# Spectral stability of shock-fronted travelling waves under viscous
relaxation

Ian Lizarraga and Robert Marangell School of Mathematics and Statistics, The
University of Sydney<EMAIL_ADDRESS>(corresponding author)School
of Mathematics and Statistics, The University of Sydney,
<EMAIL_ADDRESS>
###### Abstract

Reaction-nonlinear diffusion partial differential equations can exhibit shock-
fronted travelling wave solutions. Prior work by Li et al. (2021) has
demonstrated the existence of such waves for two classes of regularisations,
including viscous relaxation (see [22]). Their analysis uses geometric
singular perturbation theory: for sufficiently small values of a parameter
$\varepsilon>0$ characterising the ‘strength’ of the regularisation, the waves
are constructed as perturbations of a singular heteroclinic orbit. Here we
show rigorously that these waves are spectrally stable for the case of viscous
relaxation. Our approach is to show that for sufficiently small
$\varepsilon>0$, the ‘full’ eigenvalue problem of the regularised system is
controlled by a reduced slow eigenvalue problem defined for $\varepsilon=0$.
In the course of our proof, we examine the ways in which this geometric
construction complements and differs from constructions of other reduced
eigenvalue problems that are known in the wave stability literature.

###### Contents

  1. 1 Introduction
  2. 2 Existence of travelling waves
    1. 2.1 The travelling wave equations
    2. 2.2 Fast and slow singular limits
    3. 2.3 Travelling waves for $0<\varepsilon\ll 1$
  3. 3 Geometric preliminaries
  4. 4 Spatial eigenvalue problems
    1. 4.1 Asymptotic and far-field hyperbolicity
    2. 4.2 Complex 2-plane dynamics induced by the eigenvalue problem
    3. 4.3 Reduced eigenvalue problems
  5. 5 Essential & Absolute Spectrum
  6. 6 Augmented unstable bundles
    1. 6.1 Construction of the augmented unstable bundle $\mathcal{E}_{\varepsilon}(K)$ for $\varepsilon>0$
    2. 6.2 An outline of the proof of the stability theorem
    3. 6.3 Slow eigenvalues and the jump map
    4. 6.4 The construction of $\mathcal{E}_{0}(K)$
  7. 7 Comparison of the bundles $\mathcal{E}_{\varepsilon}(K)$ and $\mathcal{E}_{0}(K)$
    1. 7.1 Preliminary estimates
      1. 7.1.1 Fast and slow elephant trunks over $S^{a,\pm}_{\varepsilon}$
      2. 7.1.2 Exchange lemma-type estimates
    2. 7.2 Estimates near the slow subbundle
    3. 7.3 The jump map as a singular limit
    4. 7.4 Control from the reduced dynamics
    5. 7.5 Continuing $\mathcal{E}_{\varepsilon}(K)$ to $\mathcal{E}_{0}(K)$
  8. 8 Counting the slow eigenvalues
  9. 9 Concluding remarks
  10. A Construction of the fast and slow elephant trunks
  11. B Lemmas for estimates near the slow subbundle
  12. C Proof of Lemma 7.7
  13. D Example for modified exchange lemma estimates

## 1 Introduction

Regularised PDEs exhibiting travelling wave solutions can frequently be
regarded as singularly perturbed dynamical systems, since the regularisation
is modelled by the inclusion of strictly higher-order partial derivatives
multiplied by a small constant. The singular perturbation in turn has a
physical/dynamical interpretation reflecting an inherent hierarchy of
spatiotemporal scales, which we refer to more succinctly as a fast-slow
structure. We consider the following reaction-nonlinear diffusion PDE with a
viscous relaxation term, exhibiting (regularised) shock-fronted travelling
waves:111Throughout the paper we use bar notation to refer to phase space
variables. Later on we will use unbarred variables to denote variables defined
on the linearized subspaces along the curve, and carets to denote
projectivisations of these variables.

$\displaystyle\frac{\partial\bar{U}}{\partial t}$
$\displaystyle=\frac{\partial}{\partial
x}\left(D(\bar{U})\frac{\partial\bar{U}}{\partial
x}\right)+R(\bar{U})+\varepsilon\frac{\partial^{3}\bar{U}}{\partial
x^{2}\partial t}$ (1)

for $(x,t)\in\mathbb{R}\times\mathbb{R}$ and $\varepsilon\geq 0$ the singular
perturbation parameter characterising the strength of the the regularisation.
The nonlinear diffusion and reaction terms given by the quadratic resp. cubic
polynomials $D(\bar{U})$ and $R(\bar{U})$. We suppose that $D(\bar{U})<0$
within an interval $(a,b)\subset(0,1)$; the potential function
$F(\bar{U}):=\int D(\bar{U})d\bar{U}$ is hence nonmonotone. Nonlinear
diffusion processes with nonmonotone potential functions serve as prototypical
models for the formation of coherent solutions with sharp fronts; see e.g.
[37, 38]. Arguably the canonical system from a physical context is the Cahn-
Hilliard model [27]. The system (1) for $\varepsilon=0$ is derived as a
particular continuum limit of stochastic agent-based models of invasion
processes [31, 32]. The choice of viscous regularisation in (1) appears in the
shock regularisation literature [25, 26, 38], but there are other physically-
motivated high-order regularising terms that give rise to (smoothed families
of) shock-fronted travelling waves. We refer the reader to [22] for a
comparison of the existence problems under viscous resp. nonlocal regularising
terms relative to system (1).

In the case of existence, the effectiveness of geometric singular perturbation
theory (GSPT) is now well established. When (1) is written in a coordinate
frame that follows the travelling wave, it takes the structure of a closed
fast-slow system of ODEs. A one-parameter family of travelling waves for
$0<\varepsilon\ll 1$ is then constructed rigorously as a perturbation of a
singular heteroclinic orbit, composed of segments defined by the slow flow
along a so-called critical manifold, concatenated with fast jumps along the
fast fibres, according to the dynamics of the layer problem. Li et. al. follow
this approach in [22], numerically demonstrating the existence of smooth
travelling waves of (1) for small values of $\varepsilon>0$. These one-
parameter families of waves limit to ‘genuine’ (piecewise continuous) shock-
fronted waves as $\varepsilon\to 0$; furthermore, such singular limits are
nonunique, strongly depending on the regularisation chosen. Li and coauthors
have also studied smooth travelling-wave solutions of the unperturbed problem
arising as so-called ‘hole in the wall’ solutions; see [23] and the footnote
on page 9. Generally speaking, one can attempt to construct shock-fronted
limiting solutions directly as weak solutions, but there may be infinitely
many; see eg. [13].

It is worth re-emphasizing the effect of the singular perturbation in (1), in
terms of coordinate representations of the travelling wave. With respect to a
natural Liénard-type representation, the unperturbed problem can be thought of
as modelling the slow dynamics on the critical manifold itself. The
introduction of a regularisation term is then tantamount to embedding this
slow flow within a higher-dimensional space and introducing a rule (i.e. the
fast layer flow) for connecting the wavefront smoothly. In [22], it is
explicitly shown that viscous relaxation adds a one-dimensional layer flow,
while a fourth-order nonlocal regularization term adds a two-dimensional layer
flow.

The objective of this paper is to demonstrate the spectral stability of
travelling waves arising in (1). To determine the spectral stability along a
wave, we must find the spectrum $\sigma(L)$ of a corresponding linearized
operator $L$. The total spectrum is decomposed into its continuous and point
components as $\sigma(L)=\sigma_{e}(L)\cup\sigma_{p}(L)$. Typically, most of
the work involves deducing the point spectrum $\sigma_{p}(L)$, by recasting
the eigenvalue problem $(L-\lambda I)v=0$ as a bifurcation problem posed on
the underlying linearized subspaces along the wave. The major point that we
want to highlight is that although the existence problem remains amenable to
the usual GSPT techniques, the analysis of the stability problem (in
particular, the computation of the point spectrum) is quite distinct.

The literature on geometric and analytic techniques for linearized stability
problems is substantial (see e.g. [18] and [28] for comprehensive surveys). In
this paper we follow the geometric framework developed in a series of seminal
papers by Alexander, Gardner, and Jones [1, 8, 14]. The key point is that $L$
also inherits a slow-fast structure, enabling a reduction via linearized
subsystems defined along either the slow manifold or along the fast fibres.
The typical approach is then to hope that the spectrum of the full system, for
$\varepsilon>0$ sufficiently small, is close to the ‘fast’ and/or ‘slow’
spectra of these linearized subsystems, which are defined for $\varepsilon=0$.

Our principal result is Theorem 2.11. As a consequence of the abovementioned
high order of the singular perturbation term representing the regularisation,
the stability problem for small $\varepsilon>0$ is now close to a slow
eigenvalue problem defined on the critical manifold. We give an outline of the
proof in Section 6.2. To summarize, we show in Sections 6 and 7 that the
eigenvalues for the perturbed problem are the same as the eigenvalues for the
reduced problem, which are calculated in Section 8. Together with the fact
that the essential spectrum lies in the left half complex plane as shown in
Section 5, we can conclude that there is no spectrum of solutions to (1) in
any fixed contour $K$ in the right half complex plane.

The principal estimate for constructing these reduced eigenvalue problems is
the elephant trunk lemma, which characterises an attracting set over a fast
unstable subbundle in the linearized space when $\lambda$ is not a fast
eigenvalue; see the construction in the paper of Gardner & Jones [8]. We
highlight that a nondegenerate fast eigenvalue problem is a requirement to
construct linked elephant trunks over the entire wave. Furthermore, the
construction of a slow eigenvalue problem, corresponding to the singular limit
of some slow subbundle, is essentially auxiliary to that of the fast problem,
and many of the key properties of the slow subbundle are indirectly enforced
by the elephant trunk over the fast subbundle.

In order for us to define the promised slow eigenvalue problem, it is
essential that we determine how exactly slow linear data on one branch of the
slow manifold is transported across fast fibres to another branch of the slow
manifold. When an elephant trunk lemma is available, it enforces certain nice
properties, such as continuity of the slow subbundle across the fast layer in
the singular limit. In other words, slow linear data from one branch of the
slow manifolds is transported across fast fibres to the other branch
identically, allowing a straightforward definition of slow eigenvalue problems
across disjoint subsets of the slow manifolds.

We will show that our problem does not allow for elephant trunk-type estimates
over the fast layer. The main obstruction is that, relative to a natural set
of coordinates for the linearized system so that setting $\lambda=0$ recovers
the usual variational equations along the flow, the parameter $\lambda$ enters
the equations ‘weakly,’ i.e. only through $\mathcal{O}(\varepsilon\lambda)$
terms. The singular limit of the fast linearized system therefore degenerates
such that there is ‘always a fast eigenvalue,’ i.e. there is a connection made
between unstable and stable directions across the wavefront for all
$\lambda\in\mathbb{C}$ as $\varepsilon\to 0$. In the present case the
eigenvalue problem is low-dimensional enough that we can partly avoid this
problem by concentrating on the slow problem only, but we return to this issue
in the conclusion when we consider fourth-order regularizations.

Consequently, it must be the case that the slow linearized dynamics is
responsible for the generation of eigenvalues near the tails of the wave. In
this context, we think of our work as complementing Jones’ early result on the
stability of a travelling pulse in the FitzHugh-Nagumo system, in which the
eigenvalue problem of the full system is strongly controlled by the fast
eigenvalue problems along the wave front and back [14]; as well as the result
of Gardner and Jones on the stability of travelling waves in predator-prey
systems, where both fast and slow eigenvalue problems are constructed [8].

While we do not have access to a fast elephant trunk over the entire wave, we
nevertheless retain control of the linearized dynamics for small values of
$\varepsilon$ since $\lambda$ enters the equations only weakly. The key
observation here is that the eigenvalue problem can be thought of as a ‘weak’
($\mathcal{O}(\varepsilon)$)-perturbation of the standard variational problem.
We will show that the corresponding transport of slow data is not the identity
map, but instead is replaced by a nontrivial jump map that we can write down
explicitly in the limit as $\varepsilon\to 0$. The key technical tools that we
use are desingularized linearized slow flows; $\varepsilon$-dependent
rescalings along the fast fibres; and a continuous differentiability criterion
that holds across the fold. We remark that this analysis holds also for the
$\lambda=0$ case (corresponding to the regular variational dynamics carried by
the wave), but to our knowledge such a statement about jump compatibility is
new in the GSPT literature.

The exchange lemma, another fundamental GSPT technique (see e.g. [16,
17]),222Strictly speaking, in this paper we consider the case of an
inclination lemma (see e.g. [5, 30]). We also point out that the adapation of
the standard $(k+\sigma)$-exchange lemma to the case of zero unstable fast
directions is straightforward. can also be adapted to our eigenvalue problem,
since it is $\mathcal{O}(\varepsilon)$-close to the standard variational
problem. Exchange lemma-type estimates allow us to track solutions of the
eigenvalue problem as they leave the fast layer and enter small neighborhoods
of the slow manifolds, where they enter partial elephant trunks and remain
well-controlled along the tails of the wave. The key result here is that
solutions entering neighborhoods of the slow manifolds in a generic way will
be aligned closely to a slow subbundle defined on the slow manifold after
sufficiently long times. This subbundle is defined from the eigenvalue
problem, and is therefore not generally equivalent to the tangent bundle of
the slow manifold (but it is nearby).

These new techniques cause no extra trouble for the topological arguments that
allow us to calculate the point spectrum of the perturbed problem: the
corresponding evaluation of first Chern numbers of certain augmented unstable
bundles (defined in [1]) follows, as usual. Following the approach of Gardner
& Jones in [8], we construct a homotopy between the augmented unstable bundle
$\mathcal{E}_{\varepsilon}(K)$ of the ‘full’ problem with $\varepsilon>0$ and
that of a reduced problem $\mathcal{E}_{0}(K)$, when $\varepsilon$ is
sufficiently small. The reduced vector bundle is defined over separate
hemispheres using the slow eigenvalue problem. The jump map now plays an
essential role in defining the appropriate clutching function, which glues
these vector bundles together along the edges of the hemispheres. We also
highlight that the translational eigenvalue at $\lambda=0$ is now counted by
the slow problem.

In the course of proving stability in our problem, we also demonstrate
explicitly the geometric consequence of eigenvalue crossings in the full
system for $0<\varepsilon\ll 1$: the projectivized solutions wind around
according to the number of eigenvalues crossed, and these winds occur entirely
within the slow dynamics of the critical manifold. The propagation of winding
in the reduced problem, as well as the generation of new winds (eigenvalues)
can be identified by locating the poles and zeroes, respectively, of a
meromorphic Riccati-Evans function [10], defined with respect to a judiciously
chosen section of the projective dynamics. This also gives us another
opportunity to demonstrate the computational utility of the Riccati-Evans
function to count eigenvalues via the argument principle.

We wish to highlight some results in the literature in the context of our
present problem. Slow eigenvalues appear in a doubly-diffusive Fitzhugh-Nagumo
system which is considered as an early application of the augmented unstable
bundle theory in [1], but here the extra fast dynamics is relatively trivial;
furthermore, the ‘slow’ problem is in fact fast-slow relative to another small
parameter, with its eigenvalues arising from the corresponding fast front and
back. The occurrence of such a nontrivial slow eigenvalue problem as described
in our current analysis appears to be quite atypical; we highlight the work of
Bose [3] in this vein as one of the only comparable examples we have found in
the literature. Bose provides similar geometrical results for pulse
(meta)stability in singularly-perturbed nonlocal RDEs. The singular
perturbation occurs at the second order, so the resulting nonlocal eigenvalue
problem can then be studied with traditional Sturm-Liouville techniques, and
an oscillation theorem is proven. Relative to Bose’s result, we have two
additional technical issues to settle: (i) we must first ‘reduce’ our
singularly-perturbed third-order eigenvalue problem (see (24)) by showing
closeness to a second-order eigenvalue problem defined on the critical
manifold, and (ii) we must also find the corresponding compatibility condition
to connect the eigenvalue problem across disjoint branches of the critical
manifold.

We also wish to highlight de Rijk et. al.’s recent work on analytical slow-
fast factorizations of Evans functions [6]; the relationship between the
geometric obstructions discussed above and analytic conditions to produce
reduced Evans functions in de Rijk et. al.’s paper bears further exploration,
but we do not attempt this in our paper.

We find it sensible to closely follow the general structure in [8] in
organizing our paper. In Sec. 2, we describe the construction of a one-
parameter family of travelling waves limiting onto a singular shock-fronted
travelling wave using geometric singular perturbation theory. In Sec. 3, we
define the relevant geometric spaces in which our objects of interest lie and
write down some key facts about them. In Sec. 4 we write down the spatial
eigenvalue problem for the associated linearized operator of (1), and we write
down some leading-order estimates for the eigenvalues and eigenvectors of the
associated asymptotic systems. We also define the linearized slow and fast
subsystems that we will use often to estimate the linear dynamics of the
‘full’ system. In Sec. 5, we describe some general stability theory for
travelling waves and show that the essential spectrum is bounded away from the
imaginary axis in the left-half complex plane. In Sec. 6, we construct the
augmented unstable vector bundles over a complex contour enclosing the
remaining point spectrum, and we also take the opportunity to outline the
proof of the main stability theorem in the paper. In Sec. 7, we prove the main
uniform estimates for the linearized dynamics in terms of the singular
subsystems. We use these estimates to construct a homotopy between some
augmented unstable bundles. In Sec. 8, we characterize the point spectrum of
the reduced subsystem, finally allowing us to deduce the stability of the wave
of the full system for $\varepsilon>0$ sufficiently small. We conclude in Sec.
9, and the most technical lemmas are relegated to the Appendices.

## 2 Existence of travelling waves

Our first task is to construct constant-speed travelling waves for the PDE
(1). In this section we summarise the setup and analysis in [22]. We highlight
that the techniques and terminology used here are entirely standard in the
context of GSPT; we point to the usual references [15, 21] for general
definitions.

### 2.1 The travelling wave equations

The PDE (1) is expressed in terms of the frame $(\zeta,t^{\prime})=(x-ct,t)$
(where $c$ is a constant parameterizing the wavespeed) is given by

$\displaystyle\bar{U}_{t^{\prime}}$
$\displaystyle=c\bar{U}_{\zeta}+\frac{\partial}{\partial\zeta}\left(D(\bar{U})\frac{\partial\bar{U}}{\partial\zeta}\right)+R(\bar{U})+\varepsilon\left(\frac{\partial^{3}\bar{U}}{\partial\zeta^{2}\partial
t^{\prime}}-c\frac{\partial^{3}\bar{U}}{\partial\zeta^{3}}\right).$ (2)

Li et al. (2021) provide the precise conditions necessary on $D(\bar{U})$ and
$R(\bar{U})$ for the existence of a shock-fronted travelling wave solution. We
refer the reader to [22] for an in-depth discussion about the modelling
assumptions and potential generalisations underlying these definitions. Here
we continue with the definitions used by [22] in their computations for the
sake of consistency: we take a quadratic nonlinear diffusion term

$\displaystyle D(\bar{U})$
$\displaystyle=6\left(\bar{U}-7/12\right)\left(\bar{U}-3/4\right)$ (3)

and cubic reaction term

$\displaystyle R(\bar{U})$ $\displaystyle=5\bar{U}(1-\bar{U})(\bar{U}-1/5).$
(4)

We also record the following potential function (i.e. integral) of
$D(\bar{U})$, which will be used to define the vector field in the travelling
wave frame:

$\displaystyle F(\bar{U})$
$\displaystyle=2\bar{U}^{3}-4\bar{U}^{2}+\frac{21}{8}\bar{U}.$ (5)

Note that the diffusion term is negative within the range $7/12<\bar{U}<3/4$,
and the reaction term is pinned at $\bar{U}=0$ and $\bar{U}=1$.

A travelling wave solution is found as a steady-state to the above equation:

$\displaystyle-R(\bar{U})$ $\displaystyle=\left(c\bar{U}+\frac{\partial
F(\bar{U})}{\partial\zeta}-\varepsilon
c\frac{\partial^{2}\bar{U}}{\partial\zeta^{2}}\right)_{\zeta}.$ (6)

Letting

$\displaystyle\bar{P}$ $\displaystyle=c\bar{U}+\frac{\partial
F}{\partial\zeta}-\varepsilon c\frac{\partial^{2}\bar{U}}{\partial\zeta^{2}}$
$\displaystyle\bar{V}$ $\displaystyle=F(\bar{U})-\varepsilon
c\bar{U}_{\zeta},$

we arrive at the following slow travelling wave equations (where
$\dot{~{}}:=d/d\zeta$)

$\displaystyle\varepsilon\dot{\bar{U}}$
$\displaystyle=\frac{1}{c}\left(F(\bar{U})-\bar{V}\right)$
$\displaystyle\dot{\bar{P}}$ $\displaystyle=-R(\bar{U})$ (7)
$\displaystyle\dot{\bar{V}}$ $\displaystyle=\bar{P}-c\bar{U}.$

We also record the equivalent fast travelling wave equations in terms of the
stretched variable $\xi:=\zeta/\varepsilon$:333From now on we use lowercase
letters throughout to distinguish the variables and functions defined with
respect to the stretched scaling, following the convention in [8].

$\displaystyle\bar{u}^{\prime}$
$\displaystyle=\frac{1}{c}\left(F(\bar{u})-\bar{v}\right)$
$\displaystyle\bar{p}^{\prime}$ $\displaystyle=-\varepsilon R(\bar{u})$ (8)
$\displaystyle\bar{v}^{\prime}$ $\displaystyle=\varepsilon(\bar{p}-c\bar{u}),$

where ${~{}}^{\prime}:=d/d\xi$. From the definitions of $R$ and $F$, the
systems (7)–(8) evidently admit three fixed points for $\varepsilon>0$. Two of
them are

$\displaystyle\bar{z}^{-}$
$\displaystyle=(\bar{u}^{-},\bar{p}^{-},\bar{v}^{-})=(0,0,0)$ (9)
$\displaystyle\bar{z}^{+}$
$\displaystyle=(\bar{u}^{+},\bar{p}^{+},\bar{v}^{+})=(1,c,5/8),$

lying on $S^{a,-}_{0}$ and $S^{a,+}_{0}$, respectively, and the third is the
point $\bar{z}^{m}=(0.2,0.2c,0.381)$. Travelling waves of (1) linking the
states $\bar{U}=0$ and $\bar{U}=1$ correspond to heteroclinic connections
between the fixed points $\bar{z}^{-}$ and $\bar{z}^{+}$.

Since the systems (7)–(8) are odd-dimensional, it is convenient to choose an
orientation that minimizes the dimension of the unstable manifolds
$W^{u}(\bar{z}^{-})$ and $W^{u}(\bar{z}^{+})$. We therefore apply the
orientation reversal $\zeta\mapsto-\zeta$ and $\xi\mapsto-\xi$ in the
subsequent analysis and work henceforth with the pair of systems

$\displaystyle\varepsilon\dot{\bar{U}}$
$\displaystyle=\frac{1}{c}\left(\bar{V}-F(\bar{U})\right)$
$\displaystyle\dot{\bar{P}}$ $\displaystyle=R(\bar{U})$ (10)
$\displaystyle\dot{\bar{V}}$ $\displaystyle=c\bar{U}-\bar{P}$

and

$\displaystyle\bar{u}^{\prime}$
$\displaystyle=\frac{1}{c}\left(\bar{v}-F(\bar{u})\right)$
$\displaystyle\bar{p}^{\prime}$ $\displaystyle=\varepsilon R(\bar{u})$ (11)
$\displaystyle\bar{v}^{\prime}$ $\displaystyle=\varepsilon(c\bar{u}-\bar{p}).$

Let us now assess the linear stability of the fixed points in (9).

######  Lemma 2.1.

Let $c>0$ be fixed. For each sufficiently small $\varepsilon>0$, the fixed
points $\bar{z}^{-}$ and $\bar{z}^{+}$ of (11) are both saddle-type
equilibria. Furthermore, the three eigenvalues of the linearisation of (11)
evaluated at both $\bar{z}^{-}$ and $\bar{z}^{+}$ are all real, having the
hierarchy

$\displaystyle\mu_{f}\ll\mu_{s,1}<0<\mu_{s,2}.$ (12)

Specifically, with respect to the scaling in (11) we have
$\mu_{f}=\mathcal{O}(1)$ and $\mu_{s,i}=\mathcal{O}(\varepsilon)$ where
$i=1,2$.

Proof: We verify the spectral hierarchy for $\bar{z}^{-}$; the steps for
$\bar{z}^{+}$ are identical. Let $f(\bar{u},\bar{p},\bar{v})$ denote the
vector field of (11). The associated Jacobian matrix is

$\displaystyle Df(\bar{u},\bar{p},\bar{v})$
$\displaystyle=\begin{pmatrix}-\frac{D(\bar{u})}{c}&0&\frac{1}{c}\\\
\varepsilon R^{\prime}(\bar{u})&0&0\\\ \varepsilon
c&-\varepsilon&0\end{pmatrix}.$ (13)

At $\bar{z}^{-}$ we have $D(0)=21/8$ and $R^{\prime}(0)=-1$. The
characteristic polynomial of $Df(\bar{z}^{-})$ is

$\displaystyle p(\mu)$
$\displaystyle=-\mu^{3}-\frac{21}{8c}\mu^{2}+\varepsilon\mu+\frac{1}{c^{2}}\varepsilon^{2}.$

Therefore, the eigenvalue of largest magnitude has the expansion

$\displaystyle\mu_{f}(\varepsilon)$
$\displaystyle=\frac{-21}{8c}+\mathcal{O}(\varepsilon).$ (14)

Thus $\mu_{f}(\varepsilon)<0$ and uniformly bounded away from 0 for
$\varepsilon$ sufficiently small when $c>0$. To estimate the remaining two
$\mathcal{O}(\varepsilon)$ eigenvalues $\mu_{s,i}$, $i=1,2$, we consider the
scaling $\mu=\varepsilon\nu$. Then

$\displaystyle p(\nu)$
$\displaystyle=-\varepsilon^{3}\nu^{3}-\frac{21}{8c}\varepsilon^{2}\nu^{2}+\varepsilon^{2}\nu+\frac{1}{c^{2}}\varepsilon^{2}.$

Dividing a factor of $\varepsilon^{2}$, regular perturbation theory again
gives the solution of $p(\nu)=0$ in orders of $\varepsilon$ as

$\displaystyle\nu_{\pm}$
$\displaystyle=\frac{2}{21}\left(2\pm\sqrt{4+\frac{42}{c^{2}}}\right)+\mathcal{O}(\varepsilon).$
(15)

Since the roots limit to distinct real values as $\varepsilon\to 0$, the pair
must perturb to distinct real values. Furthermore
$\mu_{s,1}=\varepsilon\nu_{-}<0<\varepsilon\nu_{+}=\mu_{s,2}$ for each
$\varepsilon>0$ sufficiently small. $\Box$

Following standard GSPT, we now define two subproblems which characterize the
slow and fast singular limits of (10) and (11). These are used to define a
singular heteroclinic orbit by concatenating solutions of these subsystems.

### 2.2 Fast and slow singular limits

We first characterize the fast dynamics by considering the singular limit
$\varepsilon\to 0$ of (11).

######  Definition 2.2.

The layer problem of (11) is given by

$\displaystyle\bar{u}^{\prime}$
$\displaystyle=\frac{1}{c}\left(\bar{v}-F(\bar{u})\right)$
$\displaystyle\bar{p}^{\prime}$ $\displaystyle=0$ (16)
$\displaystyle\bar{v}^{\prime}$ $\displaystyle=0.$

$\triangle$

In this limit, the slow variables $\bar{p},\bar{v}$ are constant,
parametrising the fast fiber bundle; away from the zero set of $u^{\prime}$,
the layer problem then specifies the one-dimensional fast dynamics fiberwise.

######  Definition 2.3.

The critical manifold is
$S_{0}=\\{(\bar{u},\bar{p},\bar{v}):\,\bar{v}=F(\bar{u})\\}$. We decompose
$S_{0}$ into the pieces

$\displaystyle S_{0}$ $\displaystyle=S^{a,-}_{0}\cup F_{-}\cup S^{r}_{0}\cup
F_{+}\cup S^{a,+}_{0},$

with

$\displaystyle S^{a,-}_{0}$ $\displaystyle=S_{0}\cap\\{\bar{u}<7/12\\}$
$\displaystyle F_{-}$ $\displaystyle=S_{0}\cap\\{\bar{u}=\bar{u}_{F}:=7/12\\}$
$\displaystyle S^{r}_{0}$ $\displaystyle=S_{0}\cap\\{7/12<\bar{u}<3/4\\}$
$\displaystyle F_{+}$ $\displaystyle=S_{0}\cap\\{\bar{u}=3/4\\}$
$\displaystyle S^{a,+}_{0}$ $\displaystyle=S_{0}\cap\\{\bar{u}>3/4\\}.$

Here, $F_{-}$ and $F_{+}$ denote resp. the left and right folds of the cubic
manifold $S_{0}$, where $D(\bar{U})$ vanishes. We also define the jump-on
curve $J_{a}\subset S^{a,+}_{0}$, which is the horizontal projection of the
fold $F_{-}$ onto $S^{a,+}_{0}$:

$\displaystyle J_{a}$ $\displaystyle=S_{0}\cap\\{\bar{u}=\bar{u}_{J}:=5/6\\}.$

These sets are drawn in Fig. 1(a). $\triangle$

The Jacobian matrix of (16) has two trivial zero eigenvalues for each $x\in
S_{0}$, corresponding to the locally two-dimensional tangent space at each
point. The sign of the remaining nontrivial eigenvalue determines the dynamics
along the fast fibers nearby.

######  Definition 2.4.

We say that $S_{0}$ is normally hyperbolic at $x\in S_{0}$ if its nontrivial
eigenvalue $\lambda$ does not lie on the imaginary axis. Furthermore, we say
that $S_{0}$ is normally hyperbolic attracting if $\lambda<0$ and normally
hyperbolic repelling if $\lambda>0$. $\triangle$

The nontrivial eigenvalue of the linearization of (16) is

$\displaystyle\lambda$ $\displaystyle=-\frac{D(\bar{u})}{c},$

from which it immediately follows that $S^{a,-}_{0}$ and $S^{a,+}_{0}$ are
normally hyperbolic attracting and $S^{r}_{0}$ is normally hyperbolic
repelling when $c>0$; see Fig. 1(b). Well-known theorems of Fenichel [7]
specify the existence of normally hyperbolic invariant slow manifolds
$S^{a,\pm}_{\varepsilon}$ and $S^{r}_{\varepsilon}$ and fast fiber bundles
near to compact normally hyperbolic subsets of their singular counterparts
$S^{a,\pm}_{0}$ resp. $S^{r}_{0}$.

(a)
(b)

Figure 1: (a) Depiction of the critical manifold $S_{0}$ from Def. 2.3 in
$(\bar{u},\bar{p},\bar{v})$-space. (b) Dynamics of the layer problem (11) away
from $S_{0}$, projected onto the $(\bar{u},\bar{p})$ plane.

The critical manifold loses normal hyperbolicity via a simple zero eigenvalue
crossing at the folds $F_{\pm}$. Slow dynamics may be defined on $S_{0}$ away
from $F_{\pm}$ by considering the singular limit of (10). Differentiating both
sides of $\bar{V}=F(\bar{U})$ with respect to $\zeta$ we have

$\displaystyle\dot{\bar{V}}$ $\displaystyle=D(\bar{U})\dot{\bar{U}}.$ (17)

Solutions of the reduced problem blow up as they approach $F_{\pm}$, and so we
rescale $\zeta$ by the function $D(\bar{U})$ to extend solutions of the
reduced problem smoothly across the folds. Altogether, we define the following
two systems.

######  Definition 2.5.

The reduced problem defined on $S_{0}$ is given by

$\displaystyle D(\bar{U})\dot{\bar{U}}$ $\displaystyle=c\bar{U}-\bar{P}$ (18)
$\displaystyle\dot{\bar{P}}$ $\displaystyle=R(\bar{U}).$

Rescaling $\zeta$ by the diffusion $D(\bar{U})$, we obtain the
(desingularized) slow flow

$\displaystyle\dot{\bar{U}}$ $\displaystyle=c\bar{U}-\bar{P}$ (19)
$\displaystyle\dot{\bar{P}}$ $\displaystyle=R(\bar{U})D(\bar{U}).$

$\triangle$

The slow flow extends the solutions of the reduced problem across the lines of
singularities given by $F_{\pm}$ in finite time. Note that although
topological equivalence of solutions of (18) and (19) is assured on
$S^{a,\pm}_{0}$ and $S^{r}_{0}$, there is an orientation reversal on
$S^{r}_{0}$ owing to the change in sign of $D(\bar{U})$.

The slow flow (19) admits five fixed points corresponding to the zeroes of
$R(\bar{U})$ and $D(\bar{U})$. A calculation of the eigenvalues of the
Jacobian of (19) shows that the fixed points

$\displaystyle\bar{Z}^{-}$ $\displaystyle=(0,0)$ (20)
$\displaystyle\bar{Z}^{+}$ $\displaystyle=(1,c)$

are saddle-type when $c>0$. These correspond to the singular limits of the
fixed points (9) of the full system as $\varepsilon\to 0$.

######  Remark 2.6.

The set $F_{-}$ consists mostly of so-called regular (or generic) fold points;
we refer the reader to the precise definition in [21].444There are also
isolated canard points, corresponding to folded singularities where
trajectories beginning near an attracting branch of a slow manifold are able
to cross over to the repelling branch, tracking it for $\mathcal{O}(1)$ time.
In the travelling wave literature such points are referred to as ‘holes in the
wall,’ relating to the observation that $D(\bar{U})=0$ defines ‘walls’ of
singularities corresponding to the fold lines; see [9], [23], and [36]. One of
the seminal achievements of GSPT is to rigorously characterise the flow of
typical orbits across folds, where Fenichel theory breaks down. This requires
extending the slow manifolds (which a priori exist only over compact normally
hyperbolic subsets of the critical manifold) across such folded singularities
using geometric blow-up theory; see e.g. [20, 34, 35]. $\triangle$

We are interested in the existence of a singular heteroclinic orbit that
connects $\bar{Z}^{-}$ to $\bar{Z}^{+}$ by jumping across a regular fold point
on $F_{-}$. This singular orbit is a hybrid object defined by concatenating
solutions of (16) and (19).

######  Hypothesis 2.7.

There exists a singular heteroclinic orbit $\Gamma_{0}$ for $c=c_{0}>0$,
connecting $\bar{U}=0$ to $\bar{U}=1$. This orbit is formally defined as the
concatenation of the following solution segments:

  * •

the portion of the unstable manifold $W^{u}(\bar{Z}^{-})$ which connects
$\bar{Z}^{-}$ to a regular fold point $(7/12,\bar{p}_{f})$ on $F_{-}$,
corresponding to a unique trajectory $X_{R}(\zeta)$ of (19) defined on
$\zeta\in(-\infty,0]$

  * •

the horizontal fast fiber connecting $W^{u}(\bar{Z}^{-})\cap F_{-}$ to a point
$(5/6,\bar{p}_{f},\bar{v}_{f})$ on the jump curve $J_{a}$, corresponding to a
unique trajectory $x_{R}(\xi)$ of (16) defined on $\xi\in(-\infty,\infty)$,
and

  * •

the portion of the stable manifold $W^{s}(\bar{Z}^{+})$ which connects
$(5/6,\bar{p}_{f})$ to $\bar{Z}^{+}$, corresponding to a unique trajectory
$X_{R}(\zeta)$ of (19) defined on $\zeta\in[0,\infty)$. We also suppose that
the slow flow is transverse to $J_{a}$ at $W^{s}(\bar{Z}^{+})\cap J_{a}$.

We furthermore suppose that the heteroclinic connection $\Gamma_{0}$ is
transversal with respect to variation in $c$; i.e. extending the reduced
systems with the equation $c^{\prime}=0$, we have
$\Gamma_{0}=W^{u}(\bar{Z}^{-},c)\mathrel{\text{\vbox{
\halign{#\cr\smash{$-$}\crcr$\pitchfork$\crcr} }}}W^{s}(\bar{Z}^{+},c)$, where
$W^{u}(\bar{Z}^{-},c)=\cup_{c}W^{u}(\bar{Z}^{-}(c))$ and
$W^{s}(\bar{Z}^{+},c)=\cup_{c}W^{s}(\bar{Z}^{+}(c))$ denote, respectively, the
extended two-dimensional center-unstable (resp. three-dimensional center-
stable) manifolds, continued across the fast jump in the usual way.

Figure 2: Segments of $W^{u}(\bar{Z}^{-})$ and $W^{s}(\bar{Z}^{+})$ arising
from the slow flow (19)–(20), as $c$ is varied within the range $[0.19,0.23]$.
The stable manifolds are extended slightly past $J_{a}$ to illustrate the
transversality of the slow flow at $J_{a}$. The singular homoclinic orbit
$\Gamma_{0}$ is depicted by the red curve for $c_{0}\approx 0.199362$. The
blue arrows indicate how the (un)stable manifolds vary as $c$ is increased.
The magenta dashed line denotes the $\bar{U}$-nullcline for $c=c_{0}$.

Hypothesis 2.7 can be readily verified numerically; this was done already in
[22]. Here we report a singular heteroclinic connection for $c_{0}\approx
0.199362$ which is clearly transversal with respect to variation in $c$; see
Fig. 2. Based on the relative location of $\Gamma_{0}$ with respect to the
$\bar{U}$-nullcline, we also record the following monotonicity hypothesis.

######  Hypothesis 2.8.

The portions of $\Gamma_{0}$ (from Hypothesis 2.7) lying in $S^{a,\pm}_{0}$
are strictly monotone in $\bar{U}$; i.e. $\dot{\bar{U}}(\zeta)>0$
(equivalently, $\dot{\bar{V}}(\zeta)>0$) for all $\zeta<0$ on $S_{0}^{a,-}$
and for all $\zeta>0$ on $S_{0}^{a,+}$.

This monotonicity condition turns out to be useful when analyzing the reduced
eigenvalue problem introduced in Sec. 8: it is equivalent to the statement
that the projectivized dynamics of the slow coordinates lives entirely on one
chart of $\mathbb{CP}$ specified by $S=(P/V)$, $V\neq 0$, when the temporal
eigenvalue parameter $\lambda=0$. Here $P$ and $V$ denote respectively the
linearisations of $\bar{P}$ and $\bar{V}$.555Our notation is equivalent to the
“$\delta$” convention to denote linearized variables, i.e. $V=\delta\bar{V}$.

### 2.3 Travelling waves for $0<\varepsilon\ll 1$

We now use hypothesis 2.7, to show the existence of a one-parameter family of
travelling waves $\\{\Gamma_{\varepsilon}\\}$ for sufficiently small values of
$\varepsilon>0$; we write it in such a way as to make clear the uniform
convergence onto the singular wave $\Gamma_{0}$.

######  Lemma 2.9.

For any $\delta>0$, there exists $\bar{\varepsilon}>0$ such that for each
$\varepsilon\in(0,\bar{\varepsilon})$, there exists a wavespeed
$c(\varepsilon)$ and a heteroclinic orbit $\Gamma_{\varepsilon}(\xi)$ for (11)
connecting $\bar{z}^{-}$ to $\bar{z}^{+}$ with
$d_{H}(\Gamma_{\varepsilon},\Gamma_{0})<\delta$ and
$|c(\varepsilon)-c_{0}|<\delta$.

Figure 3:  Numerical approximation of a heteroclinic orbit (black solid curve)
connecting the saddle equilibria $(0,0,0)$ and $(1,c,F(1))$ for
$(\varepsilon,c)=(0.001,0.20637)$, together with the singular heteroclinic
orbit $\Gamma_{0}$ (red dashed curve) from Hypothesis 2.7.

Shock-fronted travelling waves for small fixed values of $\varepsilon>0$ are
numerically approximated in [22]; see also Fig. 3. We sketch a geometrical
proof showing that Hypothesis 2.7 implies that Lemma 2.9 holds. Our sketch
follows standard arguments for constructing heteroclinic orbits by estimating
the relative orientations of slow manifolds and their fast fiber bundles for
$\varepsilon>0$ small, see eg. [33]. Here we also address the issue that the
jump occurs across a regular fold, necessitating a slightly different
estimate.

Proof: Fix $\alpha>0$ small enough that $\Gamma_{0}$ is the unique transverse
singular heteroclinic orbit for $c\in
B_{\alpha}:=(c_{0}-\alpha,c_{0}+\alpha)$. We also fix a section which
intersects $\Gamma_{0}$ in its interior in the middle of the jump, say
$\Sigma=\\{\bar{u}=0.7\\}$. For $\varepsilon>0$ sufficiently small, Fenichel
theory provides the existence of slow manifolds $S^{a,-}_{\varepsilon}$ and
$S^{a,+}_{\varepsilon}$, which are $\mathcal{O}(\varepsilon)$-close in
Hausdorff distance to compact normally hyperbolic subsets of $S^{a,-}_{0}$ and
$S^{a,+}_{0}$.

The slow manifold $S^{a,-}_{\varepsilon}$ can be continuously extended across
a regular fold of a two-dimensional critical manifold in $\mathbb{R}^{3}$
using blow-up theory; after crossing the fold, $S^{a,-}_{\varepsilon}$ lies
$\mathcal{O}(\varepsilon^{2/3})$ close to the singular fast fiber subbundle
$\mathscr{J}$ extending from the fold $F_{-}$ to the jump-on curve $J_{a}$, at
the section $\Sigma$ (see Theorem 1 in [34] for details). The slow manifold
$S^{a,-}$ then flows along the fast fiber bundle, reaching $\Sigma$ in
$\mathcal{O}(1)$ time with respect to the scaling (11) (this intersection must
exist for $\varepsilon>0$ sufficiently small since $\Gamma_{0}$ intersects
$\Sigma$ by construction). Thus, $S^{a,-}_{\varepsilon}$ remains
$\mathcal{O}(\varepsilon^{2/3})$-close to $\mathscr{J}$ at the point of
intersection on $\Sigma$ .

On the other hand, consider a ball $V$ inside $W=\cup_{c\in
B_{\alpha}}W^{s}(\bar{z}^{-}(c))$ which straddles $J_{a}$. Fenichel theory
implies that the (nonlinear) fast fiber bundle over the basepoint set $V$
perturbs to an $O(\varepsilon)$-close fast fiber bundle over the corresponding
basepoints now given on $S^{a,+}_{\varepsilon}$ [7]. Extend this fiber bundle
backwards to $\Sigma$. Therefore at $\Sigma$, the transverse intersection of
the projection of $F$ with the stable fast fiber bundle over $S^{a,+}_{0}$
persists over a small range in $\varepsilon$, and for each such $\varepsilon$
the intersection occurs at a corresponding wavespeed $c(\varepsilon)$.
Evaluated at $\Sigma$, the perturbed intersection is
$\mathcal{O}(\varepsilon^{2/3})$-close to the original intersection in the
extended $(\bar{u},\bar{p},\bar{v},c)$-space, i.e. $c(\varepsilon)\to c_{0}$
as $\varepsilon\to 0$. $\Box$

######  Remark 2.10.

We highlight a high-dimensional generalization of this construction, due to
Lin and Wechselberger [24]. They prove a generalization of Theorem 1 in [34],
and they also require exchange/lambda lemma-type estimates to pick out unique
waves from two-parameter families. In our case, the transversal intersection
$\Gamma_{\varepsilon}$ for each corresponding wavespeed $c(\varepsilon)$ must
be unique, which can be verified directly with dimension counting. $\triangle$

We are now able to state the main theorem proved in this paper.

######  Theorem 2.11.

Suppose Hypotheses 2.7, and 2.8 hold. Fix a contour $K$ in the right half of
the complex plane. Then there exists $\bar{\varepsilon}>0$ so that for each
$0<\varepsilon\leq\bar{\varepsilon}$, the wave $\Gamma_{\varepsilon}$ does not
have spectrum inside of $K$.

The proof of this theorem is developed mostly in the second half of the paper;
see Sec. 6.2 for a general outline of the strategy. A characterization of the
spectral stability of a travelling wave must first be developed. This is
accomplished by the end of Sec. 6.

We finish this section with a preliminary estimate showing that we have strong
control over the dynamics of the wave via the reduced subsystems away from the
transition sets $F_{-}$ and $J_{a}$; c.f. Corollary 2.2 in [8].

######  Corollary 2.12.

Given $\varepsilon>0$ sufficiently small, let
$X(\zeta,\varepsilon)=x(\xi,\varepsilon)$ parametrize the travelling wave
solution from Hypothesis 2.9 corresponding to the wavespeed
$c=c(\varepsilon)$. The following are true for each fixed $a>0$:

$\displaystyle\lim_{\varepsilon\to 0}x(\xi,\varepsilon)$
$\displaystyle=x_{R}(\xi)\,\text{ uniformly for }|\xi|\leq a$
$\displaystyle\lim_{\varepsilon\to 0}x(\zeta,\varepsilon)$
$\displaystyle=X_{R}(\zeta)\text{ uniformly for }|\zeta|\geq a.$

In the second limit, we take $X_{R}$ to mean the embedding of the
corresponding parametrised solutions of $\Gamma_{0}|_{S^{a,-}_{0}\cup
S^{a,+}_{0}}$ into $\mathbb{R}^{3}$ via the map
$(\bar{U},\bar{P})\mapsto(\bar{U},\bar{P},F(\bar{U}))\in S_{0}$.

Proof: We sketch the argument for the first uniform limit. Fix an $a>0$. Then
$|x(\xi,\varepsilon)-x_{R}(\xi)|$ can be uniformly bounded through a
combination of Gronwall’s inequality and the distance estimate in Hypothesis
2.9. The argument for the second uniform limit is similar. $\Box$

## 3 Geometric preliminaries

Ahead of our analysis in the remaining sections, we explicitly describe the
geometric spaces in which our objects of interest live. We will track the
dynamics of fibers of complex vector bundles, corresponding to points in
$\mathbf{Gr}(k,n)$, the Grassmannian of complex $k$-planes in
$\mathbb{C}^{n}$. The general set-up is standard: we will study dynamics on
the Grassmannian indirectly via embedded systems on complex projective space.

We first introduce some essential notation for complex projective space. The
space $\mathbb{CP}^{n}$ is the quotient space
$(\mathbb{C}^{n+1}-\\{0\\})/\sim{}$ subject to the equivalence relation that
identifies complex rays, i.e. $y_{1},y_{2}\in\mathbb{C}^{n+1}-\\{0\\}$ are
identified if $y_{1}=\alpha y_{2}$ for some $\alpha\in\mathbb{C}$. We denote
the projection map by $\pi:\mathbb{C}^{n+1}-\\{0\\}\to\mathbb{CP}^{n}$, and
for each $y\in\mathbb{C}^{n}$ we use the notation $[y]$ or $\hat{y}$ for the
image $\pi(y)\in\mathbb{CP}^{n}$ (this notation also holds for sets
$S\subset\mathbb{C}^{n}-\\{0\\}$).

Later, in Section 7.2 we will use the following elementary lemma to interpret
metric neighborhoods on $\mathbb{CP}^{2}$ in terms of those on
$\mathbb{C}^{3}$.

######  Lemma 3.1.

Let $\rho$ be any metric on $\mathbb{CP}^{n}$. Then there is a constant $K>0$,
depending only on $\rho$, such that the following is true for each $\delta>0$:

If $y\in\mathbb{C}^{n+1}$ is a complex vector with unit modulus and if
$P\subset\mathbb{C}^{n+1}$ is a complex $n$-plane passing through the origin
with $\hat{y}\notin N_{\delta}(\hat{P})$, then

$|w|\geq K\delta,$

where $w$ is the component of $y$ lying orthogonal to $P$, and
$N_{\delta}(\hat{P})$ denotes the $\delta$-neighborhood of $\hat{P}$ with
respect to $\rho$, i.e.
$N_{\delta}(\hat{P})=\\{\hat{x}\in\mathbb{CP}^{n}:\rho(\hat{x},\hat{p})<\delta\text{
for some }\hat{p}\in\hat{P}\\}.$

######  Remark 3.2.

The norm $|x|:=\sqrt{\bar{x}^{T}x}$ in Lemma 3.1 is the usual Euclidean norm,
and the orthogonal component is defined using the Hermitian inner product
$x\cdot_{\mathbb{C}}y:=\sum_{i=1}^{n}x_{i}\bar{y}_{i}$. Essentially the lemma
says that if a vector is inclined far away from a plane, then the component of
the vector lying orthogonal to the plane cannot be too small, up to a scaling
factor that depends on the details of how this inclination is measured. This
lemma was implicitly used in Lemma 5.4 in [8] for the case $n=3$, so we
provide an explicit proof for arbitrary $n$. $\triangle$

Proof of Lemma 3.1: Suppose that there is no such uniform constant for some
fixed choice of $\delta>0$. Then we can find a sequence $\\{y_{k}\\}$ lying on
the unit sphere in $\mathbb{C}^{n+1}$ such that for each $k$, we have
$\hat{y}_{k}\notin N_{\delta}(\hat{P})$ and $|w_{k}|<\delta/k$, where $w_{k}$
denotes the orthogonal projection of $y_{k}$ with respect to $P$. Choose a
convergent subsequence $y_{k}\to y_{*}$, relabeling $k$ if necessary. Then
$y_{*}$ clearly has zero component in the direction orthogonal to $P$,
implying that $y_{*}\in P$ and hence that $\hat{y}_{*}\in
N_{\delta}(\hat{P})$. On the other hand, we have
$\hat{y}_{k}\to\hat{y}_{*}\notin N_{\delta}(\hat{P})$ since the quotient map
is continuous and the complement of $N_{\delta}(\hat{P})$ is closed, giving a
contradiction. $\Box$

Let $v_{1},\cdots,v_{k}\in\mathbb{C}^{n}$ denote a spanning set of vectors for
the $k$-plane $W$. The well-known Plücker embedding

$\displaystyle\mathbf{Gr}(k,n)$
$\displaystyle\xhookrightarrow{\psi}\mathbb{P}(\bigwedge^{k}\mathbb{C}^{n})\cong\mathbb{CP}^{\binom{n}{k}-1}$
$\displaystyle\psi(W)$ $\displaystyle:=[v_{1}\wedge\cdots\wedge v_{k}]$

provides homogeneous coordinates for the Grassmannian, called Plücker
coordinates. The image $\psi(W)$ can be coordinatized by writing down the
maximal minors/cofactors of the $(n\times k)$ matrix
$M_{W}=[v_{1}~{}\cdots~{}v_{k}]$.

Example. Consider a complex 2-plane $W\in\mathbf{Gr}(2,3)$ with spanning
complex vectors $v_{1}=(a_{1},b_{1},c_{1})^{T}$ and
$v_{2}=(a_{2},b_{2},c_{2})^{T}$. A coordinate representation of $[v_{1}\wedge
v_{2}]\in\mathbb{C}\mathbb{P}^{2}$ is given by the list of $2\times 2$
cofactors of the $3\times 2$ matrix $[v_{1}~{}~{}v_{2}]$, namely

$\displaystyle N_{W}:=\begin{pmatrix}b_{1}c_{2}-c_{1}b_{2}\\\
a_{2}c_{1}-a_{1}c_{2}\\\ a_{1}b_{2}-a_{2}b_{1}\end{pmatrix}.$

Observe that the Plücker coordinate representation $N_{W}$ can be interpreted
as a parametrisation of the plane $W$ by a (nontrivial) normal vector.

######  Remark 3.3.

The Plücker coordinates are generally redundant: we have $\binom{n}{k}$ of
them, whereas the dimension of $\mathbf{Gr}(k,n)$ is $k(n-k)$. In this paper
our analysis is restricted to the cases $\mathbf{Gr}(1,3)$ and
$\mathbf{Gr}(2,3)$, since we will be occupied with tracking complex line
(resp. plane) bundles. In the case $\mathbf{Gr}(1,3)$, the redundancy arises
from nonunique scalings of the complex ray. In the case $\mathbf{Gr}(2,3)$, an
analogous scaling nonuniqueness arises when choosing a normal vector to
represent a complex plane (see the example above). $\triangle$

The Grassmannian spaces are metrizable. Here we construct an explicit metric,
because we will need a way to measure ‘angles’ between lines and planes. Using
a complex version of the construction given in [17], we take advantage of the
Plücker embedding to equip $\mathbf{Gr}(k,n)$ with the Fubini-Study metric.
Complex projective space can be embedded into the space of Hermitian
$(n+1)\times(n+1)$ matrices $\mathcal{H}$ via the following isometric
immersion:

$\displaystyle\mathbb{CP}^{n}$
$\displaystyle\xhookrightarrow{\varphi}\mathcal{H}$
$\displaystyle\varphi([x])$ $\displaystyle:=\frac{x\bar{x}^{T}}{|x|^{2}},$

where $x\in\mathbb{C}^{n+1}-\\{0\\}$ represents the ray $\mathbb{C}x$. The
space of Hermitian matrices is in turn endowed with an inner product given by

$\displaystyle d(P,Q)$ $\displaystyle=\frac{1}{2}\text{Tr}(PQ),$

which allows us to define a norm $\lVert P\rVert=d(P,P)^{1/2}$, and hence a
metric induced by the norm given by $\text{dist}(P,Q):=\lVert P-Q\rVert$. We
can pull back this metric on $\mathcal{H}$ to one on $\mathbb{CP}^{n}$, and
hence on the Grassmannian:

$\displaystyle\text{dist}([x],[y])$
$\displaystyle:=\lVert\frac{x\bar{x}^{T}}{|x|^{2}}-\frac{y\bar{y}^{T}}{|y|^{2}}\rVert$
$\displaystyle=\sqrt{\frac{1}{2}\text{Tr}\left[\left(\frac{x\bar{x}}{|x|^{2}}-\frac{y\bar{y}^{T}}{|y|^{2}}\right)^{2}\right]}$
$\displaystyle=\sqrt{1-\frac{|(x\cdot_{\mathbb{C}}y)|^{2}}{|x|^{2}|y|^{2}}}.$

We observe that $\text{dist}([x],[y])$ recovers the sine of the so-called
Hermitian angle [29] between a corresponding pair of nontrivial complex
vectors representing $[x],\,[y]$. For example, this metric on
$\mathbf{Gr}(2,3)$ measures the Hermitian angle between a pair of normal
vectors parametrising a corresponding pair of complex $2$-planes.

We will use the Plücker embedding to study (linear) dynamics which describe
the evolution of $k$-planes in the Grassmannian. Concretely, let
$y^{\prime}=b(\xi)y$ denote a nonautonomous linear system on
$\mathbb{C}^{n+1}$. This linear system induces a nonlinear flow on
$\mathbb{CP}^{n}$, and hence on the corresponding Grassmannian, which we
denote by

$\displaystyle\hat{y}^{\prime}$ $\displaystyle=\hat{b}(\hat{y},\xi).$ (21)

We mention two elementary facts about about these induced flows. The first is
that $y$ is an eigenvector of $b(\xi)$ if and only if $\hat{y}$ is a critical
point of $\hat{b}(\hat{y},\xi)$. The second is that if $\\{\lambda_{i}\\}$
denotes the eigenvalues of the linearization of a (constant) matrix $b$ and if
$y_{i}$ is an eigenvector corresponding to the eigenvalue $\lambda_{i}$, then
the eigenvalues of the linearization of $\hat{b}$ near the corresponding
critical point $\hat{y}_{i}$ are $\lambda_{j}-\lambda_{i}$ for $j\neq i$. As a
consequence of these facts, if the largest eigenvalue of a linear system is
simple, then the corresponding eigendirection in the projectivized system is a
stable fixed point.

We will find it useful to study the system (21) in its own right as a
nonlinear flow on appropriate charts on $\mathbb{CP}^{n}$, but it is also
convenient to treat the natural system $y^{\prime}=b(\xi)y$ as the Plücker
coordinate representation of the system on projective space. To clarify this,
we introduce the notion of induced variational fields. We repeat the general
treatment of [17], Sec. 3.2. Let $V$ be a vector space of dimension $n$ and
let $1\leq\sigma\leq n$. Given a linear map $A:V\to V$, an induced derivation
of order $\sigma$ is the multilinear map
$A^{\sigma}:\bigwedge^{\sigma}V\to\bigwedge^{\sigma}V$ defined by

$\displaystyle v_{1}\wedge\cdots\wedge
v_{\sigma}\mapsto\sum_{i=1}^{\sigma}v_{1}\wedge\cdots\wedge
Av_{i}\wedge\cdots\wedge v_{\sigma}.$ (22)

We have

######  Lemma 3.4.

Let $\\{\lambda_{i}\\}_{i=1}^{n}$ denote the set of eigenvalues of
$A^{(1)}=A$. Then the set of eigenvalues of $A^{(2)}$ is given by
$\\{\lambda_{i}+\lambda_{j}\\}_{i<j,i=1}^{i=n}$.

Proof: Choose an ordered basis for $V$ consisting of eigenvectors of $A$, and
use it to construct an ordered basis for $\bigwedge^{2}V$. The conclusion
follows from a straightforward calculation using the definition (22). $\Box$

If $x^{\prime}=Ax$ is a linear vector field, then $P^{\prime}=A^{(\sigma)}P$
defines a linear vector field on the $\sigma$th exterior power space. By using
the Plücker embedding, the latter induced vector field can be interpreted as
evolving points in $\mathbf{Gr}(\sigma,n)$ (see Sec. 3.2 in [17] for a
complete description of this construction in the real setting; the extension
to complex $k$-planes is direct).

## 4 Spatial eigenvalue problems

We now assume that Hypotheses 2.7, 2.8, and 2.9 hold, and we proceed to define
the spatial eigenvalue problem associated with the one-parameter family of
waves $(\Gamma_{\varepsilon},c({\varepsilon}))$. Let us fix $\varepsilon>0$
sufficiently small and represent $\Gamma_{\varepsilon}$ by the solution
$\bar{X}(\zeta,\varepsilon)=(\bar{U}(\zeta,\varepsilon),\bar{P}(\zeta,\varepsilon),\bar{V}(\zeta,\varepsilon))$.
We consider perturbations of the form

$\displaystyle\tilde{U}(\zeta,t)$
$\displaystyle=\bar{U}(\zeta,\varepsilon)+\delta e^{\lambda
t}U(\zeta)+\mathcal{O}(\delta^{2}).$ (23)

Assuming that $\tilde{U}(\zeta,t)$ solves (1) and applying the fact that
$\bar{U}(\zeta,\varepsilon)$ is a stationary solution of (2), we arrive at the
eigenvalue problem by collecting $\mathcal{O}(\delta)$ terms:

$\displaystyle
cU_{\zeta}+(D(\bar{U})U)_{\zeta\zeta}+R^{\prime}(\bar{U})U+\varepsilon(\lambda
U_{\zeta\zeta}-cU_{\zeta\zeta\zeta})$ $\displaystyle=\lambda U.$ (24)

Collecting derivatives and defining

$\displaystyle V$ $\displaystyle=D(\bar{U})U+\varepsilon(\lambda
U-cU_{\zeta})$ $\displaystyle P$ $\displaystyle=cU+V_{\zeta},$

we obtain the following closed three-dimensional system:

$\displaystyle\varepsilon\dot{U}$
$\displaystyle=\frac{1}{c}\left((\varepsilon\lambda+D(\bar{U}))U-V\right)$
$\displaystyle\dot{P}$ $\displaystyle=(\lambda-R^{\prime}(\bar{U}))U$ (25)
$\displaystyle\dot{V}$ $\displaystyle=P-cU.$

In the stretched scaling $\xi=\zeta/\varepsilon$, we obtain

$\displaystyle u^{\prime}$
$\displaystyle=\frac{1}{c}\left(-v+(\varepsilon\lambda+D(\bar{u}))u\right)$
$\displaystyle p^{\prime}$
$\displaystyle=\varepsilon(\lambda-R^{\prime}(\bar{u}))u$ (26) $\displaystyle
v^{\prime}$ $\displaystyle=\varepsilon(p-cu).$

In view of the orientation reversal giving the pair of wave equations
(10)–(11), we will instead work with the following pair of linearized systems
throughout the paper:

$\displaystyle\varepsilon\dot{U}$
$\displaystyle=\frac{1}{c}(V-(\varepsilon\lambda+D(\bar{U}))U)$
$\displaystyle\dot{P}$ $\displaystyle=(R^{\prime}(\bar{U})-\lambda)U$ (27)
$\displaystyle\dot{V}$ $\displaystyle=cU-P.$

and

$\displaystyle u^{\prime}$
$\displaystyle=\frac{1}{c}(v-(\varepsilon\lambda+D(\bar{u}))u)$ $\displaystyle
p^{\prime}$ $\displaystyle=\varepsilon(R^{\prime}(\bar{u})-\lambda)u$ (28)
$\displaystyle v^{\prime}$ $\displaystyle=\varepsilon(cu-p).$

######  Remark 4.1.

Setting $\lambda=0$ in (27)–(28) gives the standard variational equations of
(10)–(11) along the wave $(\Gamma_{\varepsilon},c(\varepsilon))$ for each
$\varepsilon>0$. In particular, we can account for the existence of the
translational eigenvalue; the variational equations are satisfied by a
nontrivial uniformly bounded solution, namely the derivative of the wave (i.e.
the vector field). $\triangle$

Collecting $y=(u,p,v)$ and $Y=(U,P,V)$, we write the pair of systems (27)–(28)
more compactly in terms of the matrices $A,a$:

$\displaystyle\dot{Y}$ $\displaystyle=A(\bar{U},\lambda,\varepsilon)Y$ (29)
$\displaystyle y^{\prime}$ $\displaystyle=a(\bar{u},\lambda,\varepsilon)y.$
(30)

We end this section by recording the fast projectivized eigenvalue problem
$\hat{y}=\hat{a}(\bar{u},\lambda,\varepsilon)$ corresponding to the linear
eigenvalue problem (28). This system is defined on the space
$\mathbb{CP}^{2}$; with respect to the chart $(\beta_{1},\beta_{2})=(p/u,v/u)$
with $u\neq 0$, we have

$\displaystyle\beta_{1}^{\prime}$
$\displaystyle=\frac{1}{c}(D(\bar{u})\beta_{1}-\beta_{1}\beta_{2})+\varepsilon(R^{\prime}(\bar{u})+\lambda(\beta_{1}/c-1))$
(31) $\displaystyle\beta_{2}^{\prime}$
$\displaystyle=\frac{1}{c}(D(\bar{u})\beta_{2}-\beta_{2}^{2})+\varepsilon\left(c-\beta_{1}+\frac{\lambda}{c}\beta_{2}\right).$

Slow projectivized eigenvalue problems can be similarly defined using (27). We
record for future convenience the following system defined with respect to the
chart $(\beta_{1},\beta_{2})=(U/V,P/V)$ with $V\neq 0$:

$\displaystyle\varepsilon\dot{\beta_{1}}$
$\displaystyle=\frac{1}{c}\left(1-D(\bar{U})\beta_{1}\right)+\varepsilon(\beta_{1}\beta_{2}-c\beta_{1}^{2}-(\lambda/c)\beta_{1})$
(32) $\displaystyle\dot{\beta_{2}}$
$\displaystyle=\beta_{2}^{2}-c\beta_{1}\beta_{2}+(R^{\prime}(\bar{U})-\lambda)\beta_{1}.$

### 4.1 Asymptotic and far-field hyperbolicity

We turn to the asymptotic systems associated with (29)–(30). Broadly speaking,
these systems encode ‘far-field’ information about the wave; for instance,
they are useful in determining the boundaries of the essential spectrum (see
[18]). The matrices $a(\xi,\lambda,\varepsilon)$ and
$A(\zeta(\xi),\lambda,\varepsilon)$ tend to the limits
$a^{\pm}(\lambda,\varepsilon)$ and $A^{\pm}(\lambda,\varepsilon)$ as
$\xi\to\pm\infty$. Here the minus (resp. plus) superscript corresponds to an
evaluation at $\bar{z}^{-}$ (resp. $\bar{z}^{+}$). We have

$\displaystyle a^{\pm}(\lambda,\varepsilon)$
$\displaystyle=\begin{pmatrix}-\frac{1}{c}(D(u^{\pm})+\varepsilon\lambda)&0&\frac{1}{c}\\\
\varepsilon(R^{\prime}(u^{\pm})-\lambda)&0&0\\\ \varepsilon
c&-\varepsilon&0\end{pmatrix},$ (33) $\displaystyle
A^{\pm}(\lambda,\varepsilon)$
$\displaystyle=(1/\varepsilon)a^{\pm}(\lambda,\varepsilon).$

Note from (4) that $R^{\prime}(1)<R^{\prime}(0)<0$. Let $\beta$ be some fixed
real number with $R^{\prime}(0)<\beta<0$ and define

$\displaystyle\Omega$
$\displaystyle=\\{\lambda\in\mathbb{C}:\text{Re}\,\lambda>\beta\\}.$ (34)

We will show in Sec. 5 that the essential spectrum lies entirely in the left-
half complex plane, and bounded away from $\Omega$. We now record some
estimates for the eigenvalues and eigenvectors of the asymptotic matrices (33)
for any $\lambda\in\Omega$, which will be important in determining the point
spectrum.

######  Lemma 4.2.

Let $\lambda\in\Omega$. Then for $\varepsilon>0$ sufficiently small, the
asymptotic matrices (33) have three distinct eigenvalues
$\mu^{\pm}_{j}(\lambda,\varepsilon)$ for $a^{\pm}(\lambda,\varepsilon)$ (resp.
$\Delta^{\pm}_{j}(\lambda,\varepsilon)$ for $A^{\pm}(\lambda,\varepsilon)$),
which satisfy

$\displaystyle\text{Re}\,\mu^{\pm}_{f}\ll\text{Re}\,\mu^{\pm}_{s,1}<$
$\displaystyle\,0<\text{Re}\,\mu^{\pm}_{s,2}$ (35)
$\displaystyle\Delta^{\pm}_{j}$ $\displaystyle=(1/\varepsilon)\mu^{\pm}_{j}.$

In particular, we have $\mu^{\pm}_{f}=\mathcal{O}(1)$ and
$\mu^{\pm}_{s,j}=\mathcal{O}(\varepsilon)$, $j=1,2$. Furthermore, let
$e_{f}^{\pm}(\lambda,\varepsilon)$ and $e_{s,j}^{\pm}(\lambda,\varepsilon)$
denote the eigenvectors associated with $\mu^{\pm}_{f}$ and $\mu^{\pm}_{s,j}$,
respectively, where $j=1,2$. Then each $e^{\pm}(\lambda,\varepsilon)$ limits
to a ‘reduced’ eigenvector $r^{\pm}(\lambda)$ as $\varepsilon\to 0$. These
reduced eigenvectors are given explicitly by

$\displaystyle r_{f}^{\pm}$ $\displaystyle=(1,0,0)^{\top}$ (36) $\displaystyle
r_{s,1}^{\pm}$ $\displaystyle=(1/D(\bar{u}^{\pm}),\nu_{p,\pm},1)^{\top}$
$\displaystyle r_{s,2}^{\pm}$
$\displaystyle=(1/D(\bar{u}^{\pm}),\nu_{m,\pm},1)^{\top},$

where

$\displaystyle\nu_{p,\pm}$
$\displaystyle=\frac{c_{0}+\sqrt{c_{0}^{2}+4D(\bar{u}^{\pm})(\lambda-R^{\prime}(\bar{u}^{\pm}))}}{2D(\bar{u}^{\pm})}$
(37) $\displaystyle\nu_{m,\pm}$
$\displaystyle=\frac{c_{0}-\sqrt{c_{0}^{2}+4D(\bar{u}^{\pm})(\lambda-R^{\prime}(\bar{u}^{\pm}))}}{2D(\bar{u}^{\pm})}.$

######  Remark 4.3.

The smallness of $\varepsilon$ required for Lemma 4.2 is dependent on
$\lambda$, but a uniform bound $\varepsilon\leq\bar{\varepsilon}$ over closed,
bounded contours in $\mathbb{C}$ can be extracted using compactness.
$\triangle$

Proof of Lemma 4.2: We outline the proof as it is similar to the proof of
Lemma 2.1. The characteristic polynomial of $a^{\pm}(\lambda,\varepsilon)$ is

$\displaystyle p^{\pm}(\mu)$
$\displaystyle=-\mu^{3}-\mu^{2}\left(\frac{D(u^{\pm})+\varepsilon\lambda}{c}\right)+\mu\varepsilon+\varepsilon^{2}\frac{\lambda-R^{\prime}(u^{\pm})}{c},$
(38)

from which we obtain the expansion
$\mu^{\pm}_{f}=-D(\bar{u}^{\pm})/c_{0}+\mathcal{O}(\varepsilon)$. Leading-
order expansions of the $\mathcal{O}(\varepsilon)$ eigenvalues are obtained by
using the scaling $\mu=\varepsilon\nu$ as before. Their leading order terms
are given by $\nu_{p,\pm}$ and $\nu_{m,\pm}$; this can be checked by direct
calculation. The remaining small eigenvalues therefore fall into the hierarchy
stated by the lemma at both $u^{-}$ and $u^{+}$ provided that
$\text{Re}\,\lambda>R^{\prime}(0)$. The expressions for the singular limits of
the eigenvectors can also be checked with algebra. $\Box$

For fixed $a>0$, there exists $\bar{\varepsilon}>0$ sufficiently small so that
the wave $x(\xi,\varepsilon)$ lies near $S^{a,-}_{0}\cup S^{a,+}_{0}$ for
$|\xi|\geq a/\varepsilon$ for all $0<\varepsilon\leq\bar{\varepsilon}$. The
corresponding coefficient matrix $a(\xi,\lambda,\varepsilon)$ has three
eigenvalues $\mu_{i}$ satisfying the hierarchy
$\mu_{f}\ll\mu_{s,1}<0<\mu_{s,2}$, inherited from the asymptotic hyperbolicity
described in Lemma 4.2. Let
$f_{1}(\xi,\lambda,\varepsilon),\,f_{2}(\xi,\lambda,\varepsilon),\,f_{f}(\xi,\lambda,\varepsilon)$
denote a choice of eigenvectors corresponding to the eigenvalues
$\mu_{s,1},\,\mu_{s,2},\,\mu_{f}$ (note: if $\mu_{s,1}$ and $\mu_{s,2}$
coalesce, we let $f_{1},\,f_{2}$ denote the generalized eigenvectors instead).
We furthermore suppose that the $f_{i}$ are normalized so that
$|f_{i}|_{\infty}=1$ for all $(\xi,\lambda,\varepsilon)$.

######  Definition 4.4.

The slow subbundle $\sigma_{s}$ of the linearized system is

$\displaystyle\sigma_{s}(\xi,\lambda,\varepsilon)$
$\displaystyle=\bigcup_{|\xi|\geq
a/\varepsilon}\text{span}\\{f_{1}(\xi,\lambda,\varepsilon),\,f_{2}(\xi,\lambda,\varepsilon)\\}$
(39)

where the base space is $|\xi|\geq a/\varepsilon$ and $\lambda\in\Omega$. We
denote the slow subbundle with respect to the timescale
$\zeta=\xi/\varepsilon$ by $\Sigma_{s}$, i.e.

$\displaystyle\Sigma_{s}(\zeta,\lambda,\varepsilon)$
$\displaystyle=\sigma_{s}(\xi/\varepsilon,\lambda,\varepsilon).$ (40)

$\triangle$

### 4.2 Complex 2-plane dynamics induced by the eigenvalue problem

We now write down a concrete representation of the evolution equations of
complex $2$-planes that are induced by the eigenvalue problem (28), and
recording some essential properties about this induced system using the
formulation described in Sec. 3. We have

$\displaystyle\begin{pmatrix}u\wedge p\\\ p\wedge v\\\ v\wedge
u\end{pmatrix}^{\prime}$
$\displaystyle=\begin{pmatrix}-\frac{1}{c}\left(\varepsilon\lambda+D(\bar{u})\right)&-\frac{1}{c}&0\\\
-\varepsilon c&0&-\varepsilon(R^{\prime}(\bar{u})-\lambda)\\\
\varepsilon&0&-\frac{1}{c}(\varepsilon\lambda+D(\bar{u}))\end{pmatrix}\begin{pmatrix}u\wedge
p\\\ p\wedge v\\\ v\wedge u\end{pmatrix}.$ (41)

The coupled system (11)–(41) acts on the complexified tangent bundle
$T^{\mathbb{C}}U\cong\cup_{p\in U}T_{p}(U)\otimes_{\mathbb{R}}\mathbb{C}$,
where $U\subset\mathbb{R}^{3}$. In view of Corollary (2.12) and Lemmas 4.2 and
3.4, there exists $\bar{\varepsilon}>0$ and $a>0$ so that for each
$\varepsilon\in(0,\bar{\varepsilon})$ and $|\xi|>a/\varepsilon$, the
eigenvalues $\mu^{(2)}_{1}(\xi),\,\mu^{(2)}_{2}(\xi),\mu^{(2)}_{3}(\xi)$ of
the linearization of (41) can be ordered so that

$\displaystyle\text{Re}(\mu^{(2)}_{1})$ $\displaystyle=\mathcal{O}(1)<0$
$\displaystyle\text{Re}(\mu^{(2)}_{2})$ $\displaystyle=\mathcal{O}(1)<0$
$\displaystyle\text{Re}(\mu^{(2)}_{3})$
$\displaystyle=\mathcal{O}(\varepsilon).$

Let us provide some geometric intuition to clarify these estimates. When the
wave lies near the slow manifolds, the eigenvalue problem (28) provides an
eigenvector lying near the direction of strong contraction onto the slow
manifolds, and a pair of eigenvectors lying near to the tangent space of the
slow manifold. These eigenvectors can be chosen pairwise to construct an
eigenbasis for the linearisation of system (41), consisting of three complex
$2$-planes. Two of the constructed 2-planes have an axis lying near the fast
fibers of the slow manifolds, so they strongly contract in forward time under
the flow; these are the eigenvectors corresponding to the two eigenvalues of
$\mathcal{O}(1)$ negative real part. The remaining 2-plane lies near the
tangent space of the slow manifold. Depending on the relative magnitudes of
the singular eigenvalues along the wave, this eigenvector provides a weakly
(un)stable direction.

We can further clarify the local dynamics near the tails of the wave, in view
of the elementary facts about projectivized systems discussed just after (21).
The projectivized system induced by (41) on $\mathbb{CP}^{2}$, written in
terms of the coordinate representation
$(\beta_{1},\beta_{2})=\left(\frac{p\wedge v}{u\wedge p},\frac{v\wedge
u}{u\wedge p}\right)$, $u\wedge p\neq 0$, is given by

$\displaystyle\beta_{1}^{\prime}$ $\displaystyle=-\varepsilon
c+\frac{\left(\varepsilon\lambda+D(\bar{u})\right)}{c}\beta_{1}-\varepsilon(R^{\prime}(\bar{u})-\lambda)\beta_{2}+\frac{1}{c}\beta_{1}^{2}$
(42) $\displaystyle\beta_{2}^{\prime}$
$\displaystyle=\varepsilon+\frac{1}{c}\beta_{1}\beta_{2},$

where we remind the reader of the nonautonomous nature of the flow due to the
inclusion of the phase space variable $\bar{u}=\bar{u}(\xi)$. associated with
the system (42) is a family of frozen systems, defined for each fixed
$\varepsilon$ by replacing the nonautonomous system by a one-parameter family
of autonomous ODEs where the variable $\xi$ is formally replaced by an
independent parameter $\gamma$ on the right hand side (see Sec. 7.1.1 for the
explicit construction). A straightforward perturbation argument shows that for
$\varepsilon>0$ sufficiently small with $a>0$ defined as above and for each
fixed $\gamma$ with $|\gamma|>a/\varepsilon$, there is an (isolated)
attracting fixed point of the frozen family for each such $\gamma$, denoted
$\beta_{0}(\gamma,\varepsilon)$, which is $\mathcal{O}(\varepsilon)$-close to
$(\beta_{1},\beta_{2})=(-D(\bar{u}(\gamma)),0)$. For fixed $\varepsilon>0$,
the function $\beta_{0}(\gamma,\varepsilon)$ defines curves of attracting
critical points on the left and right branches of the slow manifold as
$\gamma$ is varied, corresponding precisely to the family of eigenvector
$2$-planes near the slow manifolds whose construction was just described. This
is exactly the slow subbundle in Def. 4.4.

These curves of attracting critical points serve as the organizing centers for
so-called relatively invariant sets (see [8]), which we construct over the
wave tails in Sec. 7.1.1.

### 4.3 Reduced eigenvalue problems

The dynamics of the linearized systems (25) and (26) are equivalent when
$\varepsilon>0$, but they limit to distinguished problems as $\varepsilon\to
0$. In analogy to the definitions of the layer and reduced problems for the
wave, we now define two linearized subsystems using (27)–(28). The
desingularized slow eigenvalue problem will be used in Sec. 6.3 to define a
slow eigenvalue. This will require the derivation of a jump condition for two
disjoint solution segments, defined separately on $S^{a,-}_{0}$ and
$S^{a,+}_{0}$.

######  Definition 4.5.

The fast eigenvalue problem is the linear subsystem defined by the singular
limit of (28), along the fast fiber bundle:

$\displaystyle u^{\prime}$ $\displaystyle=-\frac{1}{c}D(\bar{u})u$ (43)

with $p=v=0$. $\triangle$

######  Remark 4.6.

The limiting system (43) is the variational equation of (11), and thus is
degenerate as an eigenvalue problem. There is always a nontrivial, uniformly
bounded solution to (43). $\triangle$

######  Definition 4.7.

The slow eigenvalue problem is the linear subsystem on
$S^{a,-}_{\varepsilon}\cup S^{a,+}_{\varepsilon}$ defined by the singular
limit of (27), subject to the constraint $V=D(\bar{U})U$:

$\displaystyle\dot{P}$
$\displaystyle=(R^{\prime}(\bar{U})-\lambda)\frac{V}{D(\bar{U})}$ (44)
$\displaystyle\dot{V}$ $\displaystyle=\frac{cV}{D(\bar{U})}-P,$

or more compactly as

$\displaystyle\dot{W}$ $\displaystyle=A_{0}(\bar{U},\lambda,c)W.$ (45)

It is convenient to record here the projectivized slow eigenvalue problem on
the chart $S=P/V$ with $V\neq 0$:

$\displaystyle\dot{S}$
$\displaystyle=\frac{1}{D(\bar{U})}\left(R^{\prime}(\bar{U})-\lambda-
cS+D(\bar{U})S^{2}\right).$ (46)

We also record here a desingularized slow eigenvalue problem, which is defined
by first appending (44) to (18) and then rescaling the frame variable of the
resulting autonomous system by $D(\bar{U})$:

$\displaystyle\dot{P}$ $\displaystyle=(R^{\prime}(\bar{U})-\lambda)V$ (47)
$\displaystyle\dot{V}$ $\displaystyle=cV-D(\bar{U})P.$

$\triangle$

Projectivizations of the desingularised problem (47) will be written down and
used in Sec. 8 to identify the slow eigenvalues.

## 5 Essential & Absolute Spectrum

Our goal is to determine spectral stability of each member of the one-
parameter family of waves
$\\{\Gamma_{\varepsilon}\\}_{\varepsilon\in(0,\bar{\varepsilon}]}$ for some
sufficiently small value of $\bar{\varepsilon}>0$. Following the standard
approach, we will compute the essential and point spectra, defined as in [18],
and compute them separately. It is more convenient to write the eigenvalue
problem (24) as a first order system. Following [28], we consider the
(equivalent) family $\mathcal{L}$ of linear operators from (27) and (28)

$\displaystyle\mathcal{L}(\lambda)$
$\displaystyle:=\frac{d}{d\zeta}-A(\bar{U},\lambda,\varepsilon)=\frac{d}{d\zeta}-\begin{pmatrix}-\frac{1}{c\varepsilon}(D(\bar{U})+\varepsilon\lambda)&0&\frac{1}{c\varepsilon}\\\
(R^{\prime}(\bar{U})-\lambda)&0&0\\\ c&-1&0\end{pmatrix}$ (48)

where $\mathcal{L}(\lambda):H^{1}(\mathbb{R},\mathbb{C}^{3})\to
L^{2}(\mathbb{R},\mathbb{C}^{3})$.

The essential spectrum $\sigma_{e}(\mathcal{L})$ of the family of linear
operators $\mathcal{L}(\lambda)$ defined in (48) is the set of
$\lambda\in\mathbb{C}$ such that either $\mathcal{L}(\lambda)$ is not
Fredholm, or $\mathcal{L}(\lambda)$ is Fredholm but the (Fredholm) index is
not 0. The point spectrum $\sigma_{p}(\mathcal{L})$ is the set of values
$\lambda\in\mathbb{C}$ where $\mathcal{L}(\lambda)$ is not invertible, but
does have index 0. We will denote by $\sigma(\Gamma)$ or
$\sigma(\mathcal{L})$, the union $\sigma_{p}\cup\sigma_{e}$ of the spectrum of
the family $\mathcal{L}$ associated with the travelling wave $\Gamma$.

######  Definition 5.1.

(Def. 4.1.7 in [18]) A wave $\Gamma$ corresponding to a stationary solution of
equation (2) is called spectrally stable if
$\sigma(\Gamma)\cap\\{\lambda\in\mathbb{C}:\text{Re}(\lambda)\geq
0\\}=\emptyset$, except possibly at $\lambda=0$. $\triangle$

For nonzero $\varepsilon$, because $\bar{U}$ approaches its end states
exponentially in $\zeta$ (equivalently $\xi$), determining the essential
spectrum amounts to determining when the matrices
$A^{\pm}(\lambda,\varepsilon)$ from (33) have different signatures (this is
the content of Weyl’s essential spectrum theorem, Theorem 2.2.6 from [18] as
well as and Lemma 3.1.10, also from [18]). The edges of the essential spectrum
are called dispersion relations or the Fredholm borders as well as the
continuous spectrum. To find the dispersion relations we look for values where
$A^{\pm}$ has a purely imaginary (spatial) eigenvalue $ik$, and solve for the
temporal eigenvalue parameter $\lambda$. This gives a pair of curves
$\lambda_{\pm}(k;\varepsilon)$ in the complex plane, parametrised by the
spatial modes, where the asymptotic matrices fail to be hyperbolic. Working
with (27), and denoting $\lim_{\zeta\to\pm\infty}\bar{U}$ by $\bar{U}^{\pm}$,
we have that the dispersion relations are:

$\displaystyle\lambda_{\pm}(k;\varepsilon)$
$\displaystyle=-ick+\frac{R^{\prime}(\bar{U}^{\pm})-D(\bar{U}^{\pm})k^{2}}{1+k^{2}\varepsilon}$
(49)

We have a pair of curves partitioning $\mathbb{C}$ into five disjoint
sections: the set to the right of both curves (where we’re going to look for
point spectrum) which we denote $\Omega$, together with four more regions
which we denote by $\mathcal{A}_{j}$ for $j=1,2,3,4$. (See Fig. 4).

Since the characteristic polynomials of $A^{\pm}$ are both cubics:

$\displaystyle
P_{A^{\pm}}(\mu)=-\mu^{3}-\frac{1}{c}\left(\lambda+\frac{D(\bar{U}^{\pm})}{\varepsilon}\right)\mu^{2}+\frac{\mu}{\varepsilon}+\frac{1}{c\varepsilon}\left(\lambda-R^{\prime}(\bar{U}^{\pm})\right),$
(50)

everything can be checked explicitly. We are working in the slow variables,
but everything that follows in this section can be computed for the
(equivalent) fast systems, since

$P_{A^{\pm}}(\mu)=\frac{1}{\varepsilon^{3}}p^{\pm}(\varepsilon\mu)$

from (38). Defining the transformation

$\eta:=\mu-\frac{1}{3}\left(\lambda+\frac{D(\bar{U}^{\pm})}{c\varepsilon}\right),$

we have

$P_{A^{\pm}}(\eta)=-\eta^{3}+H_{\pm}\eta+K_{\pm},$

where

$H_{\pm}:=\frac{\lambda^{2}}{3c^{2}}+\lambda\frac{2D(\bar{U}^{\pm})}{3c^{2}\varepsilon}+\frac{D(\bar{U}^{\pm})^{2}+3c^{2}\varepsilon}{3c^{2}\varepsilon^{2}},$

and

$\displaystyle K_{\pm}$
$\displaystyle:=-\frac{2}{27c^{3}}\lambda^{3}-\frac{2D(\bar{U}^{\pm})}{9c^{3}\varepsilon}\lambda^{2}+\frac{18c^{2}\varepsilon^{2}-6\varepsilon
D(\bar{U}^{\pm})^{2}}{27c^{3}\varepsilon^{3}}\lambda$
$\displaystyle\qquad-\frac{9c^{2}\varepsilon
D(\bar{U}^{\pm})+D(\bar{U}^{\pm})^{3}+27c^{2}\varepsilon^{2}R^{\prime}(\bar{U}^{\pm})}{27c^{3}\varepsilon^{3}}.$

We note that the discriminants $\Delta_{H_{\pm}}$ of $H_{\pm}$ are polynomials
in $\lambda$ that are independent of $\bar{U}^{\pm}$:

$\Delta_{H_{\pm}}=-\frac{4}{3c^{2}\varepsilon}<0$

which means that for $\lambda\in\mathbb{R}$, we have $H_{\pm}>0$. Hence for
each $\lambda\in\mathbb{R}$,

$\frac{H_{\pm}^{3}}{27}+\frac{K_{\pm}^{2}}{4}\geq 0.$

We thus infer that for $\lambda\in\mathbb{R}$, $P_{A^{\pm}}(\mu)$ has three
real roots (counting multiplicity in edge cases). We can then use Descartes’
rule of signs to see that for $\lambda=0$ (and subsequently for all
$\lambda\in\Omega$), we have that the signature of both $A^{+}$ and $A^{-}$ is
$(-,-,+)$. That is $A^{\pm}$ has two eigenvalues with negative real part, and
one with positive real part. Checking the signatures in the rest of the
regions $\mathcal{A}_{j}$ similarly produces the following table:

Region | $\textrm{sgn}(A^{-})$ | $\textrm{sgn}(A^{+})$
---|---|---
$\Omega$ | $(-,-,+)$ | $(-,-,+)$
$\mathcal{A}_{1}$ | $(-,+,+)$ | $(-,-,+)$
$\mathcal{A}_{2}$ | $(-,-,+)$ | $(-,+,+)$
$\mathcal{A}_{3}$ | $(-,-,+)$ | $(-,+,+)$
$\mathcal{A}_{4}$ | $(-,+,+)$ | $(-,+,+)$

Thus we find that the essential spectrum is the union of regions
$\mathcal{A}_{2,3,4}$ together with their boundaries. This set in the complex
plane with a bounded real part and an unbounded imaginary component (see Fig.
4).

The dispersion relations are symmetric about $k$, and they cross the real axis
at $R^{\prime}(\bar{U}^{\pm})$ when $k=0$. The dispersion relations are
vertical lines when
$\varepsilon=\varepsilon^{\pm}_{*}:=-\frac{D(\bar{U}^{\pm})}{R^{\prime}(\bar{U}^{\pm})}$,
and are parabolic in $k$ near $k=0$, opening to the left when
$\varepsilon<\varepsilon_{*}^{\pm}$ and to the right when
$\varepsilon>\varepsilon_{*}^{\pm}$ before ‘flaring’ up to asymptote towards
vertical lines.

It is worthwhile to investigate whether the family $\mathcal{L}$ is sectorial
when $\varepsilon>0$, since the property of sectoriality allows us to
strengthen spectral stability to linear stability. (see eg. Henry [12]). When
$\varepsilon=0$ the dispersion relations define a pair of parabolas opening
leftward, and as such the linearized operator associated with the unperturbed
problem is sectorial. On the other hand, for any fixed $\varepsilon>0$, the
following asymptotics characterize the behavior of the Fredholm bolders as
$k\to\pm\infty$:

$\displaystyle\lambda_{\pm}\sim-\frac{D(\bar{U}^{\pm})}{\varepsilon}+-ick.\quad$
(51)

Hence, the essential spectrum is contained in the left half plane, but the
family $\mathcal{L}$ is not sectorial since the borders are vertically
asymptotic; see Fig. 4.

We note that as $\varepsilon\to 0$, the continuous spectrum (Fredholm borders)
of the perturbed problem will converge to the continuous spectrum (Fredholm
borders) of the linearised unperturbed problem, linearised about the steady-
state solutions $\bar{U}^{\pm}$. That is we set $\varepsilon=0$ in (2), and
consider the dispersion relations of the steady state solutions
$\bar{U}^{\pm}$. We see that these are given by

$\lambda_{\pm}(k;0)=-{k^{2}}{D(\bar{U}^{\pm})}+R^{\prime}(\bar{U}^{\pm})+ick,$

which corresponds to setting $\varepsilon=0$.

We highlight the subtlety underlying the asymptotic convergence of the
essential spectrum of the linearized operator as $\varepsilon\to 0$: for any
large bounded set intersecting the essential spectrum of the operator for
$\varepsilon>0$ and for any fixed $\delta>0$, there exists a sufficiently
small uniform bound $\bar{\varepsilon}>0$ so that for each
$\varepsilon\in(0,\bar{\varepsilon})$, the Fredholm borders for the operator
when $\varepsilon>0$ can be made $\delta$-close to the (parabolic) borders of
the reduced operator within this bounded set; on the other hand, the uniform
bound $\bar{\varepsilon}$ must be made smaller and smaller as we take larger
and larger bounded sets and demand $\delta$-closeness. See Fig. 4.

######  Remark 5.2.

In view of the nonsectoriality of the family $\mathcal{L}$, spectral stability
does not immediately imply linear exponential stability, i.e. it is unknown
whether nearby translates of the spectrally stable wave decay exponentially
onto the wave in forward time. $\triangle$

######  Remark 5.3.

Lastly, we note that as the essential spectrum is entirely contained in the
left half plane, any absolute spectrum will also be contained in the left half
plane, and so will play no role in destabilising the underlying wave.
$\triangle$

Figure 4: Left: a plot of the essential spectrum (shaded regions) of the
family $\mathcal{L}$ illustrating its qualitative features, and noting the
regions $\Omega$ and the $\mathcal{A}_{i}$ partitioning the complex plane via
the Fredholm borders (continuous spectrum). The asymptotics of the essential
spectrum for large spatial modes means that the family of operators
$\mathcal{L}$, for non-zero $\varepsilon$ is not sectorial. Right: a plot
comparing the Fredholm border of $A^{-}(\lambda,\varepsilon)$ (solid line -
blue online) to the Fredholm border of the reduced problem linearised about
the constant solution $\bar{U}^{-}=0$ with $\varepsilon=0$ (dashed line - red
online). As $\varepsilon\to 0$, the solid line follows the dashed line for a
longer region in the left half plane, before eventually ‘flaring’ up. For the
figures, $\varepsilon=0.1$ (where it is present) and $c=1$, with $D$ and $R$
as in (3) and (4).

## 6 Augmented unstable bundles

Having shown that the essential spectrum lies to the left of $\Omega$ in the
left-half complex plane, we now focus on finding the point spectrum
$\sigma_{p}(\mathcal{L})$ consisting of isolated eigenvalues of finite
multiplicity. To that end, we fix a a simple, closed curve $K\subset\Omega$,
such that $\sigma_{p}\cap K$ is empty. In view of the hyperbolicity of the
asymptotic systems $a^{\pm}(\lambda,\varepsilon)$, we henceforth refer to the
values in the point spectrum as eigenvalues. Specifically, we rely on the
following geometric characterization of the eigenvalues of (24).

######  Definition 6.1.

The number $\lambda\in\Omega$ is an eigenvalue of $\mathcal{L}(\lambda)$ if
there exists a nontrivial uniformly bounded solution $y(\xi)$ of (27)
(equivalently, if the bounded solution $y(\xi)$ decays to zero as
$\xi\to\pm\infty$; see [1]). $\triangle$

In this section we construct two augmented unstable bundles
$\mathcal{E}_{\varepsilon}(K)$ and $\mathcal{E}_{0}(K)$, over a cylindrical
base space formed by extending $K$ along a compactified time variable. As we
will outline in Sec. 6.2, our goal is to use a particular correspondence
theorem that relates a certain topological invariant of these complex vector
bundles to the eigenvalue count of $\mathcal{L}(\lambda)$ inside $K$.

### 6.1 Construction of the augmented unstable bundle
$\mathcal{E}_{\varepsilon}(K)$ for $\varepsilon>0$

Following [8], we now use general facts in [1] together with Lemma 4.2 to
construct (un)stable complex vector bundles of solutions along the wave for
each sufficiently small $\varepsilon>0$.

######  Definition 6.2.

Let $\lambda\in\Omega$. For $\varepsilon>0$ sufficiently small, the unstable
bundle and stable bundle $\varphi^{-}(\xi,\lambda,\varepsilon)$ resp.
$\varphi^{+}(\xi,\lambda,\varepsilon)$ are the linear subspaces of solutions
along the wave $\Gamma_{\varepsilon}$ defined by the conditions

$\displaystyle\varphi^{-}(\xi,\lambda,\varepsilon)$
$\displaystyle\to\text{span}\\{e_{s,2}^{-}(\lambda,\varepsilon)\\}$
$\displaystyle\text{as~{}~{}}\xi\to-\infty\hskip 28.45274pt$ (52)
$\displaystyle\varphi^{+}(\xi,\lambda,\varepsilon)$
$\displaystyle\to\text{span}\\{e_{s,1}^{+}(\lambda,\varepsilon),e_{f}^{+}(\lambda,\varepsilon)\\}$
$\displaystyle\text{as~{}~{}}\xi\to+\infty.\hskip 28.45274pt$

The asymptotic convergence to the corresponding (un)stable subspaces of
$a^{\pm}(\lambda,\varepsilon)$ is defined with respect to the topologies of
the appropriate Grassmannians $G_{1,3}$ and $G_{2,3}$. $\triangle$

The linear subspace $\varphi^{-}$ (resp. $\varphi^{+}$) can be viewed as a
complex line bundle (resp. 2-plane bundle) over the base space
$(\xi,\lambda)\in\mathbb{R}\times\Omega$. In view of Definitions 6.1 and 6.2,
$\lambda\in\Omega$ is an eigenvalue if and only if
$\varphi^{-}(\xi,\lambda,\varepsilon)$ and
$\varphi^{+}(\xi,\lambda,\varepsilon)$ intersect nontrivially for some (and
hence all) $\xi\in\mathbb{R}$.

We now describe a general construction for obtaining an augmented unstable
bundle $\mathcal{E}_{\varepsilon}(K)$ over the real 2-sphere $S^{2}$, using
$\varphi^{-}$; we refer to [1] for details. The approach is to first
compactify $\xi$ using the change of variable $\xi\to\tau$, where

$\displaystyle\tau^{\prime}$ $\displaystyle=\varepsilon\kappa(1-\tau^{2}),$
$\displaystyle\tau(0)$ $\displaystyle=0,$

for $\kappa>0$ chosen such that $\tau(\xi)\to\pm 1$ as $\xi\to\pm\infty$. With
respect to this compactified ‘timescale’,
$\varphi^{-}(\tau,\lambda,\varepsilon)$ then specifies a line bundle over
$(-1,1)\times\Omega$; a bundle can then be defined over the restricted base
space $(-1,1)\times K$. This base space is homeomorphic to an infinite
cylinder (equivalently, a finite cylinder with its caps removed). Our first
goal is to continuously extend the line bundle over the closure of the
cylinder (i.e. over the base space $[-1,1]\times K$). After this we fill in
the caps, altogether producing a complex vector bundle over the closed
cylinder, i.e. over $S^{2}$.

Let us explain this continuous extension. First recall that every complex
vector bundle can be realized as a pullback of the universal bundle over a
Grassmannian, as follows. Given a (complex) vector bundle
$\mathcal{E}:E\subset B\times\mathbb{C}^{n}\to B$ with projection map $\pi$,
there is a natural map $\hat{e}:B\to G_{k,n}$, which assigns to each $b\in B$
the complex $k$-dimensional linear fiber $\pi^{-1}(b)$, considered as a
$k$-plane in $G_{k,n}$. On the other hand, the universal bundle
$\Gamma_{k}(\mathbb{C}^{n})\to G_{k,n}$ is a trivial (tautological)
construction: the fiber above each element in $G_{k,n}$ is defined to be the
corresponding $k$-plane. Then $\mathcal{E}$ can be realized as
$\hat{e}^{*}\Gamma_{k}(\mathbb{C}^{n})$:

${E}$${\Gamma_{k}(\mathbb{C}^{n})}$${B}$${G_{n,k}}$$\scriptstyle{\pi}$$\scriptstyle{\hat{e}}$

Thus, we say that $\varphi^{-}|_{(-1,1)\times K}$ continuously extends to a
bundle over $[-1,1]\times K$ if the natural map $\hat{e}:(-1,1)\times K\to
G_{1,3}$ extends continuously to $\tilde{e}:[-1,1]\times K\to G_{1,3}$. The
extension to $\tau=-1$ is clear: for each $\lambda\in K$, $\varphi^{-}$
continuously extends to $e^{-}_{s,2}(\lambda,\varepsilon)$ as $\tau\to-1$ by
definition. At the other end, we need to specify that $\lambda\in K$ is not an
eigenvalue of (27). General considerations about $\omega$-limit sets relative
to hyperbolic fixed points (see [1]) then imply that
$\varphi^{-}(\tau,\lambda,\varepsilon)$ continuously extends to the
corresponding unstable eigenvector $e^{+}_{s,2}(\lambda,\varepsilon)$ as
$\tau\to+1$. We note that this limit can be made uniform in $\lambda$ over the
closed contour $K$.

It remains to fill in the interiors of the caps $C_{-}=\\{\tau=-1\\}\times
K^{0}$ and $C_{+}=\\{\tau=+1\\}\times K^{0}$ at either end of the cylinder.
For each $\lambda\in K^{0}$, this is achieved for both caps by gluing the
corresponding unstable eigenvector $e^{\pm}_{s,2}(\lambda,\varepsilon)$.

######  Definition 6.3.

The augmented unstable bundle $\mathcal{E}_{\varepsilon}(K)$ is the complex
line bundle over the cylindrical base space $C_{-}\cup[-1,1]\times K\cup
C_{+}$, constructed from $\varphi^{-}$ as described above. $\triangle$

This base space is homeomorphic to $S^{2}$. Our goal will be to relate the
eigenvalue count to a particular characteristic class defined for complex
vector bundles.

### 6.2 An outline of the proof of the stability theorem

We make use of the main theorem proved in [1]: the total number of eigenvalues
(counted with algebraic multiplicity) inside $K$ is equal to the first Chern
number, $c_{1}(\mathcal{E}_{\varepsilon}(K))$, of the complex vector bundle
constructed above. The main advantage of this topological characterisation
lies in the homotopy invariance of $c_{1}$.

We can now outline the steps of the proof of Theorem 2.11:

  1. 1.

Construct a reduced augmented unstable vector bundle $\mathcal{E}_{0}(K)$
using a complex two-dimensional reduced (or slow) eigenvalue problem defined
on the critical manifold $S_{0}$. This is accomplished in Sec. 6.4.

  2. 2.

Show that for sufficiently small $\varepsilon>0$, we can continue
$\mathcal{E}_{\varepsilon}(K)$ to $\mathcal{E}_{0}(K)$; i.e. we can construct
a homotopy between them. By homotopy invariance,
$c_{1}(\mathcal{E}_{\varepsilon}(K))=c_{1}(\mathcal{E}_{0}(K))$. This homotopy
is constructed at the end of Sec. 7 only after several key estimates are
proven; see Theorem 7.1.

  3. 3.

Use the correspondence theorem in [1] in the ‘other direction,’ i.e. compute
$c_{1}(\mathcal{E}_{0}(K))$ by finding the slow eigenvalues using Sturm-
Liouville techniques. We will verify that the reduced problem admits only a
simple translational eigenvalue $\lambda_{0}=0$, and another simple eigenvalue
$\lambda_{1}$ with $R^{\prime}(0)<\text{Re}\,\lambda_{1}<0$. The slow
eigenvalues are computed in Sec. 8.

Steps 1 and 3 turn out to be relatively straightforward; most of the technical
issues lie in proving the uniform estimates required to construct the homotopy
in step 2. To summarize, we will show that for any fixed contour
$K\subset\mathbb{C}$ to the right of the essential spectrum, there exists
$\bar{\varepsilon}>0$ so that for each $\varepsilon\in(0,\bar{\varepsilon}]$,
the following statements hold:

$\displaystyle\\{\\#\text{ of eigenvalues of }\mathcal{L}\text{ inside }K\\}$
$\displaystyle=c_{1}(\mathcal{E}_{\varepsilon}(K))=c_{1}(\mathcal{E}_{0}(K))\leq
2,$

where the first equality follows from the general theory in [1]; the second
follows from the Step 2 above; and the third follows from Steps 1 & 3\. By
choosing smaller contours $K_{0}$ and $K_{1}$ surrounding only $\lambda_{0}$
resp. $\lambda_{1}$ (and shrinking $\bar{\varepsilon}$ finitely many times if
necessary so that the corresponding homotopies can be constructed for all
three contours $K,\,K_{0}$, and $K_{1}$ simultaneously), we obtain linearized
stability of the one-parameter family of waves $\Gamma_{\varepsilon}$ for
$0<\varepsilon<\bar{\varepsilon}$.

### 6.3 Slow eigenvalues and the jump map

We now proceed with step 1 of the stability proof. In order to construct the
reduced bundle, we must first define the appropriate geometric criteria for a
slow eigenvalue $\lambda\in\mathbb{C}$ for (25). Let
$(\bar{U}(\zeta),\bar{P}(\zeta))$ parametrize the segments of the singular
heteroclinic orbit $\Gamma_{0}$ from Hypothesis 2.7 on $S^{a,-}_{0}\cup
S^{a,+}_{0}$, so that the fast jump occurs at $\zeta=0$.

######  Definition 6.4.

The complex number $\lambda\in\Omega$ is called a slow eigenvalue of (25) if
there exists a pair of nontrivial, uniformly bounded solutions on
$S^{a,-}_{0}\cup F_{-}$ resp. $S^{a,+}_{0}$, denoted

$\displaystyle Y_{L}(\lambda,\zeta)$ $\displaystyle\text{ for }\zeta\leq
0\text{ and}$ (53) $\displaystyle Y_{R}(\lambda,\zeta)$ $\displaystyle\text{
for }\zeta\geq 0$

of the desingularized slow eigenvalue problem (47), such that

$\displaystyle Y_{R}(\lambda,0)$
$\displaystyle=J_{\lambda}\left(Y_{L}(\lambda,0)\right),$ (54)

where $J_{\lambda}:\mathbb{C}^{2}\to\mathbb{C}^{2}$ is the (linear) jump map
defined by

$\displaystyle J_{\lambda}(P,V)$
$\displaystyle=\begin{pmatrix}1&\frac{R(\bar{U}_{J})-R(\bar{U}_{F})-\lambda(\bar{U}_{J}-\bar{U}_{F})}{c\bar{U}_{F}-\bar{P}_{F}}\\\
0&\frac{c\bar{U}_{J}-\bar{P}_{F}}{c\bar{U}_{F}-\bar{P}_{F}}\end{pmatrix}\begin{pmatrix}P\\\
V\end{pmatrix}.$ (55)

$\triangle$

The jump map will be derived rigorously as a singular limit in Sec. 7.3.

######  Remark 6.5.

Suppose that $\lambda$ is a slow eigenvalue. Owing to the asymptotic
hyperbolicity of (47) as $\zeta\to\pm\infty$, the pair of nontrivial,
uniformly bounded solutions (53) must decay to zero in forward (resp. reverse)
time. Given two nontrivial, uniformly bounded ‘half-solutions’
$Y_{L}(\lambda,\zeta)$ and $Y_{R}(\lambda,\zeta)$ as given in (53), any
nontrivial complex scalar multiple of these solutions
$\tilde{Y}_{L}(\lambda,\zeta)=KY_{L}(\lambda,\zeta)$ and
$\tilde{Y}_{R}(\lambda,\zeta)=KY_{R}(\lambda,\zeta)$ also satisfy the jump
condition (54) owing to the linearity of the jump map (55).

It is instructive to consider the case $\lambda=0$, corresponding to a
reduction of the translational eigenvalue. The reduced vector field provides
(complex scalar multiples of) nontrivial, uniformly bounded linear solutions
on $S^{a,-}_{0}\cup F_{-}$ and $S^{a,+}_{0}$ as required in (53). Indeed, any
such solution pair $Y_{L},\,Y_{R}$, separately defined along the segments of
the wave lying in $S^{a,-}_{0}\cup F_{-}$ and $S^{a,+}_{0}$, must satisfy

$\displaystyle Y_{L}(\lambda,0)$
$\displaystyle=(P_{F},V_{F})=K_{1}(R(\bar{U}_{F}),c\bar{U}_{F}-\bar{P}_{F})$
$\displaystyle Y_{R}(\lambda,0)$
$\displaystyle=(P_{J},V_{J})=\,K_{2}(R(\bar{U}_{J}),c\bar{U}_{J}-\bar{P}_{F}),$

where $K_{1},\,K_{2}\neq 0$ denote complex scalars. On the other hand,

$\displaystyle J_{0}(P_{F},V_{F})$
$\displaystyle=K_{1}\begin{pmatrix}R(\bar{U}_{F})+(R(\bar{U}_{J})-R(\bar{U}_{F}))\\\
(c\bar{U}_{J}-\bar{P}_{F})\end{pmatrix}$
$\displaystyle=K_{1}\begin{pmatrix}R(\bar{U}_{J})\\\
c\bar{U}_{J}-\bar{P}_{F}\end{pmatrix}$
$\displaystyle=(K_{1}/K_{2})(P_{J},V_{J}).$

The jump condition 54 is therefore satisfied by rescaling one of the solutions
if necessary, and hence $0$ is a slow eigenvalue. Observe that the reduced
vector field provides a natural scaling $K_{1}=K_{2}=1$ such that the jump
condition is automatically satisfied, but this is a special constraint that
arises from the relationship between the reduced vector field along the
singular limit of the wave and the variational equations, i.e. the linearised
problem when $\lambda=0$. We do not generally expect (or need) to find such a
natural scaling when $\lambda\neq 0$.

We end this remark by comparing our definition of a slow eigenvalue to that in
[8]. Motivated by the behavior of their slow line subbundle in the fast inner
layer as $\varepsilon\to 0$, their slow eigenvalue problem is defined by
asking for continuous solutions across the jump. Hence, 0 is necessarily not a
slow eigenvalue in their construction, since the tangent vector of a
continuous solution at the end of the jump is transverse to the reduced vector
field at the jump-on point on $S^{a,+}_{0}$. The translational eigenvalue is
instead counted by their fast eigenvalue problem. $\triangle$

### 6.4 The construction of $\mathcal{E}_{0}(K)$

We work with the projectivization of (47) on $\mathbb{CP}$:

$\displaystyle\dot{\hat{Z}}$
$\displaystyle=\hat{W}_{0}(\hat{Z},\zeta,\lambda).$ (56)

Let $\hat{Z}_{L}(\zeta,\lambda)$ denote the (unique) solution of (56) on
$S^{a,-}_{0}$ which tends to the image of the unstable eigenvector of the
(projective) asymptotic reduced system $\hat{W}^{-}_{0}(\hat{Z},\lambda)$ as
$\zeta\to-\infty$. We write
$Z_{L}(\zeta,\lambda)=(P_{L}(\zeta,\lambda),V_{L}(\zeta,\lambda))\in\pi^{-1}(\hat{Z}_{L})$.
Let us denote by
$Z_{R}(\zeta,\lambda)=(P_{R}(\zeta,\lambda),V_{R}(\zeta,\lambda))$ the
corresponding solution on $S^{a,+}_{0}$ for $\zeta\geq 0$ which satisfies the
jump condition (54) (note that this gives a concrete initial condition for the
solution on $S^{a,+}_{0}$). We find the corresponding embedded solutions in
$\mathbb{R}^{3}$ using

$\displaystyle E_{L}(\zeta,\lambda)$
$\displaystyle=\iota_{0}(\zeta,\lambda)Z_{L}(\zeta,\lambda)$ $\displaystyle
E_{R}(\zeta,\lambda)$
$\displaystyle=\iota_{0}(\zeta,\lambda)Z_{R}(\zeta,\lambda),$

where $\iota_{0}$ denotes the inclusion map into $TS_{0}$.

Finally, we consider the projectivizations
$\hat{E}_{L,R}(\zeta,\lambda)\in\mathbb{CP}^{2}$. In analogy to the
construction of $\mathcal{E}_{\varepsilon}(K)$, we compactify the time
variable and work instead with $T(\zeta)$, where

$\displaystyle T^{\prime}$ $\displaystyle=\kappa(1-\tau^{2})$ (57)
$\displaystyle T(0)$ $\displaystyle=0,$ (58)

with $-1\leq T\leq 1$. When there are no slow eigenvalues within $\lambda\in
K$, we can then continuously extend each of $\hat{E}_{L,R}$ to the appropriate
cap of the compactified cylinder by taking

$\displaystyle\hat{E}_{L}(-1,\lambda)$
$\displaystyle=\hat{e}_{2,0}^{-}(\lambda)$
$\displaystyle\hat{E}_{R}(+1,\lambda)$
$\displaystyle=\hat{e}_{2,0}^{+}(\lambda)$

for each $\lambda\in K\cup K^{0}$.

We construct $\mathcal{E}_{0}(K)$ by gluing together vector bundles over two
hemispheres, defined separately over the singular wave on $S^{a,\pm}_{0}$. In
particular, let

$\displaystyle i^{\pm}_{0}(\lambda)$ $\displaystyle=\lim_{\zeta\to
0^{\pm}}i_{0}(\zeta,\lambda)$ (59)

and take

$\displaystyle B_{-}$ $\displaystyle=B\cap\\{\tau\leq 0\\}$ $\displaystyle
B_{+}$ $\displaystyle=B\cap\\{\tau\geq 0\\},$

denoting the two sets splitting the base space over left and right
hemispheres. By pulling back $\hat{E}_{L}$ and $\hat{E}_{R}$ with respect to
these restricted base sets, we can obtain two bundles $\mathcal{E}^{\pm}_{R}$.
It remains to glue these bundles together over the hemisphere $B_{-}\cap
B_{+}$.

The fibers over $B_{-}\cap B_{+}$ of each bundle $\mathcal{E}^{\pm}_{0}$ is
specified by

$\displaystyle\mathcal{E}^{-}_{0}|(B_{-}\cap B_{+})$
$\displaystyle=\text{span}\\{i_{0}^{-}(\lambda)Z_{L,0}(\lambda)\\}$
$\displaystyle\mathcal{E}^{+}_{0}|(B_{-}\cap B_{+})$
$\displaystyle=\text{span}\\{i_{0}^{+}(\lambda)Z_{R,0}\\}=\text{span}\\{i_{0}^{+}(\lambda)J_{\lambda}(Z_{L,0})(\lambda)\\},$

where $Z_{L,0}\in\pi^{-1}\hat{Z}_{L}(0,\lambda)$ and
$Z_{R,0}\in\pi^{-1}\hat{Z}_{R}(0,\lambda)$. Then there is a bundle isomorphism

$\displaystyle\varphi^{H}:\mathcal{E}^{-}_{0}|\\{0\\}\times
K\to\mathcal{E}^{+}_{0}|\\{0\\}\times K$ (60)

defined by (the linear extension of)

$\displaystyle\varphi^{H}(\lambda)i_{0}^{-}(\lambda)Z_{L,0}$
$\displaystyle=i_{R}^{+}(\lambda)Z_{R,0}=i_{R}^{+}(\lambda)J_{\lambda}Z_{L,0}.$
(61)

The basic properties of bundle isomorphism can be checked by using the
linearity of the embeddings and the jump map.

######  Definition 6.6.

The reduced augmented unstable bundle $\mathcal{E}_{0}(K)$ is defined by the
clutching operation

$\displaystyle\mathcal{E}_{0}$
$\displaystyle=\mathcal{E}^{-}_{0}\cup_{\varphi^{H}}\mathcal{E}^{+}_{0},$ (62)

i.e. it is the bundle over the closed cylinder ($\cong S^{2}$) obtained by
gluing together the hemispheric bundles along their common boundary via the
equivalence relation

$\displaystyle y_{-}$ $\displaystyle\sim\varphi^{H}(\lambda)y_{-}.$

using the clutching map (60). $\triangle$

######  Remark 6.7.

We refer the reader to the standard treatments in differential topology for a
general overview of clutching operations (see e.g. [2, 11]). We will find it
useful to work instead with an equivalent bundle
$\tilde{\mathcal{E}}_{0}(K)\cong\mathcal{E}_{0}(K)$ obtained by projecting the
fiber at each point over the base set onto the corresponding two components.
It is straightforward to show that these two bundles are indeed homotopy
equivalent by constructing the explicit homotopy that continuously bends each
fiber as a homotopy parameter goes from $0$ to $1$; see the remark below Lemma
6.2 of [8] for details. $\triangle$

## 7 Comparison of the bundles $\mathcal{E}_{\varepsilon}(K)$ and
$\mathcal{E}_{0}(K)$

With the reduced objects from Sec. 6.4 in hand, let
$\hat{E}(\zeta,\lambda,\varepsilon)$ denote the (unique) projective solution
of the $\varepsilon$-dependent family of solutions tending to the unstable
subspace as $\zeta\to-\infty$. By general arguments in [1], we have that
$\lambda\notin\sigma_{p}(\mathcal{L})$ if and only if
$\hat{E}(\zeta,\lambda,\varepsilon)\to\hat{e}_{f}^{+}(\lambda,\varepsilon)$
(defined in Lemma 4.2) as $\zeta\to+\infty$. We want to show that particular
solutions of the projectivization of the slow eigenvalue problem (47)
uniformly approximate $\hat{E}(\zeta,\lambda,\varepsilon)$ when
$\varepsilon>0$ is sufficiently small .

Let $\Pi:\mathbb{C}^{3}\to\mathbb{C}^{2}$ denote projection onto the slow
coordinates, i.e.

$\displaystyle\Pi(u,p,v)=(p,v).$ (63)

We can also define the inclusion maps $\iota,\iota_{R}$ in the obvious way. If
$Y,Y_{R}$ are solutions to the full system (in the slow timescale, with
$\varepsilon\neq 0$) and the linearized slow system (i.e. with
$\varepsilon=0$), respectively, then we use the variables
$Z,Z_{0}\in\mathbb{C}^{2}$ to denote the projections:

$\displaystyle Z$ $\displaystyle=\Pi(Y)$ (64) $\displaystyle Z_{0}$
$\displaystyle=\Pi(Y_{0}).$ (65)

Owing to the linearity of the eigenvalue problem, we can also induce a
dynamical system for $\hat{Z}$ on projective space:

$\displaystyle\dot{\hat{Z}}$
$\displaystyle=\hat{B}_{0}(\hat{Z},\zeta,\lambda).$ (66)

The goal now is to compare two solutions, denoted $\hat{Z}_{*}$ and
$\hat{Z}_{0}$, defined via a singular limit of a solution of the full system
and a solution of the reduced linear system, respectively (strictly speaking,
we compare their embeddings in $\mathbb{CP}^{3}$). The solution $\hat{Z}_{0}$
is the unique solution of the projectivized linearized slow system (56). On
the other hand, the solution $\hat{Z}_{*}$ is defined as the projection of the
full solution $E(\zeta,\lambda,\varepsilon)$ onto the slow manifolds in the
singular limit $\varepsilon\to 0$; i.e. take $Z=\Pi E$ and choose a
subsequence $\varepsilon_{n}\to 0$ so that the limits

$\displaystyle\hat{Z}(\pm 1,\lambda,\varepsilon_{n})$
$\displaystyle=\hat{Z}^{\pm}$

both exist, and then define two solutions $\hat{Z}^{\pm}(\zeta,\lambda)$ of
the reduced problem (56) on $\zeta<0$ and $\zeta>0$ with initial conditions
$\hat{Z}^{\pm}$, setting

$\displaystyle\hat{\zeta}_{*}(\zeta,\lambda)$
$\displaystyle=\begin{cases}\hat{\zeta}^{-}(\zeta,\lambda)&\text{if}\zeta<0\\\
\hat{\zeta}^{+}(\zeta,\lambda)&\text{if}\zeta>0.\end{cases}$

Finally, define the following two inclusions:

$\displaystyle E_{0}(\zeta,\lambda)$
$\displaystyle=\iota_{0}(\zeta,\lambda)Z_{0}(\zeta,\lambda)$ (67)
$\displaystyle E_{*}(\zeta,\lambda)$
$\displaystyle=\iota_{0}(\zeta,\lambda)Z_{*}(\zeta,\lambda).$

We re-emphasize that the starred data is obtained from approximation of the
full solution, whereas data denoted by ‘$0$’ is obtained only from solutions
of the reduced problem on $S^{a,-}_{0}\cup F$ and $S^{a,+}_{0}$, concatenated
by an algebraic jump condition.

An approximation theorem can now be stated.

######  Theorem 7.1.

There exists $\bar{\varepsilon}>0$ such that for
$0<\varepsilon<\bar{\varepsilon}$ the following are true:

  * (a)

$\lim_{\varepsilon\to
0}\hat{E}(\zeta,\lambda,\varepsilon)=\hat{E}_{*}(\zeta,\lambda)$ uniformly in
$a\leq|\zeta|\leq A$, for each $0<a<1<A$.

  * (b)

Let $Z^{0-}_{*}(\lambda):=\lim_{\zeta\to 0^{-}}Z_{*}(\zeta,\lambda)$. Then

$\displaystyle
J_{\lambda}(Z^{0-}_{*}(\lambda))=Z_{*}(0,\lambda)=\lim_{\zeta\to
0^{+}}Z_{*}(\zeta,\lambda).$ (68)

  * (c)

Suppose that $\lambda$ is not a slow eigenvalue. Then

$\displaystyle\hat{Z}_{*}(\zeta,\lambda)$
$\displaystyle=\hat{Z}_{0}(\zeta,\lambda),$

where

$\displaystyle\hat{Z}_{0}(\zeta,\lambda)$
$\displaystyle=\begin{cases}\hat{Z}_{L}(\zeta,\lambda)\text{ for }\zeta<0\\\
\hat{Z}_{R}(\zeta,\lambda)\text{ for }\zeta\geq 0\end{cases}$ (69)

is defined by projectivizations of a pair of solutions defined on
$S^{a,-}_{0}\cup F$ resp. $S^{a,+}_{0}$ that tends to the projectivization of
the unstable eigenvector in either direction. The pair also satisfies the
projectivized jump condition from (90):

$\displaystyle\hat{Z}_{R}(0,\lambda)$ $\displaystyle=s(\bar{u}_{J},\lambda).$

Furthermore,
$\hat{E}(\zeta,\lambda,\varepsilon)\to\hat{e}_{f}^{+}(\zeta,\lambda,\varepsilon)$
as $\zeta\to+\infty$.

######  Remark 7.2.

This theorem should be directly compared to the analogous convergence theorem
5.3 in [8]. Observe that the continuity condition in part (b) is replaced by a
jump condition. In general we no longer expect the projected limiting solution
$Z_{*}$ to be continuous at $\zeta=0$ (however, it is still right-continuous).
$\triangle$

The subsections 7.2, 7.3, and 7.4 are devoted to proving Theorem 7.1 (a),(b),
and (c), respectively. In the final subsection 7.5, we use this theorem to
construct a homotopy between the bundles $\mathcal{E}_{\varepsilon}(K)$ and
$\mathcal{E}_{0}(K)$.

### 7.1 Preliminary estimates

Before proving Theorem 7.1, we need some preliminary estimates that will allow
us to control the dynamics of the eigenvalue problem within the slow
manifolds, and as they connect from the fast layers to the slow manifolds. The
key points are that although we do not have access to an elephant trunk lemma,
(i) we may still define partial relatively invariant sets over the attracting
branches $S^{a,\pm}_{\varepsilon}$ of the critical manifold $S_{\varepsilon}$
for sufficiently small values of $\varepsilon>0$, and (ii) the way in which
the eigenvalue parameter $\lambda$ enters the equations suggests that we can
retain control of the dynamics using exchange lemma-type estimates. Let us
highlight that the exchange lemma estimates are apparently new relative to the
techniques introduced in [8]. Furthermore, the use of relatively invariant set
theory applied to the case of two-plane dynamics (i.e. the construction of a
so-called slow elephant trunk over a slow subbundle) also appears to be new.

####  7.1.1 Fast and slow elephant trunks over $S^{a,\pm}_{\varepsilon}$

In this section, we construct relatively invariant attracting sets for (28)
defined within compact neighborhoods of $S^{a,\pm}_{\varepsilon}$. Here we
closely follow the treatment of Sec. IV in [8]. Let

$\Omega=\cup_{\xi\in I}\Omega(\xi)\times\\{\xi\\}$

denote a subset of $\mathbb{C}^{2}\times\mathbb{R}$ such that $I$ is an open
interval in $\mathbb{R}$ and $\Omega(\xi)$ is a neighborhood in
$\mathbb{C}^{2}$ for each $\xi$, such that
$\partial\Omega\cap\mathbb{C}^{2}\times I$ is a smooth manifold (with
$\Omega(\xi)$ varying smoothly in $\xi$). Furthermore, consider a sufficiently
smooth nonautonomous system on $\mathbb{C}^{2}$ of the form

$\displaystyle\beta^{\prime}$ $\displaystyle=G(\beta,\xi,\varepsilon).$ (70)

######  Definition 7.3.

The set $\Omega$ is (positively) invariant relative to $I$ if for any solution
$\beta(\xi)$ of (70) with $\beta(\xi_{0})\in\Omega(\xi_{0})$ for some
$\xi_{0}\in I$, we have $\beta(\xi)\in\Omega(\zeta)$ for all
$\zeta\geq\zeta_{0}$ for which $\zeta\in I$. $\triangle$

We will define two distinct collections of relatively invariant sets over the
disjoint branches $S^{a,\pm}_{\varepsilon}$, referring to one type as fast
elephant trunks, denoted $\Omega_{\pm}^{f}$, and to the other as slow elephant
trunks, denoted $\Omega_{\pm}^{s}$.

The starting point is to consider the auxiliary autonomous family of frozen
systems associated with (70):

$\displaystyle\frac{d\beta}{d\xi}$ $\displaystyle=G(\beta,\gamma,\varepsilon)$
(71)

and to assume that (71) admits a smooth curve of critical points
$\beta_{0}(\gamma,\varepsilon)$ (i.e. for each $\varepsilon>0$, $\beta_{0}$
depends smoothly on $\gamma$). Thinking of (71) as the frozen family
corresponding to a projectivization of a linearized system over a reference
chart, such curves arise naturally by continuing the eigenvectors of the
linear system along the travelling wave, which is parametrized by $\gamma$. We
will consider subsets of the following form as candidates for elephant trunks:

$\displaystyle\Omega(d,\eta,\varepsilon)$
$\displaystyle=\\{(\beta,\eta):|\beta-\beta_{0}(\gamma,\varepsilon)|_{\gamma}<\eta,\gamma\in
I(d)\\},$ (72)

where the real parameter $d$ parametrizes a nested family of subintervals
$I(d)$ (i.e. $I(d_{1})\subset I(d_{2})$ whenever $d_{1}<d_{2}$), and the
metric $|\cdot|_{\gamma}$ is defined according to the construction in Sec. IV.
B. in [8]. As noted there, the parameter $d$ can be thought of as
characterizing the slowly varying character of (70) (with respect to $\xi$
when $\varepsilon>0$ is sufficiently small), while $\eta$ characterizes the
width of the tube. See Fig. 5 for a sketch of the elephant trunks we
construct.

(a) (b)

Figure 5: Sketches of the (a) fast resp. (b) slow elephant trunks constructed
in Lemmas 7.4–7.5.

We begin with the construction of the fast elephant trunks which are defined
for (26), i.e. (28) with the frame variable having reversed orientation
$\xi\mapsto-\xi$. Strictly speaking, we work with the (orientation-reversed)
projectivized system (31).

######  Lemma 7.4.

For each $\lambda\in\Omega$ (see (34)), there exist $\varepsilon_{0}>0$ and
$d_{0}>0$ such that for each $\varepsilon\in(0,\varepsilon_{0}]$ and
$d\in(0,d_{0}]$, there is a pair of fast elephant trunks
$\Omega_{-}^{f}:=\Omega_{-}^{f}(d,\eta,\lambda,\varepsilon)$ (over
$S^{a,-}_{\varepsilon}$) and
$\Omega_{+}^{f}:=\Omega_{+}^{f}(d,\eta,\lambda,\varepsilon)$ (over
$S^{a,+}_{\varepsilon}$); namely, $\Omega_{-}^{f}$ (resp. $\Omega_{+}^{f}$) is
positively invariant relative to the nested families of sub-intervals
$I_{-}(d)$ (resp. $I_{+}(d)$), where

$I_{-}(d)=\\{\xi:\xi\leq\xi_{-}(\delta(d))<0\\}$

and

$I_{+}(d)=\\{\xi:\xi\geq\xi_{+}(\delta(d))>0\\},$

where $\delta:[0,d_{0})\to[0,\infty)$ varies smoothly in $d$ with
$\delta(0)=0$, and $\xi_{\pm}(\delta(d))$ are defined so that:

  * •

the travelling wave $\bar{x}(\xi,\varepsilon)$ lies $\delta$-close to the
singular limit of the travelling wave for all
$\xi\leq\xi_{-}(\delta)\leq-\bar{\xi}$ and for all
$\xi\geq\xi_{+}(\delta)\geq\bar{\xi}$, where $\bar{\xi}$ is defined so that:

  * •

the matrix
$A(\xi,\lambda,\varepsilon):=G_{\beta}(\beta_{0}(\xi,\lambda,\varepsilon),\xi,\lambda,\varepsilon)$
has eigenvalues $\mu_{2}-\mu_{1}$ and $\mu_{3}-\mu_{1}$ uniformly negative and
bounded away from 0 for all $|\xi|\geq\bar{\xi}$, where $G$ denotes the
orientation reversal of the projectivized system (31).

Proof: See Appendix A.

Lemma 7.4 provides elephant trunks about the fast unstable directions near the
slow manifolds in reverse time. In fact we will only require the fast elephant
trunk near the fixed point at $\bar{u}=0$ (i.e. over the branch
$S^{a,-}_{\varepsilon}$), in order to verify one part of a uniform closeness
estimate of the unique solution $E(\xi,\lambda,\varepsilon)$ of (28) with
$|E(0,\lambda,\varepsilon)|=1$ and $E(\xi,\lambda,\varepsilon)\to u^{-}$ as
$\xi\to-\infty$. As we track $E(\xi,\lambda,\varepsilon)$ for $\xi$
sufficiently large, we will also require a slow elephant trunk over
$S^{a,+}_{\varepsilon}$ which guarantees uniform closeness of $E$ near the
slow subbundle. We work with the projectivized system (42).

######  Lemma 7.5.

For each $\lambda\in\Omega$ (see (34)), there exist $\varepsilon_{0}>0$ and
$d_{0}>0$ such that for each $\varepsilon\in(0,\varepsilon_{0}]$ and
$d\in(0,d_{0}]$, there is a pair of slow elephant trunks
$\Omega_{-}^{s}:=\Omega_{-}^{s}(d,\eta,\lambda,\varepsilon)$ (over
$S^{a,-}_{\varepsilon}$) and
$\Omega_{+}^{s}:=\Omega_{+}^{s}(d,\eta,\lambda,\varepsilon)$ (over
$S^{a,+}_{\varepsilon}$) for system (42); namely, $\Omega_{-}^{s}$ (resp.
$\Omega_{+}^{s}$) is positively invariant relative to the nested families of
sub-intervals $I_{-}(d)$ (resp. $I_{+}(d)$), where

$I_{-}(d)=\\{\xi:\xi\leq\xi_{-}(\delta(d))<0\\}$

and

$I_{+}(d)=\\{\xi:\xi\geq\xi_{+}(\delta(d))>0\\},$

where $\delta:[0,d_{0})\to[0,\infty)$ varies smoothly in $d$ with
$\delta(0)=0$, and $\xi_{\pm}(\delta(d))$ are defined so that:

  * •

the travelling wave $\bar{x}(\xi,\varepsilon)$ lies $\delta$-close to the
singular limit of the travelling wave for all
$\xi\leq\xi_{-}(\delta)\leq-\bar{\xi}$ and for all
$\xi\geq\xi_{+}(\delta)\geq\bar{\xi}$, where $\bar{\xi}$ is defined so that:

  * •

the matrix
$A(\xi,\lambda,\varepsilon):=G_{\beta}(\beta_{0}(\xi,\lambda,\varepsilon),\xi,\lambda,\varepsilon)$
has eigenvalues uniformly negative and bounded away from 0 for all
$|\xi|\geq\bar{\xi}$.

Proof: The steps are essentially identical to those shown in Appendix A for
Lemma (7.4). We point out that the corresponding family of frozen systems for
(42) admits a curve of attracting critical points, as described in the
paragraph below (42). The remaining conditions are checked with direct
calculation. $\Box$

######  Remark 7.6.

The slow elephant trunks in Lemma 7.5 define attracting invariant
neighborhoods relative to a metric defined in $\mathbf{Gr}(2,3)$. Switching
instead to the Fubini-Study metric (which gives equivalent estimates up to a
constant factor depending only on the metrics), it is a rather lengthy
calculation to show that these planar neighborhoods can be in turn expressed
as neighborhoods of complex lines relative to the slow subbundle. $\triangle$

####  7.1.2 Exchange lemma-type estimates

Per Fenichel’s original setup [7], we consider a singularly perturbed system
of differential equations in $\mathbb{R}^{n}$ which admits a family of
$k$-dimensional normally hyperbolic invariant manifolds $S_{\varepsilon}$,
with $0<k<n$, for $\varepsilon\in(0,\bar{\varepsilon}]$, arising from a
critical manifold $S_{0}$ when $\varepsilon=0$. We remind the reader that
$n=3$ and $k=2$ in our model, and we restrict our interest to the case of
attracting slow invariant manifolds. We use freely a Fenichel normal form
defined over a common neighborhood $U$ of $S_{\varepsilon}$, appended with its
variational equations:

$\displaystyle b^{\prime}$ $\displaystyle=\Gamma(b,y,\varepsilon)b$ (73)
$\displaystyle y^{\prime}$ $\displaystyle=\varepsilon f(y,\varepsilon)$
$\displaystyle db^{\prime}$ $\displaystyle=\Gamma db+D_{z}\Gamma(dz\otimes b)$
$\displaystyle dy^{\prime}$ $\displaystyle=\varepsilon D_{y}f\,dy,$

where $z:=(b,y)$ and $dz:=(db,dy)$. In terms of the geometric objects
introduced in Sec. 3, the system (73) may be regarded as an induced derivation
on the tangent bundle, with the dynamics on the tangent vectors coordinatized
according to the Plücker embedding. We highlight a few properties of Fenichel
theory and the Fenichel normal form. The slow manifolds are given by
$S_{\varepsilon}=\\{b=0\\}$, so that $y\in\mathbb{R}^{k}$ may be viewed as the
slow variables. The stable manifold $W^{s}_{\varepsilon}$ is foliated by
invariant fibers, with associated projection map
$\pi^{-}:W^{s}_{\varepsilon}\to S_{\varepsilon}$.

Figure 6: Setup of the exchange lemma-type estimate using system (76). Here
the projection $\pi^{-}\times\pi^{-}_{\lambda}$ refers to the projection onto
the invariant manifold $\\{b=0\\}$, extended by the corresponding invariant
manifold given by the eigenvalue problem (76).

With respect to the straightened dynamics of the normal form, this stable
fibration is given by $(b,y)\mapsto y$. Each function in (73) is regarded as
‘sufficiently’ smooth for every succeeding statement to hold, and we further
assume that the spectra of the matrix functions $\Gamma$ and $\varepsilon
D_{y}f$ satisfy the necessary conditions for Fenichel’s Lyapunov type numbers.
Specifically, for each $y\in U$ we list the spectrum

$\displaystyle\text{spec}\,\Gamma(0,y,0)$
$\displaystyle:=\\{\gamma_{1}(y),\cdots,\gamma_{n-k}(y)\\}$

in ascending order of their real parts, i.e.

$\displaystyle\text{Re}\,(\gamma_{1}(y))\leq\cdots\leq\text{Re}\,(\gamma_{n-k}(y)),$

and we assume a spectral gap $\text{Re}\,\gamma_{n-k}(y)<0$ uniformly for
$y\in\bar{U}$. Define

$\displaystyle\gamma_{0}$ $\displaystyle=\text{sup}_{y\in U}\gamma_{n-k}(y).$
(74)

Let us summarise the geometric ideas underlying the exchange lemma. We
consider families of manifolds that enter neighborhoods of normally hyperbolic
slow invariant manifolds in a generic way. The primary goal is to track both
the position and orientation (i.e. the corresponding evolution of tangent
spaces according to the variational equations) as trajectories enter near the
stable foliation and spend sufficiently long times near the slow manifold. If
the entry manifolds intersect the stable foliation transversally, this generic
entry is ‘exchanged’ for exponential closeness, in both position and
orientation, to the unstable foliation (or in our case, to the tangent space
of the slow manifold) at exit.

This geometric result can be expressed explicitly in terms of the rectified
variables in (73). We follow the treatment by Jones and Tin in [17]. Fix a
parameter $\Delta>0$ and consider a one-parameter family of entry manifolds
$\\{M_{\varepsilon}\\}$ intersecting the stable manifold transversally at some
section $\\{|b|=\Delta\\}$. We focus on initial conditions within the entry
manifolds that remain within the box specified by

$\displaystyle B_{\Delta}$ $\displaystyle:=\\{|y|\leq\Delta,|b|\leq\Delta\\},$
(75)

leaving $B_{\Delta}$ through $\\{|y|=\Delta\\}$ only after a sufficiently long
time, say $\mathcal{T}_{\varepsilon}=1/\varepsilon^{s}$ for some $0<s<1$. Let
$z(t)$ parametrize an incoming trajectory $\gamma$ corresponding to such an
initial condition lying on $M_{\varepsilon}$. The exchange lemma assures us
that for $\kappa\in(0,|\gamma_{0}|)$ (see (74)), $z(t)$ is
$C^{1}$-$\mathcal{O}(e^{-\kappa\mathcal{T}}$)-close to $S_{\varepsilon}$ at
$t=\mathcal{T}_{\varepsilon}$, i.e.

$\displaystyle\text{dist}(z(\mathcal{T}_{\varepsilon}),S_{\varepsilon})$
$\displaystyle=\mathcal{O}(e^{-\kappa\mathcal{T}_{\varepsilon}})$
$\displaystyle\text{dist}(T_{z(\mathcal{T}_{\varepsilon})}\gamma,T_{\pi^{-}(z(\mathcal{T}_{\varepsilon}))}S_{\varepsilon})$
$\displaystyle=\mathcal{O}(e^{-\kappa\mathcal{T}_{\varepsilon}}),$

where the dist$(\cdot,\cdot)$ map is defined in the usual way, after
specifying some appropriate metrics. In view of the rectified coordinate
system for (73), $T_{x}S_{\varepsilon}=\\{db=0\\}$ (the tangent space of
$S_{\varepsilon}$ at $x$) for each $x\in S_{\varepsilon}$. Complete details
about the formulation of the problem and a statement of the Exchange Lemma are
given in [17]. This general formulation is referred to as the
$(k+\sigma)$-Exchange Lemma, where $k$ denotes the dimension of the local
unstable manifolds of the critical manifold. In the present case $k=0$, the
Exchange Lemma is referred to as an inclination lemma [5].

The first step is to write down the ‘straightened’ eigenvalue problem.
Although the coordinate change—i.e. the composition of straightening
diffeomorphisms—which defines the Fenichel normal form is usually highly
nonlinear and difficult to write down, we highlight that the linearisation of
this composition acts linearly on the dynamical system defined on the tangent
spaces.666Indeed, if the eigenvalue problem in the original coordinates is
written as $dx^{\prime}=A(x,\varepsilon)dx+\varepsilon\lambda M\cdot dx$,
where $A(x,\varepsilon,\lambda)dx$ denotes the original variational equations
and $M$ is a constant square matrix, and $z=\varphi(x,\varepsilon)$ is the
diffeomorphism giving rise to the Fenichel normal form, then
$\widetilde{dz}^{\prime}=B(z,\varepsilon)\widetilde{dz}+\varepsilon\lambda\widetilde{M}\cdot\widetilde{dz}$,
where $B\,\widetilde{dz}$ is the transformed variational equation and
$\widetilde{M}=D\varphi\circ M\circ D\varphi^{-1}$. Hence, we obtain

$\displaystyle b^{\prime}$ $\displaystyle=\Gamma(b,y,\varepsilon)b$ (76)
$\displaystyle y^{\prime}$
$\displaystyle=\varepsilon(f(y,\varepsilon)+H(b,y,\varepsilon)b)$
$\displaystyle\widetilde{db}^{\prime}$
$\displaystyle=\Gamma\widetilde{db}+D_{z}\Gamma(\widetilde{dz}\otimes
b)+\varepsilon\lambda G_{1}(b,y,\varepsilon)\cdot\widetilde{dz}$
$\displaystyle\widetilde{dy}^{\prime}$
$\displaystyle=\varepsilon(D_{y}f\,\widetilde{dy}+D_{z}H(\widetilde{dz}\otimes
b)+H\widetilde{db})+\varepsilon\lambda
G_{2}(b,y,\varepsilon)\cdot\widetilde{dz}.$

Here, $\widetilde{dz}:=(\widetilde{db},\widetilde{dy})$ and $G_{1},\,G_{2}$
are smooth functions that are defined from the $\varepsilon$-family of
diffeomorphisms used to derive the Fenichel coordinates, and so are
independent of $\lambda$. Let us write the transformed eigenvalue problem in
(76) in the more compact form

$\displaystyle\widetilde{dz}^{\prime}$
$\displaystyle=A(z,\varepsilon)\widetilde{dz}+\varepsilon\lambda\widetilde{M}(z,\varepsilon)\widetilde{dz},$
(77)

with $\widetilde{M}_{\Delta,\varepsilon_{0}}:=\max_{z\in
B_{\Delta},0\leq\varepsilon\leq\varepsilon_{0}}\widetilde{M}(z,\varepsilon)$.
We drop the tilde notation for the remainder of the section for ease of
reading. For $T>0$, let $Q_{T}$ denote the set of initial conditions within
the box of width of $\Delta>0$ so that for each $q:=z(0)=(b_{0},y_{0})\in
Q_{T}$, we have that $z(t)$ remains within the $\Delta$-box for each
$t\in[0,T]$.

######  Lemma 7.7.

Assume the hypotheses of the $(k+\sigma)$ Exchange Lemma in the singularly-
perturbed case (Theorem 6.7 in [17]) with $k=0$ (i.e. the normally hyperbolic
critical manifold does not admit fast unstable directions). Furthermore, let
$(\gamma(t,\varepsilon),dz(t,\varepsilon))$ be a one-parameter family of
trajectories of (76) (where the components of the tangent vector are given by
$dz(t,\varepsilon)=(db(t,\varepsilon),dy(t,\varepsilon)$), so that
$\gamma(t,\varepsilon)$ satisfies the hypotheses for the invariant manifolds
$M_{\varepsilon}$ in [17]. Furthermore, let
$dz(0,\varepsilon)=:(db_{0}(\varepsilon),dy_{0}(\varepsilon))$ satisfy the
estimate

$\displaystyle|dy_{0}|\geq M\varepsilon$

for some $M>0$ independent of $\varepsilon$, when $\varepsilon>0$ is
sufficiently small, and fix $\beta$ with $0<\beta<1$. Then there exists
$\bar{\varepsilon}>0$ so that for $0<\varepsilon\leq\bar{\varepsilon}$,

$\displaystyle\text{dist}(dz(\mathcal{T}_{\varepsilon}),T_{\gamma(\mathcal{T}_{\varepsilon})}S_{\varepsilon})\leq
K_{2}\varepsilon^{\beta}$

$K_{2}$ depending only on $\bar{\varepsilon}$ and $\Delta$.

Proof: See Appendix C.

######  Remark 7.8.

Let us highlight a few points about Lemma 7.7 relative to the $(k+\sigma)$
Exchange Lemma in the singularly-perturbed case. The first obvious difference
is that the estimate is weaker, being only linear in order of $\varepsilon$.
This is because we compare a generic incoming trajectory to the tangent space
of the slow manifold (i.e. the set $\\{b=0,\,db=0\\}$) instead of the slow
subbundle, where the computation becomes more difficult. For our present
purposes this weaker estimate is sufficient, since it is immediate that the
slow subbundle is $\mathcal{O}(\varepsilon)$-close to the tangent space of the
slow manifold relative to the Fubini-Study metric (note: it is crucial here
that the eigenvalue problem depends ‘weakly’ on $\lambda$, i.e. through terms
of $\mathcal{O}(\varepsilon\lambda)$ only).

However, we highlight that the system (76) is singularly-perturbed, with the
identical normally hyperbolic critical manifold $\\{b=0,\,db=0\\}$ as that of
the variational equations. By standard Fenichel theory, there exists a one-
parameter family of (real) six-dimensional locally invariant attracting slow
manifolds $\mathcal{M}_{\varepsilon}$ for sufficiently small values of
$\varepsilon>0$, which lie $\mathcal{O}(\varepsilon)$-close in Hausdorff
distance to the critical manifold; two of the dimensions come from the slow
directions in the phase space and the remaining four come from the slow
complex directions of the eigenvalue problem. We therefore expect that the
tangent spaces of incoming trajectories under the flow of the eigenvalue
problem align exponentially closely to that of $\mathcal{M}_{\varepsilon}$,
i.e. there is an analogue to the standard Exchange Lemma estimate for the
eigenvalue problem. Furthermore, the slow subbundle should provide the
$\mathcal{O}(\varepsilon)$ correction of the tangent bundle of the slow
manifold (i.e. it is $\mathcal{O}(\varepsilon^{2})$-close with respect to the
Fubini-Study metric), in analogy to the standard computations using Fenichel
theory. We illustrate this in Appendix D with a toy problem. $\triangle$

### 7.2 Estimates near the slow subbundle

In this subsection we prove Theorem 7.1 (a). Our aim is to prove a similar
result to Corollary 5.6 in [8], namely that the projectivization
$\hat{E}(\zeta,\lambda,\varepsilon)$ of any nontrivial solution
$E(\zeta,\lambda,\varepsilon)$ lies uniformly close to the slow subbundle when
the wave is near to the slow manifolds. A new technical issue here is that we
do not have an elephant trunk estimate over the fast layer. We will instead
combine our existing ‘partial’ fast and slow elephant trunks with our exchange
lemma-type estimate to achieve this uniform closeness.

In the coming analysis we will work with projectivisations of the slow
subbundle, denoted $\hat{\sigma}_{s}$ and $\hat{\Sigma}_{s}$ as usual. Now fix
any metric $\rho$ on $\mathbb{CP}^{3}$.

######  Definition 7.9.

For any set $\hat{S}\subset\mathbb{CP}^{3}$ and $\delta>0$, a
$\delta$-neighborhood of $\hat{S}$ is the set

$\displaystyle N_{\delta}(\hat{S})$
$\displaystyle=\\{\hat{y}\in\mathbb{CP}^{3}:\rho(\hat{s},\hat{y})<\delta\text{
for some }\hat{s}\in\hat{S}\\}.$ (78)

$\triangle$

We begin by estimating the closeness of $E(\zeta,\lambda,\varepsilon)$ to
$\Sigma_{s}(\zeta,\lambda,\varepsilon)$ near $S^{a,\pm}_{0}$. To do so, we
consider the following family of autonomous frozen systems (and their
projectivizations) corresponding to the linearized problem:

$\displaystyle y^{\prime}$ $\displaystyle=a(\gamma,\lambda,\varepsilon)y$ (79)
$\displaystyle\hat{y}^{\prime}$
$\displaystyle=\hat{a}(\hat{y},\gamma,\lambda,\varepsilon).$

We refer the reader to a series of technical lemmas in Appendix B, culminating
in the uniform estimate Corollary B.4 which is used in the following proof.

Proof of Theorem 7.1(a). Fix two constants $a$ and $A$ with $0<a<1<A$ and
suppose $\zeta\in[a,A]$. We select a representative solution
$E\in\pi^{-1}\hat{E}$ scaled so that slow components $|Z|\leq 1$ on
$a\leq\zeta\leq A$ for each $\varepsilon\in(0,\bar{\varepsilon})$. It quickly
follows that for $\zeta=1$ fixed, we have $Z\to Z^{+}$ as $\varepsilon\to 0$.
Our primary task is to compare a solution $Z$ representing $\hat{Z}$ to a
solution $Z_{*}$ representing the corresponding projectivisation of the
reduced problem, i.e. we show that $|Z-Z_{*}|\to 0$ uniformly on
$a\leq\zeta\leq A$ as $\varepsilon\to 0$.

Away from the jump, the projection onto the slow components $Z=(P,V)$ and
$Z_{R}=(P_{R},V_{R})$ of the full and reduced systems, respectively, may be
written as

$\displaystyle\dot{Z}$ $\displaystyle=B(\zeta,\lambda,\varepsilon)Z+G\Gamma$
$\displaystyle\dot{Z}_{R}$ $\displaystyle=B_{R}(\zeta,\lambda)Z_{R},$

where (suppressing the dependence of the phase space coordinate $\bar{U}$ on
$\zeta$ and $\varepsilon\geq 0$):

$\displaystyle B(\zeta,\lambda,\varepsilon)$
$\displaystyle=\begin{pmatrix}0&\frac{R^{\prime}(\bar{U})-\lambda}{D(\bar{U})}\\\
-1&\frac{c}{D(\bar{U})}\end{pmatrix}$ $\displaystyle
G(\zeta,\lambda,\varepsilon)$ $\displaystyle=R^{\prime}(\bar{U})-\lambda$
$\displaystyle\Gamma(\zeta,\lambda,\varepsilon)$
$\displaystyle=U-\frac{V}{D(\bar{U})}$

and

$\displaystyle B_{R}(\zeta,\lambda)$
$\displaystyle=\begin{pmatrix}0&\frac{R^{\prime}(\bar{U})-\lambda}{D(\bar{U})}\\\
-1&\frac{c}{D(\bar{U})}\end{pmatrix}.$

Here the term $G\Gamma$ can be thought of as a forcing term that is ‘turned
on’ when $\varepsilon>0$; specifically, $\Gamma$ measures how far (the slow
projection of) the solution $Y$ is from solving the reduced eigenvalue
problem. Evidently $B(\zeta,\lambda,\varepsilon)\to B_{R}(\zeta,\lambda)$ as
$\varepsilon\to 0$, uniformly in the interval $a\leq\zeta\leq A$. It remains
to estimate the forcing term $G\Gamma$ uniformly within this $\zeta$-interval.
The term $G$ has a uniform singular limit for $\varepsilon\to 0$, so we
concern ourselves with the singular limit for $\Gamma$. For $\delta>0$
arbitrarily chosen, Corollary B.4 provides a sufficiently small
$\bar{\varepsilon}>0$ such that

$\displaystyle\hat{E}(\zeta,\lambda,\varepsilon)\in
N_{\delta}(\Sigma_{s}(\zeta,\lambda,\varepsilon))$

for each $\varepsilon\in(0,\bar{\varepsilon}]$ and $\zeta\geq a$. A fixed
representative $E(\zeta,\lambda,\varepsilon)$ of the projectivised solution of
the full linearized system can thus be made arbitrarily close to a linear
combination of the basis vectors $F_{1},\,F_{2}$ by varying $\delta$ (see the
definitions of $f_{1},\,f_{2}$ in (40); we use capital letters to denote the
appropriate timescale). Each of these basis vectors in turn has a singular
limit. Indeed, we have

$\displaystyle
E(\zeta,\lambda,\varepsilon)\to\alpha_{1}(\zeta,\lambda)R_{s,1}(\zeta,\lambda)+\alpha_{2}(\zeta,\lambda)R_{s,2}(\zeta,\lambda)$
(80)

as $\varepsilon\to 0$, where $R_{s,1}(\zeta,\lambda)$ and
$R_{s,2}(\zeta,\lambda)$ are the eigenvectors spanning the reduced slow
subbundle. These are the required singular limits of the basis vectors
$F_{1},\,F_{2}$, and they admit explicit formulas:

$\displaystyle F_{1}\to R_{s,1}$ $\displaystyle=(1/D(\bar{U}),V_{p},1)^{\top}$
(81) $\displaystyle F_{2}\to R_{s,2}$
$\displaystyle=(1/D(\bar{U}),V_{m},1)^{\top},$

and $\alpha_{1}(\zeta,\lambda)$ and $\alpha_{2}(\zeta,\lambda)$ are uniformly
bounded coefficients. Note that the basis vectors $R_{s,i}$, $i=1,2$ are
derived similarly to the asymptotic case in (36), and the auxiliary quantities
$V_{p,m}$ are defined analogously to $\nu_{j,\pm}$ in (37). Finally, we write
$\Gamma$ in a convenient form by including and rearranging terms:

$\displaystyle\Gamma(\zeta,\lambda,\varepsilon)$
$\displaystyle=U(\zeta,\varepsilon)-\frac{V}{D(\bar{U}(\zeta,0))}+\left(\frac{1}{D(\bar{U}(\zeta,\varepsilon))}-\frac{1}{D(\bar{U}(\zeta,0))}\right)V.$
(82)

The term in parentheses converges uniformly to 0 and $|V|\leq 1$ by
construction, and the first to terms also converge uniformly to zero by 81;
hence $|\Gamma|\to 0$ uniformly on the required $\zeta$-interval, and thus
$Z\to Z_{*}$ uniformly on this interval by Gronwall’s inequality. Finally, the
$U$-components of $E$ also converge uniformly to those of $E_{*}$ by (80),
(82), and Corollary B.4. Altogether, the uniform convergence of $E$ to $E_{*}$
on $a\leq\zeta\leq A$ follows from Gronwall’s inequality. The case $\zeta<0$
proceeds identically, so we omit the proof. $\Box$

### 7.3 The jump map as a singular limit