}_{odd}}$ to $k+2r\in r{\mathbb{Z}_{odd}}$ the
argument of $\tan$ changes by $2r\omega\sqrt{a}\Theta\pi$ which is an odd
multiple of $\pi/2$. Since
$\tan(x+{\mathbb{Z}_{odd}}\frac{\pi}{2})=-1/\tan(x)$ we see that the sequence
$\phi_{k}^{\prime}(0)$ is alternating for $k\in r{\mathbb{Z}_{odd}}$. This
shows in particular that for any $j\in\mathbb{N}$ the sequence
$(\phi_{hr^{j}}^{\prime}(0))_{h\in{\mathbb{N}_{odd}}}$ contains infinitely
many positive and negative elements, and hence Theorem 4.1 for the existence
of infinitely many breathers is applicable. This concludes the proof Theorem
1.2 since we have shown that the potential $g$ satisfies the assumptions (C1)
and (C2) from Theorem 1.5.

###  6.3. Embedding of Hölder-spaces into Sobolev-spaces

######  Lemma 6.1.

For $0<\tilde{\nu}<\nu<1$ there is the continuous embedding
${C^{0,\nu}(\mathbb{T}_{T})}\to{H^{\tilde{\nu}}(\mathbb{T}_{T})}$.

###### Proof.

Let $z(t)=\sum_{k}\hat{z}_{k}e_{k}(t)$ be a function in
${C^{0,\nu}(\mathbb{T}_{T})}$. We need to show the finiteness of the spectral
norm $\|z\|_{H^{\tilde{\nu}}}$. For this we use the equivalence of the
spectral norm $\|\cdot\|_{H^{\tilde{\nu}}}$ with the Slobodeckij norm, cf.
Lemma 6.2. Therefore it suffices to check the estimate

$\displaystyle\int_{\mathbb{T}_{T}}\int_{\mathbb{T}_{T}}\frac{|z(t)-z(\tau)|^{2}}{|t-\tau|^{1+2\tilde{\nu}}}\differential{t}\differential{\tau}\leq\|z\|_{C^{\nu}(\mathbb{T}_{T})}^{2}\int_{\mathbb{T}_{T}}\int_{\mathbb{T}_{T}}|t-\tau|^{-1+2(\nu-\tilde{\nu})}\differential{t}\differential{\tau}\leq
C(\nu,\tilde{\nu})\|z\|_{C^{\nu}(\mathbb{T}_{T})}^{2}$

where the double integral is finite due to $\nu>\tilde{\nu}$. ∎

For $0<s<1$ recall the definition of the Slobodeckij-seminorm for a function
$z:\mathbb{T}_{T}\to\mathbb{R}$

$\displaystyle[z]_{s}\coloneqq\left(\int_{\mathbb{T}_{T}}\int_{\mathbb{T}_{T}}\frac{|z(t)-z(\tau)|^{2}}{|t-\tau|^{1+2s}}\differential{t}\differential{\tau}\right)^{1/2}.$

######  Lemma 6.2.

For functions $z\in H^{s}(\mathbb{T}_{T})$, $0<s<1$ the spectral norm
$\|z\|_{H^{s}}=(\sum_{k}(1+k^{2})^{s}|\hat{z}_{k}|^{2})^{1/2}$ and the
Solobodeckij norm ${\interleave
z\interleave}_{H^{s}}\coloneqq(\|z\|_{L^{2}(\mathbb{T}_{T})}^{2}+[z]_{s}^{2})^{1/2}$
are equivalent.

###### Proof.

The Solobodeckij space and the spectrally defined fractional Sobolev space are
both Hilbert spaces. Hence, by the open mapping theorem, if suffices to verify
the estimate ${\interleave z\interleave}_{H^{s}}\leq C\|z\|_{H^{s}}$. By
direct computation we get

$\displaystyle\int_{\mathbb{T}_{T}}\int_{\mathbb{T}_{T}}\frac{|z(t)-z(\tau)|^{2}}{|t-\tau|^{1+2s}}\differential{t}\differential{\tau}$
$\displaystyle=\int_{0}^{T}\int_{-\tau}^{T-\tau}\frac{|z(x+\tau)-z(\tau)|^{2}}{|x|^{1+2s}}\differential{x}\differential{\tau}$
$\displaystyle=\int_{0}^{T}\left(\int_{0}^{T-\tau}\frac{|z(x+\tau)-z(\tau)|^{2}}{x^{1+2s}}\differential{x}+\int_{T-\tau}^{T}\frac{|z(x+\tau)-z(\tau)|^{2}}{(T-x)^{1+2s}}\differential{x}\right)\differential{\tau}$
$\displaystyle=\int_{0}^{T}\int_{0}^{T}\frac{|z(x+\tau)-z(\tau)|^{2}}{g(x,\tau)^{1+2s}}\differential{x}\differential{\tau}$

with

$\displaystyle g(x,\tau)=\left\\{\begin{array}[]{ll}x&\mbox{ if }0\leq x\leq
T-\tau,\vspace{\jot}\\\ T-x&\mbox{ if }T-\tau\leq x\leq T.\end{array}\right.$

Since $g(x,\tau)\geq\operatorname{dist}(x,\partial\mathbb{T}_{T})$ and due to
Parseval’s identity we find

$\displaystyle\int_{\mathbb{T}_{T}}\int_{\mathbb{T}_{T}}\frac{|z(t)-z(\tau)|^{2}}{|t-\tau|^{1+2s}}\differential{t}\differential{\tau}$
$\displaystyle\leq\int_{\mathbb{T}_{T}}\frac{\|\widehat{z(\cdot+x)}-\hat{z}\|_{l^{2}}^{2}}{\operatorname{dist}(x,\partial\mathbb{T}_{T})^{1+2s}}\differential{x}$
$\displaystyle=\int_{\mathbb{T}_{T}}\sum_{k}\frac{|\exp(\mathrm{i}k\omega
x)-1|^{2}|\hat{z}_{k}|^{2}}{\operatorname{dist}(x,\partial\mathbb{T}_{T})^{1+2s}}\differential{x}$
$\displaystyle=4\int_{0}^{T/2}\sum_{k}\frac{1-\cos(k\omega
x)}{x^{1+2s}}|\hat{z}_{k}|^{2}\differential{x}$ $\displaystyle\leq
4\tilde{C}\sum_{k}k^{2s}|\hat{z}_{k}|^{2}$

with
$\tilde{C}=\int_{0}^{\infty}\frac{1-\cos(\omega\xi)}{\xi^{1+2s}}\differential{\xi}$.
This finishes the proof. ∎

## Acknowledgment

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Project-ID 258734477 – SFB 1173

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