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# Behavior of Spiral Wave Spectra with a Rank-Deficient Diffusion Matrix

Stephanie Dodson
Department of Mathematics
University of California, Davis
Davis, CA 95616, USA  Björn Sandstede
Division of Applied Mathematics
Brown University
Providence, RI 02912, USA

###### Abstract

Spiral waves emerge in numerous pattern forming systems and are commonly
modeled with reaction-diffusion systems. Some systems used to model biological
processes, such as ion-channel models, fall under the reaction-diffusion
category and often have one or more non-diffusing species which results in a
rank-deficient diffusion matrix. Previous theoretical research focused on
spiral spectra for strictly positive diffusion matrices. In this paper, we use
a general two-variable reaction-diffusion system to compare the essential and
absolute spectra of spiral waves for strictly positive and rank-deficient
diffusion matrices. We show that the essential spectrum is not continuous in
the limit of vanishing diffusion in one component. Moreover, we predict
locations for the absolute spectrum in the case of a non-diffusing slow
variable. Predictions are confirmed numerically for the Barkley and Karma
models.

##  1 Introduction

Spiral waves are frequently observed in nature, including chemical
oscillations in the Belousouv-Zhabotinsky reaction [27, 26], cell signaling
patterns in slime molds [14], and in electrical activity in cardiac dynamics
[25, 19]. Stable spiral waves are observed in these systems, but bifurcations
to complex and unstable pattens are common and these bifurcations can have
profound results. For example, rotating spiral waves in cardiac electrical
activity have been linked to dangerous tachycardiac rhythms and the transition
to break up can lead to life-threatening fibrillation and sudden cardiac death
[19, 16]. Therefore, understanding the stability and bifurcations of spiral
waves poses interesting mathematical questions and is important in
applications.

Reaction-diffusion systems are canonical pattern forming systems describing
biological and physical processes and take on the form

$\displaystyle U_{t}=D\Delta U+F(U),\qquad U=U(y,t)\in\mathbb{R}^{n},\
D\in\mathbb{R}^{n\times n},\ y\in\mathbb{R}^{2},$ (1.1)

where $\Delta$ is the Laplacian and the smooth, typically nonlinear $F(U)$
defines kinetic reaction terms. The $n$ species of $U=[u_{1},\dots,u_{n}]^{T}$
diffuse with diffusion rates given by the entries $D_{jj}$ of the positive
diagonal matrix $D$ for $j=1,\ldots,n$. Planar spiral waves have a regular
shape and rotate with angular frequency $\omega_{0}$. Thus, these pattens are
stationary solutions of (1.1) in a polar coordinate rotating frame, and
stability can be investigated by evaluating the spectrum of the reaction-
diffusion operator linearized about the spiral wave solution.

The propagation of electrical potentials in excitable media, such as neurons
and cardiac tissue, can be described by biophysically detailed ion-channel
models given by the system

$V_{t}=\Delta V+f(V,n),\qquad n_{t}=g(V,n).$

Here, $V=V(y,t)\in\mathbb{R}$ corresponds to the electrical potential and
$n=\left(n_{1}(y,t),\dots,n_{M}(y,t)\right)^{T}\in~{}\mathbb{R}^{M}$ are $M$
dynamic gating variables explaining the opening and closing of ion channels
that facilitate the voltage propagation. Ion channel models can still be
written in the general reaction-diffusion framework, with the popular
Hodgkin–Huxley [8], Noble [15], and Beeler–Reuter [5] models falling into this
category.

In this paper, we focus on two-component reaction-diffusion models. In ion
channel models, the gating variable does not diffuse and an appropriate ion
channel models is therefore of the form

$u_{t}=\Delta u+f(u,v),\qquad v_{t}=g(u,v),$

where we now use $(u,v)$ instead of $(V,n)$. Often, small unphysical diffusion
is added to these components or included in qualitative models. Throughout, we
will use $\delta$ to correspond to the small diffusion coefficient of
interest, leading to the adjusted model

$u_{t}=\Delta u+f(u,v),\qquad v_{t}=\delta\Delta v+g(u,v)$

where $0<\delta\ll 1$. We would expect that the spectra should change smoothly
as the diffusion coefficient $\delta\rightarrow 0$, and this is indeed true
for the continuous spectra of one-dimensional periodic wave trains [17]. Yet,
we observe that spiral spectra computed for $\delta>0$ do not converge as
$\delta\searrow 0$ to the spectrum computed for $\delta=0$. Figure 1 shows the
differences in the spectra for the Barkley model, a two-variable reaction-
diffusion system, with $\delta=0.2$ and $\delta=0$. For $\delta>0$, the
spectral curves are unbounded, whereas the curves remain bounded for
$\delta=0$. Furthermore at the finite limit points, adjacent curves meet and
form the cusps seen in Figure 1. Additional changes are seen in the absolute
and point spectrum, with the absolute spectrum collapsing to short branches
that align with the essential spectrum cusp points. The effect of a rank-
deficient diffusion matrix on the structure of the spectra of planar spiral
waves has previously not been analyzed.

Figure 1: Barkley Model: Differences in the essential, absolute, and point
spectra for $\delta>0$ and $\delta=0$.

In this paper, we focus on changes in the continuous spectrum for spiral waves
in the $\delta=0$ limit and investigate the mechanisms responsible for the
discontinuities of essential spectrum observed in the limit of a rank-
deficient diffusion matrix. Under mild conditions on the linearization of the
reaction terms, Theorems 1-2 provide expressions for the spectral curves near
the limit points. Our findings indicate the discontinuity occuring in the
$\delta=0$ limit is dictated by the non-diffusing species. Furthermore, we use
the result of Theorem 1 to predict locations for the rank-deficient absolute
spectrum in Theorem 3.

We proceed as follows: Mathematical preliminaries of spiral waves and the
relevant spectral properties are described in Section 2. The main results for
the continuous spectra of planar spiral waves are derived in Sections 3-4.
Implications for the locations of absolute spectrum are discussed in Section
5. Finally, the results are applied to the Barkley and Karma models in Section
6.

##  2 Mathematical preliminaries

###  2.1 Planar spiral waves and asymptotic wave trains

Planar spiral waves are solutions to the reaction-diffusion equation (1.1)
which rigidly rotate in time with a constant angular temporal frequency
$\omega_{0}$. Spiral waves are therefore stationary solutions $U_{*}(r,\psi)$
in the co-rotating frame under polar coordinates $(r,\psi)$

$\displaystyle U_{t}=D\Delta_{r,\psi}U+\omega_{0}U_{\psi}+F(U),\qquad
U\in\mathbb{R}^{n}$

where $\Delta_{r,\psi}$ denotes the Laplacian expressed in polar coordinates.
In the limit $r\rightarrow\infty$, the spiral solutions $U_{*}(r,\psi)$ limit
to $2\pi/\kappa$-periodic functions $U_{\infty}(r+\psi/\kappa)=U_{\infty}(x)$
referred to as asymptotic wave trains, which are stationary solutions to

$\displaystyle U_{t}=DU_{xx}+\omega U_{x}+F(U)$

where $\omega=\omega_{0}/\kappa$. We focus on planar spiral waves that can be
viewed as a source which emits wave trains with positive group velocity. The
spatial wave number $\kappa$ is selected by the spiral and the nonlinear
dispersion relation $\omega=\omega_{*}(\kappa)$ of the wave train connects
$\omega$ and $\kappa$. Our main assumption is the existence of a spiral-wave
solution that depends smoothly on the diffusion coefficient $\delta$ for
$0\leq\delta\ll 1$.

###### Hypothesis (H1)

The spiral-wave solution $U_{*}$ and the asymptotic wave train $U_{\infty}$
both depend smoothly on the diffusion coefficient $\delta$ for $0\leq\delta\ll
1$.

While we do not have a proof that Hypothesis (H1) holds, this hypothesis is
well supported by numerical evidence.

###  2.2 Essential spectrum of wave trains

We are interested in the continuous spectrum of the reaction-diffusion
operator linearized about a spiral wave pattern. The reader is referred to [7,
21, 20, 9] for a detailed study of spectrum of operators in nonlinear waves.

We start with the wave train spectrum as it is directly linked to the
continuous spectrum of the spiral wave. The spectrum of the linearization

$\mathcal{L}_{\infty}V=DV_{xx}+\omega
V_{x}+F_{U}(U_{\infty})V=\lambda_{\infty}V$

of the asymptotic wave trains $U_{\infty}(x)$ on the space
$L^{2}(\mathbb{R},\mathbb{R}^{n})$ is given by the set of
$\lambda_{\infty}\in\mathbb{C}$ for which the eigenvalue problem
$\mathcal{L}_{\infty}V=\lambda_{\infty}V$ has a nontrivial solution of the
form $V(x)=\mathrm{e}^{\nu x}\bar{V}(x)$ where
$\bar{V}(x+2\pi/\kappa)=\bar{V}(x)$ and $\nu=\mathrm{i}\gamma$ for
$\gamma\in\mathbb{R}$. Wave train eigenvalues are therefore determined by
nontrivial $2\pi/\kappa$-periodic solutions $\bar{V}(x)$ of

$\displaystyle\mathcal{L}_{\infty}(\lambda,\nu)\bar{V}=D\left(\partial_{x}+\nu\right)^{2}\bar{V}+\omega\left(\partial_{x}+\nu\right)\bar{V}+F_{U}(U_{\infty})\bar{V}-\lambda_{\infty}\bar{V}=0$
(2.1)

and come in curves $\lambda_{\infty}=\lambda_{\infty}(\mathrm{i}\gamma)$
parameterized by the Floquet exponent $\nu=\mathrm{i}\gamma$. From
translational symmetry, one spectral curve includes $\lambda_{\infty}(0)=0$
for $\nu=0$ with eigenfunction $V(x)=U^{\prime}_{\infty}(x)$. The orientation
of each curve is defined as the direction of increasing $\gamma$.

###  2.3 Essential spectrum of planar spiral waves

The spectrum of a planar spiral wave is given by considering the spectrum of
the operator

$\displaystyle\mathcal{L}_{*}V=D\Delta_{r,\psi}V+\omega_{0}V_{\psi}+F_{U}(U_{*})V$
(2.2)

on $L^{2}(\mathbb{R},\mathbb{R}^{n})$ which contains isolated eigenvalues in
the point spectrum and continuous curves of essential spectrum. The essential
spectrum of the spiral wave is determined by the far-field asymptotic
dynamics, and it can be shown that the spiral wave continuous spectral curves
$\lambda_{*}(\nu)$ are related to those of the wave trains via

$\displaystyle\lambda_{*}(\nu)=\lambda_{\infty}(\nu)-\frac{\omega_{0}}{\kappa}\nu+\mathrm{i}\omega_{0}\ell,\
\ \ell\in\mathbb{Z}.$ (2.3)

Therefore, the linear dispersion relation for the spiral wave is determined by
the solvability of

$\displaystyle\mathcal{L}_{*}(\lambda,\nu)\bar{V}=D\left(\partial_{x}+\nu\right)^{2}\bar{V}+\omega\bar{V}_{x}+F_{U}(U_{\infty})\bar{V}-\lambda_{*}\bar{V}=0.$
(2.4)

Since $\nu\in\mathrm{i}\mathbb{R}$, the real parts wave train and spiral wave
continuous spectra coincide. The $\mathrm{i}\omega_{0}\ell$ term provides
additional vertically periodic branches due to the rotational symmetry of the
spiral wave.

###  2.4 Computation of essential spectra

In the remainder of the paper, we focus on spiral waves described by a two-
component reaction-diffusion system $U=(u,v)^{T}$ set in a rotating frame with
polar coordinates $(r,\psi)=(r,\phi-\omega t)$,

$\displaystyle u_{t}=$ $\displaystyle\ \Delta_{r,\psi}u+\omega
u_{\psi}+f(u,v)$ (2.5) $\displaystyle v_{t}=$ $\displaystyle\
\delta\Delta_{r,\psi}v+\omega v_{\psi}+g(u,v)$

where $0\leq\delta\ll 1$. In this framework, $u$ and $v$ are typically
referred to as the fast and slow diffusing species, respectively. Many of the
reduced systems, including the Barkley, Karma, and Morris-Lecar models, have
two components.

In these systems, the equation for the essential spectrum of the planar spiral
wave is then given by

$\displaystyle\lambda_{*}u=$
$\displaystyle\left(\partial_{x}+\mathrm{i}\gamma\right)^{2}u+\omega
u_{x}+f_{u}\left(U_{\infty}(x)\right)u+f_{v}\left(U_{\infty}(x)\right)v$ (2.6)
$\displaystyle\lambda_{*}v=\delta$
$\displaystyle\left(\partial_{x}+\mathrm{i}\gamma\right)^{2}v+\omega
v_{x}+g_{u}\left(U_{\infty}(x)\right)u+g_{v}\left(U_{\infty}(x)\right)v,$

where the imaginary part $\gamma$ of the spatial Floquet exponent
parameterizes the essential spectrum curves, and the nonlinear terms are
linearized about the wave train $U_{\infty}(x)$. In Theorems 1-2, we assume
that the derivative $g_{v}\left(U_{\infty}(x)\right)=\bar{g}$ is a constant.
To simplify notation, we remove the explicit $U_{\infty}$ dependence in the
linearizations and instead write
$f_{u,v}\left(U_{\infty}(x)\right)=f_{1,2}(x)$ and
$g_{u}\left(U_{\infty}(x)\right)=g_{1}(x)$. The observed changes in the
essential spectrum are for large values of $\gamma$ so that $|\gamma|\gg 1$.
Sending $\gamma\rightarrow\pm\infty$ corresponds to traversing the curve in
opposite directions. To focus on the limiting behavior (and the area near the
cusp point for $\delta=0$),we divide the system (2.6) by $\gamma^{2}$, define
$\alpha=1/\gamma$, and group terms by orders of $\alpha$ to arrive at

$\displaystyle-u+\alpha\left(2\mathrm{i}u_{x}\right)+\alpha^{2}\left(u_{xx}+\omega
u_{x}+f_{1}(x)u+f_{2}(x)v-\lambda u\right)=0$ (2.7) $\displaystyle-\delta
v+\alpha\left(2\mathrm{i}\delta v_{x}\right)+\alpha^{2}\left(\delta
v_{xx}+\omega v_{x}+g_{1}(x)u+\bar{g}v-\lambda v\right)=0$ (2.8)

where $(u,v)(x+2\pi/\kappa)=(u,v)(x)$. The theorems and lemmas to follow
analyze the continuous spectrum assuming the conditions in Hypothesis (H1) and
using the formulation in equations (2.7)-(2.8). We consider values of
$|\alpha|\ll 1$, where $\alpha>0$ ($\alpha<0$) corresponds to the positively
(negatively) oriented tails of essential spectrum curves.

##  3 Statement of main results: essential spectrum

Throughout, we will make use of [22, Corollary 3.4], which states that the
spectrum of a planar spiral wave is invariant under the shift
$\lambda\mapsto\lambda+\mathrm{i}\omega_{0}n$ for integers $n\in\mathbb{Z}$,
and therefore restrict our analysis to rectangles of the form

$\Lambda_{R}:=\left\\{\lambda\in\mathbb{C}:\;|\mathop{\mathrm{missing}}{Im}\nolimits\lambda|\leq\frac{1}{2},\,|\mathop{\mathrm{missing}}{Re}\nolimits\lambda|<R\right\\}$
(3.1)

where $R$ is positive or $R=\infty$.

Our first result theorem focuses on the case of vanishing diffusion in the
$v$-component. For each $|\alpha|\ll 1$, we seek to find all
$\lambda\in\Lambda_{\infty}$ for which the eigenvalue problem

$\displaystyle\lambda u$
$\displaystyle=u_{xx}+\left(\frac{2\mathrm{i}}{\alpha}+\omega\right)u_{x}-\frac{1}{\alpha^{2}}u+f_{1}(x)u+f_{2}(x)v$
(3.2) $\displaystyle\lambda v$ $\displaystyle=\omega v_{x}+\bar{g}v+g_{1}(x)u$
(3.3)

has a nontrivial $2\pi/\kappa$-periodic complex-valued solution $(u,v)(x)$.

###### Theorem 1

Fix $R>0$, $\omega>0$, and $\bar{g}\in\mathbb{R}$, and assume that
$f_{1},f_{2},g_{1}$ are given $2\pi/\kappa$-periodic functions of class
$C^{4}$, then there is a constant $\alpha_{0}>0$ so that the following is
true. There is a unique function
$\lambda_{0}^{*}:[-\alpha_{0},\alpha_{0}]\to\Lambda_{R}$ so that (3.2)-(3.3)
has a nontrivial $2\pi/\kappa$-periodic solution for $\lambda\in\Lambda_{R}$
and $|\alpha|\leq\alpha_{0}$ if and only if $\lambda=\lambda_{0}^{*}(\alpha)$.
Furthermore, the function $\lambda_{0}^{*}$ is continuous, and there are
constants $\lambda_{2,3}\in\mathbb{R}$ so that

$\displaystyle\lambda_{0}^{*}(\alpha)=\bar{g}+\lambda_{2}\alpha^{2}+2\mathrm{i}\lambda_{3}\alpha^{3}+\mathrm{o}(\alpha^{3}).$

which shows that the trace of this function in the complex plane is a cusp
emerging from $\lambda_{0}^{*}(0)=\bar{g}$.

We will provide more details of the expansions of $\lambda_{0}^{*}(\alpha)$
and the associated eigenfunctions of (3.2)-(3.3) in Proposition 6 below. We
collect two extensions of our results in the following remark.

###### Remark 1

  * •

The spectrum is invariant under the shifts
$\lambda\mapsto\lambda+\mathrm{i}\omega_{0}n,\ n\in\mathbb{Z}$ [22]. Thus,
more generally, the expansion of $\lambda_{0}^{*}(\alpha)$ is

$\displaystyle\lambda_{0}^{*}(\alpha)=\bar{g}+\mathrm{i}\omega_{0}n+\lambda_{2}\alpha^{2}+\mathrm{i}\lambda_{3}\alpha^{3}+\mathrm{o}(\alpha^{4}),\qquad
n\in\mathbb{Z}.$

In particular, the spectrum of a planar spiral wave with vanishing diffusion
features infinitely many cusps that emerge from each
$\lambda_{0}=\bar{g}+\mathrm{i}\omega_{0}n$ for $n\in\mathbb{Z}$. The cusp
structure at each $\lambda_{0}$ is formed from the positive and negative tails
of two curves, corresponding to $\alpha>0$ and $\alpha<0$, meeting at
$\lambda_{0}$.

  * •

Theorem 1 also holds if we replace $\bar{g}$ by a $2\pi/\kappa$-periodic
function $g_{2}(x)$. The expressions for the coefficients in the expansion of
$\lambda_{0}^{*}(\alpha)$ remain valid provided we use the definition
$\bar{g}:=\frac{\kappa}{2\pi}\int_{0}^{2\pi/\kappa}g_{2}(y)\,\mathrm{d}y$.

Our second result focuses on the case where the equation for $v$ includes
diffusion with a small diffusion constant $0<\delta\ll 1$. For each fixed
$0<\delta\ll 1$, we will identify a range of values of $\alpha$ with
$|\alpha|\ll 1$ for which the eigenvalue problem

$\displaystyle\lambda u$
$\displaystyle=u_{xx}+\left(\frac{2\mathrm{i}}{\alpha}+\omega\right)u_{x}-\frac{1}{\alpha^{2}}u+f_{1}(x)u+f_{2}(x)v$
(3.4) $\displaystyle\lambda v$ $\displaystyle=\delta
v_{xx}+\left(\frac{2\mathrm{i}\delta}{\alpha}+\omega\right)v_{x}-\frac{\delta}{\alpha^{2}}v+\bar{g}v+g_{1}(x)u$
(3.5)

has a nontrivial $2\pi/\kappa$-periodic complex-valued solution $(u,v)(x)$.

###### Theorem 2

Fix $\omega>0$ and $\bar{g}\in\mathbb{R}$, and assume that $f_{1},f_{2},g_{1}$
are given $2\pi/\kappa$-periodic functions of class $C^{4}$, then there are
constants $\delta_{0},s_{0}>0$ and functions
$\alpha^{*},\lambda^{*}:[-s_{0},s_{0}]\times(0,\delta_{0}]\to\mathbb{C}$ with

$\displaystyle\alpha^{*}(s,\delta)$
$\displaystyle=\frac{1}{2}\left(s+\sqrt{s^{2}+4\sqrt{\delta}}\right)$ (3.6)
$\displaystyle\lambda^{*}(s,\delta)$
$\displaystyle=\left\\{\begin{array}[]{lcl}\lambda_{0}^{*}(|s|)+\mathrm{O}(\sqrt{\delta})&&s\geq
0\\\
\bar{g}-\frac{s^{2}}{\omega}(1+\mathrm{O}(\delta^{\frac{1}{4}}))+\mathrm{O}(\sqrt{\delta})&&s\leq
0\end{array}\right.$ (3.9)

so that (3.4)-(3.5) has a nontrivial $2\pi/\kappa$-periodic solution when
$(\alpha,\lambda)=(\alpha^{*},\lambda^{*})(s,\delta)$.

Theorem 2 implies that the $\delta=0$ limit of the trace of the dispersion
curve $\lambda^{*}(\cdot,\delta)$ is given by the trace of the dispersion
curve $\lambda_{0}^{*}(\cdot)$ for $\delta=0$ plus the interval
$[\lambda_{0}^{*}(0)-s_{0}^{2}/\omega,\lambda_{0}^{*}(0)]$. In particular, the
essential spectrum at $\delta=0$ differs from the $\delta=0$ limit of the
essential spectra for $\delta>0$ (Figure 2).

Figure 2: The three panels illustrate the behavior of the dispersion curves,
and their complex conjugates, of (3.2)-(3.3) for $\delta=0$ and (3.4)-(3.5)
for fixed $0<\delta\ll 1$ in a small fixed neighborhood of $\lambda=\bar{g}$.
Panel (i) shows the trace of $\lambda_{0}^{*}(\cdot)$ given in Theorem 1 for
(3.2)-(3.3), while panel (ii) illustrates the trace of
$\lambda^{*}(\cdot,\delta)$ established in Theorem 2 for (3.4)-(3.5) with
$\delta>0$ fixed. Panel (iii) shows that the traces of
$\lambda_{0}^{*}(\cdot)$ and $\lambda^{*}(\cdot,0)$ differ.

##  4 Proofs of main results: essential spectrum

To prove Theorems 1-2, we systematically construct the eigenfunction solutions
to the appropriate system. Lemmas 1-3 and Proposition 4 assume that $v$ is a
given smooth function and establish a $2\pi/\kappa$-periodic solution
$u=\mathcal{D}(\lambda,\alpha)v$ for (3.2). These results for the $u$-equation
are independent of $\delta$, and the cases $\delta=0$ and $\delta>0$ are then
handled separately in the construction of the $v$-eigenfunction. Lemma 5 and
Proposition 6 use and refine the form of $u$ to find a $2\pi/\kappa$-periodic
solution for (3.3) when $\delta=0$, and Theorem 2 follows a similar outline
for the case of $\delta>0$. Throughout the proofs, we will ignore that the
coefficient functions depend on $\delta$: we assumed in Hypothesis (H1) that
these functions depend smoothly on $\delta$ for $0\leq\delta\ll 1$ and
including this dependence does not change our arguments or expansions since
the $\delta$-dependence shows up at higher order.

We note that (3.4)-(3.5) is a singular perturbation of (3.2)-(3.3) since
$0\leq\delta\ll 1$. However, due to the presence of the additional small
parameter $0\leq\alpha\ll 1$, (3.4)-(3.5) cannot be rescaled so that there is
a well defined limit as $(\alpha,\delta)$ approaches zero. In particular,
methods such as geometric singular perturbation theory are not readily
applicable.

Throughout, we will fix a constant $T>0$ and denote the stable and unstable
spectral projections of a hyperbolic matrix $A$ by $P^{\mathrm{s}}$ and
$P^{\mathrm{u}}$, respectively. We say a function is continuously
differentiable on a closed interval if it is differentiable on the open
interval, the one-sided derivatives at the boundary points exist, and the
resulting derivative is continuous on the closed interval. We denote the
Banach space of linear bounded operators between two given Banach spaces $X$
and $Y$ by $L(X,Y)$.

###  4.1 Periodic solutions of linear systems of differential equations

For each natural number $k\geq 0$, we define the spaces

$\displaystyle
Y^{k}_{\mathrm{per}}:=C^{k}(\mathbb{R}/T\mathbb{Z},\mathbb{C}^{n}),\qquad
Y^{0}:=C^{0}([0,T],\mathbb{C}^{n}),\qquad
Y^{k+1}:=C^{k}(\mathbb{R}/T\mathbb{Z},\mathbb{C}^{n})\cap
C^{k+1}([0,T],\mathbb{C}^{n}).$

Consider the inhomogeneous linear differential equation

$U_{x}=AU+G(x),\qquad U\in\mathbb{C}^{n},\qquad 0\leq x\leq T.$ (4.1)

The following lemma establishes the existence of $T$-periodic solutions of
(4.1).

###### Lemma 1

Fix $T>0$ and assume that the matrix $A\in\mathbb{C}^{n\times n}$ is
hyperbolic, then the following is true.

  * •

For each $G\in Y^{0}$ equation (4.1) has a unique $T$-periodic solution
$U=\mathcal{S}G\in Y^{1}$, and the operator $\mathcal{S}\in L(Y^{0},Y^{1})$ is
given by

$\displaystyle[\mathcal{S}G](x)=$ $\displaystyle\
\mathrm{e}^{Ax}\left(1-\mathrm{e}^{AT}\right)^{-1}\int_{0}^{T}\mathrm{e}^{A(T-s)}P^{\mathrm{s}}G(s)\,\mathrm{d}s\
+\int_{0}^{x}\mathrm{e}^{A(x-s)}P^{\mathrm{s}}G(s)\,\mathrm{d}s\ $
$\displaystyle+\mathrm{e}^{A(x-T)}\left(1-\mathrm{e}^{-AT}\right)^{-1}\int_{T}^{0}\mathrm{e}^{-As}P^{\mathrm{u}}G(s)\,\mathrm{d}s\
+\int_{T}^{x}\mathrm{e}^{A(x-s)}P^{\mathrm{u}}G(s)\,\mathrm{d}s\ ,\qquad 0\leq
x\leq T.$

  * •

For each $k\geq 0$, we have $\mathcal{S}\in
L(Y^{k}_{\mathrm{per}},Y^{k+1}_{\mathrm{per}})$.

  * •

For each $k\geq 1$, we can write $\mathcal{S}$ as

$\displaystyle\mathcal{S}G=A^{-k}\mathcal{S}\frac{\mathrm{d}^{k}G}{\mathrm{d}x^{k}}-\sum_{j=1}^{k}A^{-j}\frac{\mathrm{d}^{j-1}G}{\mathrm{d}x^{j-1}},\qquad
G\in Y^{k}_{\mathrm{per}}.$ (4.2)

  * Proof. Let $E^{\mathrm{s,u}}:=\mathcal{R}\left(P^{\mathrm{s,u}}\right)$ denote the generalized stable and unstable eigenspaces of the matrix $A$. Using the variation-of-constants formula, the general solution of (4.1) is given by

$\displaystyle
U(x)=\mathrm{e}^{Ax}a^{\mathrm{s}}+\mathrm{e}^{A(x-T)}a^{\mathrm{u}}+\int_{0}^{x}\mathrm{e}^{A(x-s)}P^{\mathrm{s}}G(s)\,\mathrm{d}s\
+\int_{T}^{x}\mathrm{e}^{A(x-s)}P^{\mathrm{u}}G(s)\,\mathrm{d}s,\qquad 0\leq
x\leq T$

with $a^{\mathrm{s,u}}\in E^{\mathrm{s,u}}$. The coefficients
$a^{\mathrm{s,u}}$ are uniquely determined upon enforcing the periodic
boundary condition $U(0)=U(T)$, which yields the unique $T$-periodic solution
$U(x)=[\mathcal{S}G](x)$ given by

$\displaystyle[\mathcal{S}G](x):=$ $\displaystyle\
\mathrm{e}^{Ax}\left(1-\mathrm{e}^{AT}\right)^{-1}\int_{0}^{T}\mathrm{e}^{A(T-s)}P^{\mathrm{s}}G(s)\,\mathrm{d}s\
+\int_{0}^{x}\mathrm{e}^{A(x-s)}P^{\mathrm{s}}G(s)\,\mathrm{d}s\ $ (4.3)
$\displaystyle+\mathrm{e}^{A(x-T)}\left(1-\mathrm{e}^{-AT}\right)^{-1}\int_{T}^{0}\mathrm{e}^{-As}P^{\mathrm{u}}G(s)\,\mathrm{d}s\
+\int_{T}^{x}\mathrm{e}^{A(x-s)}P^{\mathrm{u}}G(s)\,\mathrm{d}s$

for $G\in Y^{0}$.

Next, it follows directly from (4.1) and periodicity of $\mathcal{S}G$ that
the derivative of $\mathcal{S}G$ is periodic. In particular, $G\in
Y^{k}_{\mathrm{per}}$ implies $\mathcal{S}G\in Y^{k+1}_{\mathrm{per}}$.

Finally, when $G\in Y^{k}_{\mathrm{per}}$ for some $k\geq 1$, we apply
integration-by-parts $k$-times in each of the integral terms that appear in
the expression for the solution operator $\mathcal{S}$. Upon each integration
by parts, most of the boundary terms cancel due to the $T$-periodicity of $G$,
and the remaining terms combine to provide the identity (4.2). We omit the
details of this calculation.

###  4.2 Periodic solutions of linear second-order equations

Next, we apply these results to second-order equations of the form

$u_{xx}=a_{21}u+a_{22}u_{x}+g(x),\qquad u\in\mathbb{C},\qquad 0\leq x\leq T.$
(4.4)

We write $P_{1}$ for the projection of $\mathbb{C}^{2}$ onto the first
component and define the vector $e_{2}:=\begin{pmatrix}0\\\
1\end{pmatrix}\in\mathbb{C}^{2}$. Finally, we modify our function spaces and
define

$\displaystyle
X^{k}_{\mathrm{per}}:=C^{k}(\mathbb{R}/T\mathbb{Z},\mathbb{C}),\qquad
X^{0}:=C^{0}([0,T],\mathbb{C}),\qquad
X^{k+1}:=C^{k}(\mathbb{R}/T\mathbb{Z},\mathbb{C})\cap
C^{k+1}([0,T],\mathbb{C})$

for any natural number $k\geq 0$.

###### Lemma 2

Fix $T>0$ and assume that $a_{21},a_{22}$ are such that the matrix
$A:=\begin{pmatrix}0&1\\\ a_{21}&a_{22}\end{pmatrix}\in\mathbb{C}^{2\times 2}$
has eigenvalues $\nu^{\mathrm{s}},\nu^{\mathrm{u}}$ with
$\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}<0<\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}$,
then the following is true.

  * •

For each $g\in X^{0}$ equation (4.4) has a unique $T$-periodic solution
$u=\mathcal{T}g\in X^{2}$ given by

$\displaystyle[\mathcal{T}g](x)=$ $\displaystyle\ [P_{1}\mathcal{S}e_{2}g](x)$
$\displaystyle=$ $\displaystyle\
P_{1}\left[\mathrm{e}^{\nu^{\mathrm{s}}x}\left(1-\mathrm{e}^{\nu^{\mathrm{s}}T}\right)^{-1}\int_{0}^{T}\mathrm{e}^{\nu^{\mathrm{s}}(T-s)}P^{\mathrm{s}}e_{2}g(s)\,\mathrm{d}s\
+\int_{0}^{x}\mathrm{e}^{\nu^{\mathrm{s}}(x-s)}P^{\mathrm{s}}e_{2}g(s)\,\mathrm{d}s\
\right.$ (4.5)
$\displaystyle\left.+\mathrm{e}^{\nu^{\mathrm{u}}(x-T)}\left(1-\mathrm{e}^{-\nu^{\mathrm{u}}T}\right)^{-1}\int_{T}^{0}\mathrm{e}^{-\nu^{\mathrm{u}}s}P^{\mathrm{u}}e_{2}g(s)\,\mathrm{d}s\
+\int_{T}^{x}\mathrm{e}^{\nu^{\mathrm{u}}(x-s)}P^{\mathrm{u}}e_{2}g(s)\,\mathrm{d}s\
\right],\qquad 0\leq x\leq T,$

and we have

$\displaystyle\|\mathcal{T}\|_{L(X^{0},X^{0}_{\mathrm{per}})}\leq\frac{2}{1-\mathrm{e}^{-\min(|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s,u}}|)T}}\left(\frac{|P_{1}P^{\mathrm{s}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}|}+\frac{|P_{1}P^{\mathrm{u}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}|}\right).$
(4.6)

  * •

For each $k\geq 1$, we have $\mathcal{T}\in
L(X^{k}_{\mathrm{per}},X^{k+2}_{\mathrm{per}})$ with

$\displaystyle\mathcal{T}g=P_{1}A^{-k}\mathcal{S}e_{2}\frac{\mathrm{d}^{k}g}{\mathrm{d}x^{k}}-\sum_{j=1}^{k}P_{1}A^{-j}e_{2}\frac{\mathrm{d}^{j-1}g}{\mathrm{d}x^{j-1}},\quad
g\in X^{k}_{\mathrm{per}},$ (4.7)

and for $g\in X^{k}_{\mathrm{per}}$

$\displaystyle\left|P_{1}A^{-k}\mathcal{S}e_{2}\frac{\mathrm{d}^{k}g}{\mathrm{d}x^{k}}\right|_{X^{0}_{\mathrm{per}}}\leq\frac{2}{1-\mathrm{e}^{-\min(|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s,u}}|)T}}\left(\frac{|P_{1}P^{\mathrm{s}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}||\nu^{\mathrm{s}}|^{k}}+\frac{|P_{1}P^{\mathrm{u}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}||\nu^{\mathrm{u}}|^{k}}\right)\left|\frac{\mathrm{d}^{k}g}{\mathrm{d}x^{k}}\right|_{X^{0}_{\mathrm{per}}}.$
(4.8)

  * Proof. The second-order equation (4.4) can be rewritten as the system

$\displaystyle U_{x}=\begin{pmatrix}0&1\\\
a_{21}&a_{22}\end{pmatrix}U+\begin{pmatrix}0\\\
g(x)\end{pmatrix}:=AU+e_{2}g(x),\qquad U=\begin{pmatrix}u\\\
u_{x}\end{pmatrix}.$ (4.9)

Since $A$ is hyperbolic by assumption, Lemma 1 shows that (4.9) has a unique
$T$-periodic solution in $Y^{1}$ for each $g\in X^{0}$ and that this solution
is given by $U=\mathcal{S}e_{2}g$. Hence,
$u=P_{1}\mathcal{S}e_{2}g=:\mathcal{T}g$ is the unique $T$-periodic solution
of (4.4) in $X^{2}$. Substituting $\mathcal{S}$ from Lemma 1 and using that
$\mathrm{e}^{Ax}P^{\mathrm{s}}=\mathrm{e}^{\nu^{\mathrm{s}}x}P^{\mathrm{s}}$
and
$\mathrm{e}^{Ax}P^{\mathrm{u}}=\mathrm{e}^{\nu^{\mathrm{u}}x}P^{\mathrm{u}}$
establishes (4.5) and shows that $\mathcal{T}\in
L(X^{0},X^{2}_{\mathrm{per}})$.

To prove (4.6), we note that

$\displaystyle\|\mathrm{e}^{Ax}|_{E^{\mathrm{s}}}\|\leq\mathrm{e}^{\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}x}\text{
for }x\geq
0,\qquad\|\mathrm{e}^{Ax}|_{E^{\mathrm{u}}}\|\leq\mathrm{e}^{\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}x}\text{
for }x\leq 0$

Hence, we obtain from (4.5) that

$\displaystyle|\mathcal{T}g|_{X^{0}}\leq$ $\displaystyle\ \max_{0\leq x\leq
T}\left[\left(\frac{\mathrm{e}^{\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}x}}{1-\mathrm{e}^{\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}T}}\int_{0}^{T}\mathrm{e}^{\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}(T-s)}\,\mathrm{d}s\
+\int_{0}^{x}\mathrm{e}^{\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}(x-s)}\,\mathrm{d}s\
\right)|P_{1}P^{\mathrm{s}}e_{2}|\right.$
$\displaystyle\qquad+\left.\left(\frac{\mathrm{e}^{\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}(x-T)}}{1-\mathrm{e}^{-\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}T}}\int_{T}^{0}\mathrm{e}^{-\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}s}\,\mathrm{d}s\
+\int_{T}^{x}\mathrm{e}^{\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}(x-s)}\,\mathrm{d}s\
\right)|P_{1}P^{\mathrm{u}}e_{2}|\right]|g|_{X^{0}}$ $\displaystyle\leq$
$\displaystyle\
\frac{2}{1-\mathrm{e}^{-\min(|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s,u}}|)T}}\left(\frac{|P_{1}P^{\mathrm{s}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}|}+\frac{|P_{1}P^{\mathrm{u}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}|}\right)|g|_{X^{0}},$

which establishes (4.6).

Finally, for $g\in X^{k}_{\mathrm{per}}$, equation (4.2) in Lemma 1 shows that
$\mathcal{T}$ has the representation given in (4.7). Furthermore, proceeding
as in the preceding paragraph and using the estimates

$\displaystyle\|A^{-k}|_{E^{\mathrm{s}}}\|\leq\frac{1}{|\nu^{\mathrm{s}}|^{k}},\qquad\|A^{-k}|_{E^{\mathrm{u}}}\|\leq\frac{1}{|\nu^{\mathrm{u}}|^{k}}$

establishes the estimate (4.8).

###  4.3 Solutions to the linearized eigenvalue problem for the $u$-component

We fix $T=2\pi/\kappa$ and consider the differential equation

$u_{xx}=-\left(\frac{2\mathrm{i}}{\alpha}+\omega\right)u_{x}+\frac{1}{\alpha^{2}}u+g(x),\qquad
u\in\mathbb{C},\qquad 0\leq x\leq 2\pi/\kappa$ (4.10)

for $|\alpha|\ll 1$. The following result establishes the existence of
$2\pi/\kappa$-periodic solutions of (4.10). We note that the results of Lemmas
3-5 and Propositions 4-6 hold for $|\alpha|\ll 1$. For clarity, we consider
the case $\alpha>0$ in the proofs.

###### Lemma 3

Fix $\omega>0$, then there are constants $\alpha_{0}>0$ and $C_{0}>0$ so that
the following is true. For each $|\alpha|\leq\alpha_{0}$ and each $g\in
X^{0}$, equation (4.10) has a unique $2\pi/\kappa$-periodic solution
$u=\mathcal{T}(\alpha)g\in X^{2}$, and the operator $\mathcal{T}(\alpha)$
satisfies $\|\mathcal{T}(\alpha)\|_{L(X^{0},X^{0}_{\mathrm{per}})}\leq
C_{0}|\alpha|$. Furthermore, we have

$\displaystyle|\mathcal{T}(\alpha)g|_{X^{0}_{\mathrm{per}}}$
$\displaystyle\leq C_{0}\alpha^{2}|g|_{X^{1}_{\mathrm{per}}},\qquad\mbox{ for
}g\in X^{1}_{\mathrm{per}}$ (4.11)
$\displaystyle\left|\mathcal{T}(\alpha)g+\alpha^{2}g+2\mathrm{i}\alpha^{3}g_{x}\right|_{X^{0}_{\mathrm{per}}}$
$\displaystyle\leq C_{0}\alpha^{4}|g|_{X^{3}_{\mathrm{per}}},\qquad\mbox{ for
}g\in X^{3}_{\mathrm{per}}.$ (4.12)

  * Proof. First, we rewrite the second order equation (4.10) as the first-order system

$U_{x}=\begin{pmatrix}0&1\\\
\frac{1}{\alpha^{2}}&-\frac{2i}{\alpha}-\omega\end{pmatrix}U+\begin{pmatrix}0\\\
1\end{pmatrix}g(x)=:A(\alpha)U+e_{2}g(x),\qquad U=\begin{pmatrix}u\\\
u_{x}\end{pmatrix}.$ (4.13)

The eigenvalues $\nu_{\pm}$ of $A(\alpha)$ are roots of the polynomial
$\nu^{2}+\left(\frac{2i}{\alpha}+\omega\right)\nu-\frac{1}{\alpha^{2}}$, which
are given by

$\displaystyle\mathop{\mathrm{missing}}{Re}\nolimits\nu_{\pm}=\pm(1+\mathrm{O}(\sqrt{\alpha}))\sqrt{\frac{\omega}{2\alpha}},\qquad\mathop{\mathrm{missing}}{Im}\nolimits\nu_{\pm}=\frac{-1+\mathrm{O}(\sqrt{\alpha})}{\alpha}.$
(4.14)

In particular, there is an $\alpha_{0}>0$ so that the matrices $A(\alpha)$ are
hyperbolic with
$|\mathop{\mathrm{missing}}{Re}\nolimits\nu_{\pm}|\geq\sqrt{\omega/4\alpha}$
for all $0<\alpha\leq\alpha_{0}$, and Lemma 2 guarantees that (4.13) has a
unique $2\pi/\kappa$-periodic solution for each $0<\alpha\leq\alpha_{0}$ and
that this solution is given by $u=\mathcal{T}(\alpha)g$.

It remains to verify the estimates for $\mathcal{T}(\alpha)$. For $0<\alpha\ll
1$, the stable and unstable eigenspaces of $A(\alpha)$ are given by

$\displaystyle E^{\mathrm{s}}=\mathbb{R}\begin{pmatrix}\frac{1}{\nu_{-}}\\\
1\end{pmatrix},\qquad
E^{\mathrm{u}}=\mathbb{R}\begin{pmatrix}\frac{1}{\nu_{+}}\\\ 1\end{pmatrix}$

and the associated spectral projections can be written as

$\displaystyle
P^{\mathrm{s}}U=\left(\frac{1}{\nu_{+}}-\frac{1}{\nu_{-}}\right)^{-1}\begin{pmatrix}\frac{1}{\nu_{-}}\\\
1\end{pmatrix}\left(-1,\frac{1}{\nu_{+}}\right)U,\qquad
P^{\mathrm{u}}U=\left(\frac{1}{\nu_{-}}-\frac{1}{\nu_{+}}\right)^{-1}\begin{pmatrix}\frac{1}{\nu_{+}}\\\
1\end{pmatrix}\left(-1,\frac{1}{\nu_{-}}\right)U.$

We conclude from (4.14) that for all $0<\alpha\ll 1$

$\displaystyle|\nu_{\pm}|=\frac{1+\mathrm{O}(\sqrt{\alpha})}{\alpha},\quad\frac{1}{|\nu_{\pm}|}=\alpha(1+\mathrm{O}(\sqrt{\alpha})),\quad\frac{1}{|\nu_{-}-\nu_{+}|}=\frac{\sqrt{\alpha}(1+\mathrm{O}(\alpha^{2}))}{2\sqrt{\omega}},\quad\frac{1}{1-\mathrm{e}^{-2\pi\min(|\mathop{\mathrm{missing}}{Re}\nolimits\nu_{\pm}|)/\kappa}}\leq
2.$

Hence, we have

$\displaystyle|P_{1}P^{\mathrm{s}}e_{2}|$
$\displaystyle=\left|\frac{1}{\nu_{-}\nu_{-}+}\left(\frac{1}{\nu_{+}}-\frac{1}{\nu_{-}}\right)^{-1}\right|=\frac{1}{|\nu_{-}-\nu_{+}|}=\frac{\sqrt{\alpha}(1+\mathrm{O}(\alpha^{2}))}{2\sqrt{\omega}}$

and similarly

$\displaystyle|P_{1}P^{\mathrm{u}}e_{2}|=\frac{\sqrt{\alpha}(1+\mathrm{O}(\alpha^{2}))}{2\sqrt{\omega}}.$

Substituting these expressions into (4.6), we see that there is an
$\alpha_{0}>0$ so that

$\displaystyle\|\mathcal{T}(\alpha)\|_{L(X^{0},X^{0}_{\mathrm{per}})}\stackrel{{\scriptstyle(\ref{e:test})}}{{\leq}}\frac{2}{1-\mathrm{e}^{-2\pi\min(|\mathop{\mathrm{missing}}{Re}\nolimits\nu_{\pm}|)/\kappa}}\left(\frac{|P_{1}P^{\mathrm{s}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}|}+\frac{|P_{1}P^{\mathrm{u}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}|}\right)\leq\frac{4\alpha}{\omega}$

for all $0<\alpha<\alpha_{0}$ as claimed.

Next, we establish (4.11) by using the estimate (4.8) in the expression (4.7)
with $k=1$. Since

$\displaystyle A^{-1}=\alpha^{2}\begin{pmatrix}\frac{2i}{\alpha}+\omega&1\\\
\frac{1}{\alpha^{2}}&0\end{pmatrix},$

we see that $P_{1}A^{-1}e_{2}=\alpha^{2}$. Hence, (4.7)-(4.8) with $k=1$ for
$g\in X^{1}_{\mathrm{per}}$ yield

$\displaystyle|\mathcal{T}(\alpha)g|_{X^{0}_{\mathrm{per}}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{d:t})}}{{\leq}}\left|P_{1}A^{-1}\mathcal{S}e_{2}g_{x}\right|_{X^{0}_{\mathrm{per}}}+|P_{1}A^{-1}e_{2}||g|$
$\displaystyle\stackrel{{\scriptstyle(\ref{d:test})}}{{\leq}}4\left(\frac{|P_{1}P^{\mathrm{s}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}|}\frac{1}{|\nu^{s}|}+\frac{|P_{1}P^{\mathrm{u}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}|}\frac{1}{|\nu^{\mathrm{u}}|}\right)\left|g_{x}\right|_{X^{0}_{\mathrm{per}}}+|P_{1}A^{-1}e_{2}||g|_{X^{0}_{\mathrm{per}}}$
$\displaystyle\leq\frac{8}{\omega}\alpha^{2}\left|g_{x}\right|_{X^{0}_{\mathrm{per}}}+\alpha^{2}|g|_{X^{0}_{\mathrm{per}}}$

as claimed.

Finally, we take $g\in X^{3}_{\mathrm{per}}$. Calculating $P_{1}A^{-j}e_{2}$
iteratively for $j=1,2,3$ shows that

$\displaystyle P_{1}A^{-1}e_{2}=\alpha^{2},\qquad
P_{1}A^{-2}e_{2}=2\mathrm{i}\alpha^{3}+\omega\alpha^{4},\qquad|P_{1}A^{-3}e_{2}|\leq
5\alpha^{4},$

and (4.7)-(4.8) with $k=3$ gives

$\displaystyle|\mathcal{T}(\alpha)g+\alpha^{2}g+2\mathrm{i}\alpha^{3}g_{x}|_{X^{0}_{\mathrm{per}}}$
$\displaystyle\leq\left|P_{1}A^{-3}\mathcal{S}e_{2}g_{xxx}\right|_{X^{0}_{\mathrm{per}}}+\omega\alpha^{4}|g_{x}|_{X^{0}_{\mathrm{per}}}+|P_{1}A^{-3}e_{2}||g_{xx}|_{X^{0}_{\mathrm{per}}}$
$\displaystyle\leq
4\left(\frac{|P_{1}P^{\mathrm{s}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{s}}|}\frac{1}{|\nu^{s}|^{3}}+\frac{|P_{1}P^{\mathrm{u}}e_{2}|}{|\mathop{\mathrm{missing}}{Re}\nolimits\nu^{\mathrm{u}}|^{3}}\frac{1}{|\nu^{\mathrm{u}}|^{3}}\right)|g_{xxx}|_{X^{0}_{\mathrm{per}}}+(5+\omega)\alpha^{4}|g|_{X^{2}_{\mathrm{per}}}$
$\displaystyle\leq C_{0}\alpha^{4}|g|_{X^{3}_{\mathrm{per}}},$

which completes the proof of (4.12).

Next, we consider the eigenvalue problem (3.2) for the $u$-component, which is
given by the differential equation

$u_{xx}=-\left(\frac{2\mathrm{i}}{\alpha}+\omega\right)u_{x}+\frac{1}{\alpha^{2}}u+(\lambda-
f_{1}(x))u-f_{2}(x)v\qquad u\in\mathbb{C},\qquad 0\leq x\leq 2\pi/\kappa.$
(4.15)

We will vary $\lambda$ in the rectangle $\Lambda_{R}$ of width $2R$ and height
one that we defined in (3.1). For fixed functions $f_{1},f_{2}\in
X^{0}_{\mathrm{per}}$, we define the multiplication operators
$B_{1}(\lambda),B_{2}\in L(X^{0})$ via $B_{1}(\lambda)v:=(\lambda-f_{1}(x))v$
and $B_{2}v:=-f_{2}v$ and note that the map

$\displaystyle B_{1}:\Lambda_{R}\longrightarrow
L(X^{0}),\quad\lambda\longmapsto B_{1}(\lambda)$

is analytic. The next result establishes the existence of
$2\pi/\kappa$-periodic solutions of (4.15).

###### Proposition 4

For each fixed choice of $R>0$, $\omega>0$, and $f_{1},f_{2}\in
X^{0}_{\mathrm{per}}$, there are constants $\alpha_{1}>0$ and $C_{1}>0$ so
that the following is true. For each
$(\lambda,\alpha)\in\Lambda_{R}\times[-\alpha_{1},\alpha_{1}]\setminus\\{0\\}$
and $v\in X^{0}$, equation (4.15) has a unique $2\pi/\kappa$-periodic solution
$u=\mathcal{D}(\lambda,\alpha)g\in X^{2}$, where

$\displaystyle\mathcal{D}(\lambda,\alpha)=(1-\mathcal{T}(\alpha)B_{1}(\lambda))^{-1}\mathcal{T}(\alpha)B_{2}.$

The function

$\displaystyle\mathcal{D}(\cdot,\alpha):\quad\Lambda_{R}\longrightarrow
L(X^{0},X^{0}_{\mathrm{per}}),\quad\lambda\longmapsto\mathcal{D}(\lambda,\alpha)$

is well defined and analytic in $\lambda$ for each fixed
$\alpha\in[-\alpha_{0},\alpha_{0}]\setminus\\{0\\}$, and we have

$\displaystyle\|\mathcal{D}(\lambda,\alpha)\|_{L(X^{0},X^{0}_{\mathrm{per}})}\leq
C_{1}\alpha,\qquad\|\mathcal{D}_{\lambda}(\lambda,\alpha)\|_{L(X^{0},X^{0}_{\mathrm{per}})}\leq
C_{1}\alpha^{2}.$ (4.16)

Finally, setting $\mathcal{D}(\lambda,0):=0$, the mapping
$\mathcal{D}:\Lambda_{R}\times[-\alpha_{1},\alpha_{1}]\to
L(X^{0},X^{0}_{\mathrm{per}})$ is analytic in $\lambda$, and $\mathcal{D}$ and
its derivatives in $\lambda$ are continuous in $\alpha$.

  * Proof. The eigenvalue problem (4.15) coincides with (4.10) provided we set $g(x)=B_{1}(\lambda)u+B_{2}v$ in (4.10). Lemma 3 therefore shows that there is an $\alpha_{0}>0$ so that (4.15) has a solution $u\in X^{2}$ for some $\alpha\in(0,\alpha_{0}]$ if and only $u\in X^{0}_{\mathrm{per}}$ satisfies

$u=\mathcal{T}(\alpha)\left(B_{1}(\lambda)u+B_{2}v\right),$ (4.17)

where $\mathcal{T}(\alpha)$ has been defined in Lemma 3. We set
$M:=\max(|f_{1}|_{X^{0}_{\mathrm{per}}}+R,|f_{2}|_{X^{0}_{\mathrm{per}}})$ and
note that Lemma 3 implies that there is a constant $C_{0}$ so that the
estimates

$\displaystyle\|\mathcal{T}(\alpha)B_{1}(\lambda)\|_{X^{0}_{\mathrm{per}}}\leq
C_{0}M\alpha,\qquad\|\mathcal{T}(\alpha)B_{2}\|_{X^{0}_{\mathrm{per}}}\leq
C_{0}M\alpha$

hold uniformly in $(\lambda,\alpha)\in\Lambda_{R}\times(0,\alpha_{0}]$.
Choosing $\alpha_{1}:=\min(\alpha_{0},\frac{1}{2C_{0}M})$, we see that
$\|\mathcal{T}(\alpha)B_{1}(\lambda)\|_{X^{0}_{\mathrm{per}}}\leq\frac{1}{2}$
for $(\lambda,\alpha)\in\Lambda_{R}\times(0,\alpha_{1}]$, and the fixed-point
equation (4.17) therefore has a unique solution

$\displaystyle
u=(1-\mathcal{T}(\alpha)B_{1}(\lambda))^{-1}\mathcal{T}(\alpha)B_{2}v=:\mathcal{D}(\lambda,\alpha)v$

in $X^{0}_{\mathrm{per}}$ for each
$(\lambda,\alpha)\in\Lambda_{R}\times(0,\alpha_{1}]$ and $v\in X^{0}$.
Furthermore, we have

$\displaystyle\mathcal{D}(\lambda,\alpha)=(1-\mathcal{T}(\alpha)B_{1}(\lambda))^{-1}\mathcal{T}(\alpha)B_{2}\in
L(X^{0},X^{0}_{\mathrm{per}})$

with

$\displaystyle\|\mathcal{D}(\lambda,\alpha)\|_{L(X^{0},X^{0}_{\mathrm{per}})}=\|(1-\mathcal{T}(\alpha)B_{1}(\lambda))^{-1}\|_{L(X^{0},X^{0}_{\mathrm{per}})}\|\mathcal{T}(\alpha)B_{2}\|_{X^{0}_{\mathrm{per}}}\leq
2\|\mathcal{T}(\alpha)B_{2}\|_{X^{0}_{\mathrm{per}}}\leq 2C_{0}M\alpha$

for $(\lambda,\alpha)\in\Lambda_{R}\times(0,\alpha_{1}]$.

Analyticity of $\mathcal{D}(\lambda,\alpha)$ in $\lambda$ follows from
analyticity of $B_{1}(\lambda)$. To estimate the derivative
$\mathcal{D}_{\lambda}(\lambda,\alpha)$, we differentiate the identity

$\displaystyle(1-\mathcal{T}(\alpha)B_{1}(\lambda))\mathcal{D}(\lambda,\alpha)=\mathcal{T}(\alpha)B_{2}$

with respect to $\lambda$ to get

$\displaystyle-\mathcal{T}(\alpha)\frac{\mathrm{d}B_{1}}{\mathrm{d}\lambda}(\lambda))\mathcal{D}(\lambda,\alpha)+(1-\mathcal{T}(\alpha)B_{1}(\lambda))\mathcal{D}_{\lambda}(\lambda,\alpha)=0.$

Since $\frac{\mathrm{d}B_{1}}{\mathrm{d}\lambda}(\lambda)=1$, we find

$\displaystyle\mathcal{D}_{\lambda}(\lambda,\alpha)=(1-\mathcal{T}(\alpha)B_{1}(\lambda))^{-1}\mathcal{T}(\alpha)\mathcal{D}(\lambda,\alpha)$

and therefore

$\displaystyle\|\mathcal{D}_{\lambda}(\lambda,\alpha)\|_{L(X^{0},X^{0}_{\mathrm{per}})}\leq\|\left(1-\mathcal{T}(\alpha)B_{1}(\lambda)\right)^{-1}\|_{L(X^{0},X^{0}_{\mathrm{per}})}\|\mathcal{T}(\alpha)\|_{X^{0}_{\mathrm{per}}}\|\mathcal{D}(\lambda,\alpha)\|_{L(X^{0},X^{0}_{\mathrm{per}})}\leq
4C_{0}^{2}M\alpha^{2}$

as claimed. Note that $\mathcal{D}(\lambda,\alpha)$ and its derivatives in
$\lambda$ converge to zero in $L(X^{0},X^{0}_{\mathrm{per}})$ as
$\alpha\searrow 0$, and we can therefore extend $\mathcal{D}(\lambda,\alpha)$
into $\alpha=0$ by setting $\mathcal{D}(\lambda,0)=0$.

###  4.4 Solutions to the linearized eigenvalue problem for the $v$-component
for $\delta=0$

It remains to find nontrivial $2\pi/\kappa$-periodic solutions of the
eigenvalue problem (3.3), given by

$\lambda v=\omega v_{x}+\bar{g}v+g_{1}(x)u$ (4.18)

where $u=\mathcal{D}(\lambda,\alpha)v$ is the unique $2\pi/\kappa$-periodic
solution of (4.15) for a given function $v$, whose existence was shown in
Proposition 4. Our first result shows existence of solutions to (4.18) without
enforcing periodicity in $x$.

###### Lemma 5

Fix $R>0$, $\omega>0$, $\bar{g}\in\mathbb{R}$, and $f_{1},f_{2},g_{1}\in
X^{0}_{\mathrm{per}}$, then there are constants $\alpha_{2}>0$ and $C_{2}>0$
so that the following is true. For each
$(\lambda,\alpha)\in\Lambda_{R}\times[-\alpha_{2},\alpha_{2}]\setminus\\{0\\}$,
equation (4.18) with $u=\mathcal{D}(\lambda,\alpha)v$ has a nontrivial
solution $v=\mathcal{C}(\lambda,\alpha)\in X^{1}$, and this solution is unique
up to a constant factor. In addition:

  * •

We have

$\displaystyle\mathcal{C}(\lambda,\alpha)$
$\displaystyle=(1+\mathcal{Q}(\lambda,\alpha))^{-1}\mathrm{e}^{(\lambda-\bar{g})x/\omega}$
(4.19) $\displaystyle[\mathcal{Q}(\lambda,\alpha)h](x)$
$\displaystyle:=\frac{1}{\omega}\int_{0}^{x}\mathrm{e}^{(\lambda-\bar{g})(x-y)/\omega}g_{1}(y)[\mathcal{D}(\lambda,\alpha)h](y)\,\mathrm{d}y,\qquad
h\in X^{0}.$ (4.20)

  * •

For each fixed $\alpha\in[-\alpha_{2},\alpha_{2}]\setminus\\{0\\}$, the
mappings

$\displaystyle\mathcal{Q}(\cdot,\alpha):\;\Lambda_{R}\to
L(X^{0}),\;\lambda\mapsto\mathcal{Q}(\lambda,\alpha),\qquad\mathcal{C}(\cdot,\alpha):\;\Lambda_{R}\to
X^{0},\;\lambda\mapsto\mathcal{C}(\lambda,\alpha)$

are analytic in $\lambda$ with

$\displaystyle\left\|\frac{\mathrm{d}^{k}}{\mathrm{d}\lambda^{k}}\mathcal{Q}(\lambda,\alpha)\right\|_{L(X^{0})}\leq
C_{2}\alpha,\quad\left|\frac{\mathrm{d}^{k}}{\mathrm{d}\lambda^{k}}\mathcal{C}(\lambda,\alpha)\right|_{X^{0}}\leq
C_{2},\quad\left|\mathcal{C}(\lambda,\alpha)-\mathrm{e}^{(\lambda-\bar{g})x/\omega}\right|_{X^{0}}\leq
C_{2}\alpha$ (4.21)

for $k=0,1$ uniformly in $(\lambda,\alpha)\in\Lambda_{R}\times(0,\alpha_{2}]$.
Furthermore, setting $\mathcal{Q}(\lambda,0)=0$ and $\mathcal{C}(\lambda,0)=0$
shows that $\mathcal{Q}$ and $\mathcal{C}$ are analytic in $\lambda$ and
continuous in $\alpha$ on $\Lambda_{R}\times[-\alpha_{2},\alpha_{2}]$.

  * Proof. Let $\alpha_{1}$ be the constant from Proposition 4. For each $0<\alpha\leq\alpha_{1}$, a function $v\in X^{1}$ is then a solution of (4.18) for $u=\mathcal{D}(\lambda,\alpha)v$ with $v(0)=1$ if and only if $v\in X^{0}$ satisfies the fixed-point equation

$\displaystyle
v(x)=\mathrm{e}^{(\lambda-\bar{g})x/\omega}-\frac{1}{\omega}\int_{0}^{x}\mathrm{e}^{(\lambda-\bar{g})(x-y)/\omega}g_{1}(y)[\mathcal{D}(\lambda,\alpha)v](y)\,\mathrm{d}y=:\mathrm{e}^{(\lambda-\bar{g})x/\omega}-[\mathcal{Q}(\lambda,\alpha)v](x).$

Using the estimates for $\mathcal{D}(\lambda,\alpha)$ from Proposition 4, we
have

$\displaystyle\|\mathcal{Q}(\lambda,\alpha)\|_{L(X^{0})}\leq\frac{1}{\omega}\int_{0}^{2\pi/\kappa}|\mathrm{e}^{(\lambda-\bar{g})(2\pi/\kappa-y)/\omega}|\,\mathrm{d}y\
\|\mathcal{D}(\lambda,\alpha)\|_{L(X^{0})}\
|g_{1}|_{X^{0}_{\mathrm{per}}}\leq\frac{2\pi}{\omega\kappa}\mathrm{e}^{2\pi(R+|\bar{g}|)/(\omega\kappa)}C_{1}\alpha=:\tilde{C}_{1}\alpha.$

In particular, setting
$\alpha_{2}:=\min(\alpha_{1},\frac{1}{2\tilde{C}_{1}})$, we can solve the
fixed-point equation

$\displaystyle
v=\mathrm{e}^{(\lambda-\bar{g})x/\omega}-\mathcal{Q}(\lambda,\alpha)v$

uniquely for $v$ for each $(\lambda,\alpha)\in\Lambda_{R}\times(0,\alpha_{2}]$
to get

$\displaystyle
v=\mathcal{C}(\lambda,\alpha):=\left(1+\mathcal{Q}(\lambda,\alpha)\right)^{-1}\mathrm{e}^{(\lambda-\bar{g})x/\omega}.$

Furthermore,

$\displaystyle|\mathcal{C}(\lambda,\alpha)|_{X^{0}}\leq
2\mathrm{e}^{2\pi(R+|\bar{g}|)/(\kappa\omega)}=:\tilde{C}_{2},\qquad|\mathcal{C}(\lambda,\alpha)-\mathrm{e}^{(\lambda-\bar{g})x/\omega}|_{X^{0}}=|\mathcal{Q}(\lambda,\alpha)\mathcal{C}(\lambda,\alpha)|_{X^{0}}\leq\tilde{C}_{1}\tilde{C}_{2}\alpha,$

and setting
$C_{2}:=\max(\tilde{C}_{1},\tilde{C}_{2},\tilde{C}_{1}\tilde{C}_{2})$
completes the proof of the estimates. Finally, analyticity of
$\mathcal{D}(\lambda,\alpha)$ implies analyticity of
$\mathcal{Q}(\lambda,\alpha)$ and $\mathcal{C}(\lambda,\alpha)$, and the
estimates for the derivatives in $\lambda$ follow in a similar fashion.

Our next result characterizes the set of $\lambda\in\Lambda_{R}$ for which the
eigenvalue problem

$\displaystyle\lambda u$
$\displaystyle=u_{xx}+\left(\frac{2\mathrm{i}}{\alpha}+\omega\right)u_{x}-\frac{1}{\alpha^{2}}u+f_{1}(x)u+f_{2}(x)v$
(4.22) $\displaystyle\lambda v$ $\displaystyle=\omega
v_{x}+\bar{g}v+g_{1}(x)u$ (4.23)

has a nontrivial $2\pi/\kappa$-periodic solution for an appropriate
$|\alpha|\ll 1$.

###### Proposition 6

Fix $\omega>0$, $\bar{g}\in\mathbb{R}$, $R>|\bar{g}|$, and
$f_{1},f_{2},g_{1}\in X^{3}_{\mathrm{per}}$, then there are constants
$\alpha_{3}>0$ and $C_{3}>0$ so that the following is true. There is a unique
function $\lambda_{0}^{*}:[-\alpha_{3},\alpha_{3}]\to\Lambda_{R}$ so that
(4.22)-(4.23) has a nontrivial $2\pi/\kappa$-periodic solution for
$(\lambda,\alpha)\in\Lambda_{R}\times[-\alpha_{3},\alpha_{3}]$ if and only if
$\lambda=\lambda_{0}^{*}(\alpha)$. Furthermore, upon defining the real
constants

$\displaystyle\lambda_{2}:=\frac{\kappa}{2\pi}\int_{0}^{2\pi/\kappa}f_{2}(x)g_{1}(x)\,\mathrm{d}x,\qquad\lambda_{3}:=\frac{\kappa}{2\pi}\int_{0}^{2\pi/\kappa}f_{2}^{\prime}(x)g_{1}(x)\,\mathrm{d}x,$

we have the expansion

$\displaystyle|\lambda_{0}^{*}(\alpha)-(\bar{g}+\lambda_{2}\alpha^{2}+2\mathrm{i}\lambda_{3}\alpha^{3})|\leq
C_{3}\alpha^{4}$

for the dispersion curve, and the associated eigenfunctions $(u,v)(x;\alpha)$
satisfy

$\displaystyle\left|u(x;\alpha)-(\alpha^{2}f_{2}(x)+2\mathrm{i}\alpha^{3}f_{2}^{\prime}(x))\right|_{X^{0}_{\mathrm{per}}}\leq
C_{3}\alpha^{4},\qquad\left|v(x;\alpha)-(1+\alpha^{2}v_{2}(x)+2\mathrm{i}\alpha^{3}v_{3}(x))\right|_{X^{0}_{\mathrm{per}}}\leq
C_{3}\alpha^{4},$

where

$\displaystyle
v_{2}(x)=\frac{1}{\omega}\left(\lambda_{2}x-\int_{0}^{x}f_{2}(y)g_{1}(y)\,\mathrm{d}y\right),\qquad
v_{3}(x)=\frac{1}{\omega}\left(\lambda_{3}x-\int_{0}^{x}f_{2}^{\prime}(y)g_{1}(y)\,\mathrm{d}y\right).$

  * Proof. We will use the notation and constants introduced in Lemma 5. Proposition 6 and Lemma 5 show that (4.22)-(4.23) has a nontrivial solution $(u,v)\in X^{2}_{\mathrm{per}}\times X^{1}_{\mathrm{per}}$ for some $(\lambda,\alpha)\in\Lambda_{R}\times(0,\alpha_{2}]$ if and only if $(u,v)=(\mathcal{D}(\lambda,\alpha)\mathcal{C}(\lambda,\alpha),\mathcal{C}(\lambda,\alpha))\in X^{0}_{\mathrm{per}}\times X^{0}_{\mathrm{per}}$ with $v(2\pi/\kappa)=v(0)$. It therefore suffices to solve $\Delta(\lambda,\alpha):=v(2\pi/\kappa)-v(0)=0$. Using the expression for $\mathcal{C}(\lambda,\alpha)$ from Lemma 5, we can write $\Delta(\lambda,\alpha)$ as

$\displaystyle\Delta(\lambda,\alpha)$
$\displaystyle=\mathrm{e}^{2\pi(\lambda-\bar{g})/(\kappa\omega)}-1-\frac{1}{\omega}\int_{0}^{2\pi/\kappa}\mathrm{e}^{(\lambda-\bar{g})(2\pi/\kappa-y)/\omega}g_{1}(y)[\mathcal{D}(\lambda,\alpha)v](y)\,\mathrm{d}y$
$\displaystyle=\mathrm{e}^{2\pi(\lambda-\bar{g})/(\kappa\omega)}-1-[\mathcal{Q}(\lambda,\alpha)\mathcal{C}(\lambda,\alpha)]\left(2\pi/\kappa\right)$
(4.24)

and conclude that $\Delta(\lambda,\alpha)$ is analytic in $\lambda$. Lemma 5
shows that

$\displaystyle\left|[\mathcal{Q}(\alpha,\lambda)\mathcal{C}(\alpha,\lambda)]\left(2\pi/\kappa\right)\right|\leq\|\mathcal{Q}(\alpha,\lambda)\|_{L(X^{0})}|\mathcal{C}(\alpha,\lambda)|_{X^{0}}\leq
C_{2}^{2}\alpha.$

Using the extensions of $\mathcal{Q}$ and $\mathcal{C}$ into $\alpha=0$, it
follows that

$\displaystyle\Delta(\lambda,0)=\mathrm{e}^{2\pi(\lambda-\bar{g})/(\kappa\omega)}-1$

so that $\lambda=\bar{g}\in\Lambda_{R}$ is the only solution in $\Lambda_{R}$
when $\alpha=0$. Since $\Delta_{\lambda}(\bar{g},0)=2\pi/(\kappa\omega)\neq
0$, we can solve $\Delta(\lambda,\alpha)=0$ near $(\bar{g},0)$ uniquely by the
implicit function theorem and conclude that there is an $\alpha_{3}>0$ and a
unique function $\lambda^{0}_{*}:[0,\alpha_{3}]\to\mathbb{C}$ with
$\lambda^{0}_{*}(0)=\bar{g}$ so that $\Delta(\lambda,\alpha)=0$ for
$(\lambda,\alpha)\in\Lambda_{R}\times[0,\alpha_{3}]$ if and only if
$\lambda=\lambda^{0}_{*}(\alpha)$.

It remains to establish the expansions for $\lambda=\lambda^{0}_{*}(\alpha)$
and the associated solutions $(u,v):=(u,v)(x;\alpha)$ of (4.22)-(4.23).
Setting $B_{1}(\lambda)v:=(\lambda-f_{1}(x))v$ and $B_{2}v:=-f_{2}v$, we can
represent $(u,v,\lambda)$ via

$\displaystyle u$
$\displaystyle=\mathcal{D}(\lambda,\alpha)v=(1-\mathcal{T}(\alpha)B_{1}(\lambda))^{-1}\mathcal{T}(\alpha)B_{2}v$
(4.25) $\displaystyle v$
$\displaystyle=\mathrm{e}^{(\lambda-\bar{g})x/\omega}-\frac{1}{\omega}\int_{0}^{x}\mathrm{e}^{(\lambda-\bar{g})(x-y)/\omega}g_{1}(y)u(y)\,\mathrm{d}y$
(4.26) $\displaystyle 0$
$\displaystyle=\mathrm{e}^{2\pi(\lambda-\bar{g})/(\kappa\omega)}-1-\frac{1}{\omega}\int_{0}^{2\pi/\kappa}\mathrm{e}^{(\lambda-\bar{g})(2\pi/\kappa-y)/\omega}g_{1}(y)u(y)\,\mathrm{d}y$
(4.27)

where $\lambda$ is always evaluated at $\lambda^{0}_{*}(\alpha)$. Throughout,
we will denote by $C$ a constant that depends only on $\omega$, $\bar{g}$,
$R$, $|f_{1}|_{X^{3}_{\mathrm{per}}}$, $|f_{2}|_{X^{3}_{\mathrm{per}}}$, and
$|g_{1}|_{X^{3}_{\mathrm{per}}}$ and that may change from estimate to
estimate. We will use Landau symbols only when we can estimate them by such a
constant multiplied by the argument of the Landau symbol.

First, equation (4.21) shows that
$v=\mathcal{C}(\lambda_{*}(\alpha),\alpha)=\mathrm{O}(1)$. Equation (4.25) and
the estimate (4.16) for $\mathcal{D}$ then imply that $u=\mathrm{O}(\alpha)$,
and using this estimate in (4.27) proves that
$\lambda^{0}_{*}(\alpha)=\bar{g}+\mathrm{O}(\alpha)$. Using these expansions
in (4.26), we conclude that

$\displaystyle v=1+\mathrm{O}(\alpha),\qquad
v_{x}=\frac{\lambda-\bar{g}}{\omega}v-\frac{g_{1}(x)}{\omega}u=\mathrm{O}(\alpha),$

and we therefore have $|v|_{X^{1}_{\mathrm{per}}}\leq C$ for a constant $C$ as
above. Using the estimate $|\mathcal{T}(\alpha)g|_{X^{0}_{\mathrm{per}}}\leq
C_{0}\alpha^{2}|g|_{X^{1}_{\mathrm{per}}}$ from Lemma 3, we obtain from (4.25)
that

$\displaystyle|u|_{X^{0}_{\mathrm{per}}}\leq\|(1-\mathcal{T}(\alpha)B_{1}(\lambda))^{-1}\|_{L(X^{0},X^{0}_{\mathrm{per}})}|\mathcal{T}(\alpha)B_{2}v|_{X^{0}_{\mathrm{per}}}\leq
2C_{0}\alpha^{2}|B_{2}v|_{X^{1}_{\mathrm{per}}}\leq 2C_{0}C\alpha^{2},$

which shows that $u=\mathrm{O}(\alpha^{2})$. Using this estimate for $u$ in
(4.27) gives

$\displaystyle\lambda^{0}_{*}(\alpha)=:\bar{g}+\lambda_{2}(\alpha),\qquad\lambda_{2}(\alpha)=\mathrm{O}(\alpha^{2})$
(4.28)

and therefore

$\displaystyle
v_{x}=\frac{\lambda-\bar{g}}{\omega}v-\frac{g_{1}(x)}{\omega}u=\mathrm{O}(\alpha^{2}).$

In particular, we can write $v=1+\tilde{v}$ with
$|\tilde{v}|_{X^{1}_{\mathrm{per}}}\leq C\alpha^{2}$.

Next, we verify the expansion for $u$. We write

$\displaystyle
u=\alpha^{2}f_{2}+2\mathrm{i}\alpha^{3}f_{2}^{\prime}+\tilde{u}$ (4.29)

and need to show that $\tilde{u}=\mathrm{O}(\alpha^{4})$. Equation (4.25) can
be written as
$u=\mathcal{T}(\alpha)B_{1}(\lambda)u-\mathcal{T}(\alpha)(f_{2}v)$ and
substituting our expression for $u$ as well as $v=1+\tilde{v}$ into this
equation gives

$\displaystyle\alpha^{2}f_{2}+2\mathrm{i}\alpha^{3}f_{2}^{\prime}+\tilde{u}=\alpha^{2}\mathcal{T}(\alpha)B_{1}(\lambda)f_{2}+2\mathrm{i}\alpha^{3}\mathcal{T}(\alpha)B_{1}(\lambda)f_{2}^{\prime}+\mathcal{T}(\alpha)B_{1}(\lambda)\tilde{u}-\mathcal{T}(\alpha)f_{2}-\mathcal{T}(\alpha)(f_{2}\tilde{v}).$
(4.30)

Since $B_{1}(\lambda)f_{2}$ and $B_{1}(\lambda)f_{2}^{\prime}$ are in
$X^{1}_{\mathrm{per}}$ and
$|f_{2}\tilde{v}|_{X^{1}_{\mathrm{per}}}=\mathrm{O}(\alpha^{2})$, we can use
the estimate (4.11) to get

$\displaystyle\left|\alpha^{2}\mathcal{T}(\alpha)B_{1}(\lambda)f_{2}+2\mathrm{i}\alpha^{3}\mathcal{T}(\alpha)B_{1}(\lambda)f_{2}^{\prime}+\mathcal{T}(\alpha)(f_{2}\tilde{v})\right|_{X^{0}_{\mathrm{per}}}\leq
C\alpha^{4}.$

Furthermore, since $f_{2}\in X^{3}_{\mathrm{per}}$, equation (4.12) shows that

$\displaystyle\left|\mathcal{T}(\alpha)f_{2}+\alpha^{2}f_{2}+2\mathrm{i}\alpha^{3}f_{2}^{\prime}\right|_{X^{0}_{\mathrm{per}}}\leq
C\alpha^{4}.$

Hence, (4.30) becomes

$\displaystyle\tilde{u}=\mathcal{T}(\alpha)B_{1}(\lambda)\tilde{u}+\mathrm{O}(\alpha^{4})$

in $X^{0}_{\mathrm{per}}$ and since
$\|\mathcal{T}(\alpha)B_{1}(\lambda)\|_{L(X^{0})}\leq\frac{1}{2}$, we conclude
that $|\tilde{u}|_{X^{0}}\leq C\alpha^{4}$ as claimed.

Our last step is to verify the expansions for $v$ and
$\lambda^{0}_{*}(\alpha)$. We substitute the expansions (4.28) and (4.29) into
(4.26) to get

$\displaystyle v(x)$
$\displaystyle=\mathrm{e}^{(\lambda-\bar{g})x/\omega}-\frac{1}{\omega}\int_{0}^{x}\mathrm{e}^{(\lambda-\bar{g})(x-y)/\omega}g_{1}(y)u(y)\,\mathrm{d}y$
$\displaystyle=\mathrm{e}^{\lambda_{2}(\alpha)x/\omega}-\frac{1}{\omega}\int_{0}^{x}\mathrm{e}^{\lambda_{2}(\alpha)(x-y)/\omega}g_{1}(y)\left(\alpha^{2}f_{2}(y)+2\mathrm{i}\alpha^{3}f_{2}^{\prime}(y)+\mathrm{O}(\alpha^{4})\right)\,\mathrm{d}y$
$\displaystyle=\left[1+\frac{\lambda_{2}(\alpha)}{\omega}x\right]-\frac{\alpha^{2}}{\omega}\int_{0}^{x}g_{1}(y)f_{2}(y)\,\mathrm{d}y-\frac{2\mathrm{i}\alpha^{3}}{\omega}\int_{0}^{x}g_{1}(y)f_{2}^{\prime}(y)\,\mathrm{d}y\
+\mathrm{O}(\alpha^{4}).$

We know that $v(0)=v(2\pi/\kappa)$ which shows that

$\displaystyle\lambda_{2}(\alpha)=\frac{\alpha^{2}\kappa}{2\pi}\int_{0}^{2\pi/\kappa}g_{1}(y)f_{2}(y)\,\mathrm{d}y\
+\frac{2\mathrm{i}\alpha^{3}\kappa}{2\pi}\int_{0}^{2\pi/\kappa}g_{1}(y)f_{2}^{\prime}(y)\,\mathrm{d}y\
+\mathrm{O}(\alpha^{4})=:\lambda_{2}\alpha^{2}+2\mathrm{i}\lambda_{3}\alpha^{3}+\mathrm{O}(\alpha^{4})$

as claimed. Substituting this expression into the expansion for $v$ gives

$\displaystyle v(x)$
$\displaystyle=1+\frac{\alpha^{2}}{\omega}\left(\lambda_{2}x-\int_{0}^{x}g_{1}(y)f_{2}(y)\,\mathrm{d}y\
\right)+\frac{2\mathrm{i}\alpha^{3}}{\omega}\left(\lambda_{3}x-\int_{0}^{x}g_{1}(y)f_{2}^{\prime}(y)\,\mathrm{d}y\right)+\mathrm{O}(\alpha^{4}).$

This completes the proof of Proposition 6 and the construction of the spectral
expansion for the case $\delta=0$.

We emphasize that the results in this section extend immediately to the case
where the coefficient $\bar{g}$ is a $2\pi/\kappa$-periodic function and not a
constant.

###  4.5 Periodic solutions of linear second-order equations with center
directions

In preparation for the case of $\delta>0$, we seek to find nontrivial periodic
solutions to second-order equations of the form

$v_{xx}=b_{21}v+b_{22}v_{x}+h(x),\qquad v\in\mathbb{C},\qquad 0\leq x\leq T,$
(4.31)

where the associated first-order system is not guaranteed to be hyperbolic.

###### Lemma 7

Fix $T>0$ and coefficients $b_{21},b_{22}$. Assume that there is an $r>0$ such
that the eigenvalues $\eta^{\mathrm{c}},\eta^{\mathrm{s}}$ of the matrix
$B:=\begin{pmatrix}0&1\\\ b_{21}&b_{22}\end{pmatrix}$ satisfy
$\mathop{\mathrm{missing}}{Re}\nolimits\eta^{\mathrm{s}}<-2r$ and
$|\mathop{\mathrm{missing}}{Re}\nolimits\eta^{\mathrm{c}}|\leq r$, then the
following is true. For each $h\in X^{0}$, equation (4.31) has a $T$-periodic
solution $v(x)\in X^{2}_{\mathrm{per}}$ if and only if (i) there exists an
$a^{\mathrm{c}}\in E^{\mathrm{c}}$ so that

$\displaystyle
v(x)=P_{1}\left(\mathrm{e}^{\eta^{\mathrm{c}}x}a^{\mathrm{c}}+[Jh](x)\right),\qquad
0\leq x\leq T$ (4.32)

where

$\displaystyle[Jh](x)$
$\displaystyle=\int_{0}^{x}\mathrm{e}^{\eta^{\mathrm{c}}(x-s)}P^{\mathrm{c}}e_{2}h(s)\,\mathrm{d}s+\frac{\mathrm{e}^{\eta^{\mathrm{s}}x}}{1-\mathrm{e}^{\eta^{\mathrm{s}}T}}\int_{0}^{T}\mathrm{e}^{\eta^{\mathrm{s}}(T-s)}P^{\mathrm{s}}e_{2}h(s)\,\mathrm{d}s\
+\int_{0}^{x}\mathrm{e}^{\eta^{\mathrm{s}}(x-s)}P^{\mathrm{s}}e_{2}h(s)\,\mathrm{d}s$

and (ii)

$\displaystyle(\mathrm{e}^{\eta^{\mathrm{c}}T}-1)a^{\mathrm{c}}+[P^{\mathrm{c}}Jh](2\pi)=0.$
(4.33)

  * Proof. We write (4.31) as the system

$V_{x}=\begin{pmatrix}0&1\\\ b_{21}&b_{22}\end{pmatrix}V+\begin{pmatrix}0\\\
h(x)\end{pmatrix}:=BV+e_{2}h(x),\qquad V=\begin{pmatrix}v\\\
v_{x}\end{pmatrix}.$ (4.34)

Enforcing periodicity in the stable direction as in Lemma 1 and using the
variation-of-constants formula in the center direction establishes the
expression (4.32) of the solution $v(x)$ in $X^{2}$. The solution $v$ lies in
$X^{2}_{\mathrm{per}}$ if and only if $V(T)-V(0)=0$. Since
$P^{\mathrm{s}}(V(T)-V(0))=0$ by construction, this condition reduces to
$P^{\mathrm{c}}(V(T)-V(0))=0$, and substituting the expressions for $V(0)$ and
$V(T)$ gives (4.33).

###  4.6 Solutions to the linearized eigenvalue problem for $\delta>0$

The next result focuses on the eigenvalue problem

$\displaystyle\lambda u$
$\displaystyle=u_{xx}+\left(\frac{2\mathrm{i}}{\alpha}+\omega\right)u_{x}-\frac{1}{\alpha^{2}}u+f_{1}(x)u+f_{2}(x)v$
(4.35) $\displaystyle\lambda v$ $\displaystyle=\delta
v_{xx}+\left(\frac{2\mathrm{i}\delta}{\alpha}+\omega\right)v_{x}-\frac{\delta}{\alpha^{2}}v+g_{1}(x)u+\bar{g}v$
(4.36)

for $0<\delta\ll 1$. For each fixed $\delta>0$, we will identify values of
$(\lambda,\alpha)$ for which (4.35)-(4.36) has a nontrivial
$2\pi/\kappa$-periodic solution.

Theorem 2. _Fix $\omega>0$ and $\bar{g}\in\mathbb{R}$, and assume that
$f_{1},f_{2},g_{1}$ are given $2\pi/\kappa$-periodic functions of class
$C^{4}$, then there are constants $\delta_{0},s_{0}>0$ and functions
$\alpha^{*},\lambda^{*}:[-s_{0},s_{0}]\times(0,\delta_{0}]\to\mathbb{C}$ with_

$\displaystyle\alpha^{*}(s,\delta)$
$\displaystyle=\frac{1}{2}\left(s+\sqrt{s^{2}+4\sqrt{\delta}}\right)$ (4.37)
$\displaystyle\lambda^{*}(s,\delta)$
$\displaystyle=\left\\{\begin{array}[]{lcl}\lambda_{*}(|s|)+\mathrm{O}(\sqrt{\delta})&&s\geq
0\\\
\bar{g}-\frac{s^{2}}{\omega}(1+\mathrm{O}(\delta^{\frac{1}{4}}))+\mathrm{O}(\sqrt{\delta})&&s\leq
0\end{array}\right.$ (4.40)

_so that ( 4.35)-(4.36) has a nontrivial $2\pi/\kappa$-periodic solution when
$(\alpha,\lambda)=(\alpha^{*},\lambda^{*})(s,\delta)$. _

  * Proof. Fix $R\geq|\bar{g}|$ and let $\lambda$ vary in $\Lambda_{R}$. Throughout, we will denote by $C$ a constant that depends only on $\omega$, $\bar{g}$, $R$, $|f_{1}|_{X^{3}_{\mathrm{per}}}$, $|f_{2}|_{X^{3}_{\mathrm{per}}}$, and $|g_{1}|_{X^{3}_{\mathrm{per}}}$ and that may change from estimate to estimate. We will use Landau symbols only when we can estimate them by such a constant multiplied by the argument of the Landau symbol.

We showed in Proposition 4 that for each given $v\in X^{0}$ the unique
solution of (4.35) is given by $u=\mathcal{D}(\lambda,\alpha)v\in
X^{2}_{\mathrm{per}}$, and it therefore suffices to solve (4.36) with
$u=\mathcal{D}(\lambda,\alpha)v$. Throughout the proof, we will use the
scaling $\delta=\alpha^{2}\beta^{2}$ with $0<\alpha,\beta\leq 1$. These
results hold for $|\alpha|\ll 1$, and as before, the proofs consider the case
$\alpha>0$ for clarity. With these definitions, (4.36) becomes

$v_{xx}=-\left(\frac{2\mathrm{i}\alpha\beta^{2}+\omega}{\alpha^{2}\beta^{2}}\right)v_{x}+\left(\frac{\lambda+\beta^{2}-\bar{g}}{\alpha^{2}\beta^{2}}\right)v-\frac{1}{\alpha^{2}\beta^{2}}g_{1}(x)[\mathcal{D}(\lambda,\alpha)v](x).$
(4.41)

We will use Lemma 7 to reformulate (4.41) as a fixed-point problem. The roots
of the characteristic equation

$\displaystyle\eta^{2}=-\left(\frac{2\mathrm{i}\alpha\beta^{2}+\omega}{\alpha^{2}\beta^{2}}\right)\eta+\left(\frac{\lambda+\beta^{2}-\bar{g}}{\alpha^{2}\beta^{2}}\right)$

associated with (4.41) are given by

$\displaystyle\eta^{\mathrm{c}}(\lambda,\alpha,\beta)=\frac{\lambda+\beta^{2}-\bar{g}}{\omega}+\mathrm{O}(\alpha\beta^{2}),\qquad\eta^{\mathrm{s}}(\lambda,\alpha,\beta)=-\frac{\omega+\mathrm{O}(\alpha\beta^{2})}{\alpha^{2}\beta^{2}}.$
(4.42)

These expressions are analytic in $\lambda\in\Lambda_{R}$ with
$|\mathop{\mathrm{missing}}{Re}\nolimits\eta^{\mathrm{c}}|\leq C$,
$\mathop{\mathrm{missing}}{Re}\nolimits\eta^{\mathrm{s}}\leq\frac{-\omega}{2\alpha^{2}\beta^{2}}$,
and $|\eta^{\mathrm{c,s}}_{\lambda}(\lambda,\alpha,\beta)|\leq C$ uniformly in
$\Lambda_{R}$. Hence, there is an $\alpha_{4}$ so that the assumptions of
Lemma 7 are met for $0<\alpha\leq\alpha_{4}$. We conclude that (4.41) has a
nontrivial solution $v\in X^{2}$ if and only if

$\displaystyle
v=\mathrm{e}^{\eta^{\mathrm{c}}(\lambda,\alpha,\beta)x}P_{1}a^{\mathrm{c}}-\mathcal{J}(\lambda,\alpha,\beta)B_{3}\mathcal{D}(\lambda,\alpha)v,\qquad\mathcal{J}(\lambda,\alpha,\beta):=\frac{1}{\alpha^{2}\beta^{2}}P_{1}J(\lambda,\alpha,\beta)$
(4.43)

for some $a^{\mathrm{c}}\in E^{\mathrm{c}}$, where $B_{3}\in L(X^{0})$ is
given by $[B_{3}v](x)=g_{1}(x)v(x)$, the operator $J(\lambda,\alpha,\beta)\in
L(X^{0})$ is defined by

$\displaystyle[J(\lambda,$
$\displaystyle\alpha,\beta)h](x)=\int_{0}^{x}\mathrm{e}^{\eta^{\mathrm{c}}(\lambda,\alpha,\beta)(x-s)}P^{\mathrm{c}}(\lambda,\alpha,\beta)e_{2}h(s)\,\mathrm{d}s$
$\displaystyle+\frac{\mathrm{e}^{\eta^{\mathrm{s}}(\lambda,\alpha,\beta)x}}{1-\mathrm{e}^{2\pi\eta^{\mathrm{s}}(\lambda,\alpha,\beta)/\kappa}}\int_{0}^{2\pi/\kappa}\mathrm{e}^{\eta^{\mathrm{s}}(\lambda,\alpha,\beta)(2\pi/\kappa-s)}P^{\mathrm{s}}(\lambda,\alpha,\beta)e_{2}h(s)\,\mathrm{d}s\
+\int_{0}^{x}\mathrm{e}^{\eta^{\mathrm{s}}(\lambda,\alpha,\beta)(x-s)}P^{\mathrm{s}}(\lambda,\alpha,\beta)e_{2}h(s)\,\mathrm{d}s,$

and the center and stable spectral projections are of the form

$\displaystyle
P^{\mathrm{c}}(\lambda,\alpha,\beta)=\frac{1}{1-\frac{\eta^{\mathrm{c}}}{\eta^{\mathrm{s}}}}\begin{pmatrix}1\\\
\eta^{\mathrm{c}}\end{pmatrix}\begin{pmatrix}1,-\frac{1}{\eta^{\mathrm{s}}}\end{pmatrix},\qquad
P^{\mathrm{s}}(\lambda,\alpha,\beta)=\frac{1}{1-\frac{\eta^{\mathrm{c}}}{\eta^{\mathrm{s}}}}\begin{pmatrix}\frac{1}{\eta^{\mathrm{s}}}\\\
1\end{pmatrix}\begin{pmatrix}-\eta^{\mathrm{c}},1\end{pmatrix}.$

To solve the fixed-point problem (4.43), it suffices to show that
$\mathcal{J}B_{3}\mathcal{D}$ lies in $L(X^{0})$ and has norm strictly less
than one. In addition to proving this contraction property, we will derive an
expansion of $\mathcal{J}$ as this will help us find expansions of the
dispersion curve. We define

$\displaystyle[\mathcal{J}^{\mathrm{c}}_{0}(\lambda)h](x):=\frac{1}{\omega}\int_{0}^{x}\mathrm{e}^{(\lambda-\bar{g})(x-s)/\omega}h(s)\,\mathrm{d}s$
(4.44)

and set

$\displaystyle\mathcal{J}(\lambda,\alpha,\beta)=\mathcal{J}^{\mathrm{c}}(\lambda,\alpha,\beta)+\mathcal{J}^{\mathrm{s}}(\lambda,\alpha,\beta),\qquad\mathcal{J}^{\mathrm{c,s}}(\lambda,\alpha,\beta):=\frac{1}{\alpha^{2}\beta^{2}}P_{1}P^{\mathrm{c,s}}J(\lambda,\alpha,\beta).$

Using the identities

$\displaystyle\frac{1}{\alpha^{2}\beta^{2}}P_{1}P^{\mathrm{c}}(\lambda,\alpha,\beta)e_{2}=\frac{-1}{\alpha^{2}\beta^{2}(\eta^{\mathrm{s}}+\eta^{\mathrm{c}})}=\frac{1}{\omega}+\mathrm{O}(\alpha\beta^{2}),\qquad\frac{1}{\alpha^{2}\beta^{2}}P_{1}P^{\mathrm{s}}(\lambda,\alpha,\beta)e_{2}=\frac{-1}{\omega}+\mathrm{O}(\alpha\beta^{2})$
(4.45)

and proceeding as in the proof of Lemma 2, we obtain

$\displaystyle\|\mathcal{J}^{\mathrm{c}}(\lambda,\alpha,\beta)\|_{L(X^{0})}$
$\displaystyle=\left\|\frac{1}{\alpha^{2}\beta^{2}}P_{1}P^{\mathrm{c}}J\right\|_{L(X^{0})}\leq
C\frac{|P_{1}P^{\mathrm{c}}e_{2}|}{\alpha^{2}\beta^{2}}\leq C$ (4.46)
$\displaystyle\|\mathcal{J}^{\mathrm{s}}(\lambda,\alpha,\beta)\|_{L(X^{0})}$
$\displaystyle=\left\|\frac{1}{\alpha^{2}\beta^{2}}P_{1}P^{\mathrm{s}}J\right\|_{L(X^{0})}\leq\frac{|P_{1}P^{\mathrm{s}}e_{2}|}{\alpha^{2}\beta^{2}}\frac{C}{|\mathop{\mathrm{missing}}{Re}\nolimits\eta^{\mathrm{s}}|}\leq\frac{C}{|\mathop{\mathrm{missing}}{Re}\nolimits\eta^{\mathrm{s}}|}\leq
C\alpha^{2}\beta^{2}$ (4.47)
$\displaystyle\|\mathcal{J}^{\mathrm{c}}(\lambda,\alpha,\beta)-\mathcal{J}^{\mathrm{c}}_{0}(\lambda)\|_{L(X^{0})}$
$\displaystyle\leq C\beta^{2}.$ (4.48)

Hence, $\|\mathcal{J}(\lambda,\alpha,\beta)\|_{L(X^{0})}\leq C$, and a similar
estimate shows that we also have
$\|\mathcal{J}_{\lambda}(\lambda,\alpha,\beta)\|_{L(X^{0})}\leq C$, which we
will use below. Using the estimate (4.16) for $\mathcal{D}$, we conclude that

$\displaystyle\|\mathcal{J}(\lambda,\alpha,\beta)B_{3}\mathcal{D}(\lambda,\alpha)\|_{L(X^{0})}\leq
C|g_{1}|_{X^{0}}C_{1}\alpha\leq C\alpha.$

Thus, there is an $\alpha_{4}>0$ such that (4.43) has a unique (up to scalar
multiples) nontrivial solution $v\in X^{2}$ for each $(\lambda,\alpha,\beta)$
with $\lambda\in\Lambda_{R}$, $0<\alpha\leq\alpha_{4}$, and $0<\beta\leq 1$,
and upon setting $a^{\mathrm{c}}=(1,\eta^{\mathrm{c}})^{t}$ this solution is
given by

$\displaystyle
v=\left(1+\mathcal{J}(\lambda,\alpha,\beta)B_{3}\mathcal{D}(\lambda,\alpha)\right)^{-1}\mathrm{e}^{\eta^{\mathrm{c}}(\lambda,\alpha,\beta)x}=\left(1+\mathcal{J}^{\mathrm{c}}_{0}(\lambda)B_{3}\mathcal{D}(\lambda,\alpha)\right)^{-1}\mathrm{e}^{(\lambda-\bar{g})x/\omega}+\mathrm{O}(\beta^{2})=$

where we used (4.42), (4.47), and (4.48).

It remains to solve the periodicity condition (4.33) with $a^{\mathrm{c}}$ as
given above. Since $P_{1}P^{\mathrm{c}}=1$ on $E^{\mathrm{c}}$, we can apply
$P_{1}$ to (4.33), which upon using again the estimates (4.42), (4.47), and
(4.48) results in the equivalent equation

$\displaystyle\Delta(\lambda,\alpha,\beta)$
$\displaystyle:=\mathrm{e}^{2\pi\eta^{\mathrm{c}}(\lambda,\alpha,\beta)/\kappa}-1-\left[\mathcal{J}^{\mathrm{c}}(\lambda,\alpha,\beta)B_{3}\mathcal{D}(\lambda,\alpha)\left(1+\mathcal{J}(\lambda,\alpha,\beta)B_{3}\mathcal{D}(\lambda,\alpha)\right)^{-1}\mathrm{e}^{\eta^{\mathrm{c}}(\lambda,\alpha,\beta)x}\right](2\pi/\kappa)$
$\displaystyle=\mathrm{e}^{2\pi(\lambda+\beta^{2}-\bar{g})/(\kappa\omega)}-1-\left[\mathcal{J}^{\mathrm{c}}_{0}(\lambda)B_{3}\mathcal{D}(\lambda,\alpha)\left(1+\mathcal{J}^{\mathrm{c}}_{0}(\lambda)B_{3}\mathcal{D}(\lambda,\alpha)\right)^{-1}\mathrm{e}^{(\lambda-\bar{g})x/\omega}\right](2\pi/\kappa)+\mathrm{O}(\alpha\beta^{2})$
(4.49) $\displaystyle=0$

that we need to solve. Comparing (4.44) and (4.49) with the expressions (4.19)
and (4.20) for the solutions of the $\delta=0$ equation, we see that we can
write (4.49) as

$\displaystyle\Delta(\lambda,\alpha,\beta)=\mathrm{e}^{2\pi(\lambda+\beta^{2}-\bar{g})/(\kappa\omega)}-1-\left[\mathcal{Q}(\lambda,\alpha)\mathcal{C}(\lambda,\alpha)\right](2\pi/\kappa)+\mathrm{O}(\alpha\beta^{2})=0.$
(4.50)

We showed in the proof of Proposition 6 that the equation
$\Delta(\lambda,\alpha,0)=0$ with $\beta=0$ has the unique solution
$\lambda=\lambda_{0}^{*}(\alpha)=\bar{g}+\mathrm{O}(\alpha^{2})$. Hence,
setting $\lambda=\lambda_{0}^{*}(\alpha)+\mu$ with $\mu$ near zero and using
the estimates (4.21) for $\mathcal{C}$ and $\mathcal{Q}$, we see that (4.50)
can be written as

$\displaystyle\Delta(\lambda_{0}^{*}(\alpha)+\mu,\alpha,\beta)$
$\displaystyle=\mathrm{e}^{2\pi(\lambda_{0}^{*}(\alpha)+\mu+\beta^{2}-\bar{g})/(\kappa\omega)}-1-\left[\mathcal{Q}(\lambda_{0}^{*}(\alpha)+\mu,\alpha)\mathcal{C}(\lambda_{0}^{*}(\alpha)+\mu,\alpha)\right](2\pi/\omega)+\mathrm{O}(\alpha\beta^{2})$
$\displaystyle=\frac{2\pi}{\kappa\omega}(\mu+\beta^{2})+\mathrm{O}(|\mu+\beta^{2}|^{2})+\mathrm{O}(\alpha\mu)+\mathrm{O}(\alpha\beta^{2})=0,$

which we can solve uniquely for $\mu=\mu^{*}(\alpha,\beta)$ with
$\mu^{*}(\alpha,\beta)=-\beta^{2}(1+\mathrm{O}(\alpha))$. In summary, we
proved that (4.49) has a solution
$(\lambda,\alpha,\beta)\in\Lambda_{R}\times(0,\alpha_{4}],\times(0,\beta_{4}]$
if and only if

$\displaystyle\lambda=\lambda^{*}(\alpha,\beta)\quad\mbox{with}\quad\lambda^{*}(\alpha,\beta)=\lambda_{0}^{*}(\alpha)-\beta^{2}\left(1+\mathrm{O}(\alpha)\right).$
(4.51)

Finally, for each fixed $\delta\geq 0$, we parameterize the curve
$\delta=\alpha^{2}\beta^{2}$ in order to write
$\lambda^{*}=\lambda^{*}(s,\delta)$. Define

$\displaystyle\left(\alpha(s,\delta),\beta(s,\delta)\right)=\frac{1}{2}\left(s+\sqrt{s^{2}+4\sqrt{\delta}},-s+\sqrt{s^{2}+4\sqrt{\delta}}\right)$
(4.52)

for $|s|\leq 1$. Then,

$\displaystyle\left(\alpha(s,0),\beta(s,0)\right)=\begin{cases}(s,0)&s\geq
0\\\ (0,|s|)&s\leq 0\end{cases},\qquad\text{ and }\qquad\max_{|s|\leq
1}\left|(\alpha,\beta)(s,\delta)-(\alpha,\beta)(s,0)\right|\leq\delta^{1/4}$
(4.53)

Using (4.52), write
$\lambda^{*}(s,\delta):=\lambda^{*}\left(\alpha(s,\delta),\beta(s,\delta)\right)$,
and from (4.51) we find

$\displaystyle\lambda^{*}(s,\delta)$
$\displaystyle=\lambda^{0}_{*}\left(\alpha(s,\delta)\right)-\beta^{2}(s,\delta)\left(1+\mathrm{O}(\alpha(s,\delta))\right).$
(4.54)

For $\delta$ small, we have

$\displaystyle\lambda_{0}^{*}(\alpha(s,\delta))-\lambda_{0}^{*}(\alpha(s,0))=\begin{cases}\lambda_{0}^{*}(\alpha(s,\delta))-\lambda_{0}^{*}(s)&s\geq
0\\\ \lambda_{0}^{*}(\alpha(s,\delta))-\bar{g}&s\leq 0\end{cases}$

hence

$\displaystyle|\lambda_{0}^{*}(\alpha(s,\delta))-\lambda_{0}^{*}(\alpha(s,0))|=\mathrm{O}(\alpha^{2})=\mathrm{O}(\delta^{1/2}),\
\ \forall|s|\leq 1.$ (4.55)

Thus, using formulation (4.54) along with estimates in (4.53) and (4.55) and
we find that there exists $s_{*},\delta_{*}>0$ such that

$\displaystyle\lambda^{*}(s,\delta)=\begin{cases}\lambda^{0}_{*}\left(s\right)+\mathrm{O}(\delta^{1/2})&s\geq
0\\\
\bar{g}-s^{2}\left(1+\mathrm{O}(\delta^{1/4})\right)+\mathrm{O}(\delta^{1/2})&s\leq
0\end{cases}$

uniformly in $|s|\leq s_{*}$ and $0<\delta<\delta_{*}$. Hence, we have proved
that (4.35)-(4.36) has a non-trivial $2\pi/\kappa$-periodic solution if and
only if $\lambda=\lambda_{*}(s,\delta)$ as defined above.

##  5 Predictions for spiral waves on bounded disks when $\delta=0$

The essential spectrum does not supply stability information for spirals on
bounded disks of radius $R$; instead the absolute spectrum is relevant. The
absolute spectrum provides limit points to which infinitely many discrete
eigenvalues converge to as $R\rightarrow\infty$ [21, 22]. For bounded domains,
the essential spectrum nevertheless furnishes useful knowledge of the spatial
eigenvalue distribution, which is exploited in Theorem 3 to predict absolute
spectrum locations. As the absolute spectrum and spatial eigenvalues are
described in detail in [7, 21, 22, 20, 9], we provide only a brief description
of these topics.

For each $\lambda\in\mathbb{C}$, there exist infinitely many spatial
eigenvalues $\nu\in\mathbb{C}$ and eigenfunctions that satisfy relation (2.4).
Spatial eigenvalues $\nu$ depend continuously on $\lambda$. For $\lambda\gg
1$, we divide $\nu$ into two disjoint groups labeled “left” for those $\nu$
with $\mathop{\mathrm{missing}}{Re}\nolimits\nu<0$ and “right” for those $\nu$
with $\mathop{\mathrm{missing}}{Re}\nolimits\nu>0$. Spatial eigenvalues $\nu$
will retain their labels even if their real part changes sign as $\lambda$
varies. The essential spectrum is defined by the set of $\lambda$ for which
there exists a purely imaginary spatial eigenvalue. When crossing a positively
(negatively) orientated essential spectrum curve from right to left, a right
(left) spatial eigenvalue crosses the imaginary axis. The absolute spectrum is
defined as values of $\lambda$ for which one right and one left spatial
eigenvalue have precisely the same real part: the absolute spectrum must
necessarily lie to the left of an essential spectrum branch.

For the scenario $\delta=0$, there are three essential spectrum curves that
influence the possible locations of the absolute spectrum. We denote the
essential spectrum curve that is tangent to the imaginary axis by
$\Sigma^{0}_{\text{ess}}$ and the two curves that form the cusp to be
$\Sigma^{c,1}_{\text{ess}}$ and $\Sigma^{c,2}_{\text{ess}}$. Next, we denote
the region of the $\lambda$-plane enclosed by $\Sigma^{0}_{\text{ess}}$,
$\Sigma^{c,1}_{\text{ess}}$, and $\Sigma^{c,2}_{\text{ess}}$ by $\Omega$.
Furthermore, we label the spatial eigenvalues that lie on
$\mathrm{i}\mathbb{R}$ along these essential spectrum curves as $\nu^{-}_{1},\
\nu^{-}_{2},$ and $\nu^{+}_{1}$ where the superscripts give the sign of Re
$\nu_{j}$ for $\lambda\gg 1$ and subscripts designate the order the spatial
eigenvalues cross the imaginary axis.

Two separate cases emerge for the orientation of essential spectrum branches
that compose the cusp structure at $\lambda_{0}$. The curve orientation is
dictated by the sign of the imaginary $\alpha^{3}$ term in
$\lambda_{*}^{0}(\alpha)$. A positive coefficient (Case 1) leads to
$\Sigma^{c,1,2}_{\text{ess}}$ having the same orientation as
$\Sigma^{0}_{\text{ess}}$, and a negative coefficient (Case 2) produces curves
of opposite orientation (see Figure 3). In both cases,
$\Sigma^{0}_{\text{ess}}$ is negatively oriented and crossing this curve from
right to left results in the spatial eigenvalue $\nu^{-}_{1}$ crossing the
imaginary axis (Figure 3). Theorem 3 predicts the locations of the absolute
spectrum for each case.

(a) Case 1

(b) Case 2

Figure 3: Schematic illustration for two cases of essential spectrum branches.
Arrows on essential spectrum branches indicate the orientation. Insets show
the distribution of spatial eigenvalues $\nu(\lambda)$: crosses (squares)
indicate spatial eigenvalues $\nu(\lambda)$ that have positive (negative) real
part for $\lambda\gg 1$. Shaded region between essential spectrum curves is
denoted $\Omega$. (a) Case 1 ($\lambda_{3}>0$): Same orientation of essential
spectrum branches, leading to the absolute spectrum falling to the left of the
cusp point. (b) Case 2 ($\lambda_{3}<0$): Opposite orientation of essential
spectrum branches, leading to the absolute spectrum to the right of the cusp
point.

###### Theorem 3

Consider the essential spectrum $\lambda_{*}^{0}(\alpha)$ for $\delta=0$. For
$\lambda_{2}<0$, predictions of absolute spectrum locations are as follows.

  1. (i)

Case 1 ($\lambda_{3}>0$): If Re $\nu^{-}_{1}(\lambda)<$ Re
$\nu^{+}_{1}(\lambda)$ for all $\lambda\in\Omega$, then there are no absolute
spectra curves within $\Omega$. Moreover, locally the absolute spectrum is
given by the set

$\displaystyle\Sigma_{\text{abs}}=\left\\{\lambda\in\mathbb{C}:\lambda=\lambda_{0}-r^{2},\
\ 0<r<r_{0}\right\\}$

for some small real $r_{0}>0$.

  2. (ii)

Case 2 ( $\lambda_{3}<0$): The set $\Omega$ contains a curve of absolute
spectrum

The proof of Theorem 3 follows from arguments of the orientation of essential
spectrum curves and crossings of spatial eigenvalues. If $\lambda_{3}=0$, the
result holds by replacing $\lambda_{3}$ with the leading order non-zero
imaginary term in $\lambda_{*}^{0}(\alpha)$.

  * Proof. Case 1: First, the orientation of essential spectrum curves and assumption on the spatial eigenvalues ensures there is no absolute spectrum curve within $\Omega$ (see schematic in Figure 3a). An absolute spectrum curve will be given by the set of $\lambda$ for which left and right spatial eigenvalues have the same positive real part.

We use $\lambda_{*}^{0}(\alpha)$ to solve locally for $\alpha=\alpha(\lambda)$
to find the spatial eigenvalues given by
$\nu=\mathrm{i}\gamma=\mathrm{i}/\alpha$. We have
$\lambda_{*}^{0}(\alpha)=\lambda_{0}-\lambda_{2}\alpha^{2}+\mathrm{i}\lambda_{3}\alpha^{3}$
which is written such that $\lambda_{2},\lambda_{3}>0$. After shifting the
curve by $\lambda_{0}$, we want to find solutions $\alpha(\lambda)$ of

$\displaystyle\mathcal{F}(\alpha,\lambda)=-\lambda_{2}\alpha^{2}+\mathrm{i}\lambda_{3}\alpha^{3}-\lambda=0$

and determine for which values of $\lambda$ two spatial eigenvalues have the
same real part. Using a Newton’s polygon, set $\alpha=\lambda^{1/2}w$ which
transforms the equation to

$\displaystyle-\lambda_{2}w^{2}+\mathrm{i}\lambda_{3}\lambda^{1/2}w^{3}-1=0.$

Near $\lambda=0$, $w^{2}=-1/\sqrt{\lambda_{2}}$ and there are two unique roots
$\alpha_{1,2}=\lambda^{1/2}w_{1,2}(\lambda^{1/2})$ where $w_{1,2}$ are
analytic in $\lambda^{1/2}$ and $w_{1,2}(0)=\pm\mathrm{i}/\sqrt{\lambda_{2}}$.
The final root is found by setting $\alpha=\lambda^{0}w(\lambda^{0})$,
resulting in $\alpha_{3}(0)=-\mathrm{i}\lambda_{2}/\lambda_{3}$. Mapping these
$\alpha$-roots to $\nu=\mathrm{i}/\alpha$ gives

$\displaystyle\nu_{1}(\lambda)$
$\displaystyle=\frac{\mathrm{i}}{\lambda^{1/2}v_{1}(\lambda^{1/2})},\ \
\nu_{1}(0)=\frac{\sqrt{\lambda_{2}}}{\lambda}$ $\displaystyle\nu_{2}(\lambda)$
$\displaystyle=\frac{\mathrm{i}}{\lambda^{1/2}v_{2}(\lambda^{1/2})},\ \
\nu_{2}(0)=-\frac{\sqrt{\lambda_{2}}}{\lambda}$ $\displaystyle\nu_{3}$
$\displaystyle=-\frac{\lambda_{3}}{\lambda_{2}}.$

The two relevant spatial eigenvalues are $\nu_{1}$ and $\nu_{2}$. Using the
leading order in $\lambda$ terms, set
$\mathop{\mathrm{missing}}{Re}\nolimits\nu_{1}=\mathop{\mathrm{missing}}{Re}\nolimits\nu_{2}$
and let $\lambda=r^{2}\mathrm{e}^{2\mathrm{i}\theta}$. Then, solving

$\displaystyle\frac{\sqrt{\lambda_{2}}}{r\cos(\theta)}=-\frac{\sqrt{\lambda_{2}}}{r\cos(\theta)}$

gives $\theta=\pi/2$ or $\lambda=r^{2}\mathrm{e}^{\mathrm{i}\pi}=-r^{2}$.
Locally, near $\lambda_{0}$, there exists a small $r_{0}>0$ such that an
absolute spectrum curve is given by

$\displaystyle\Sigma_{\text{abs}}=\left\\{\lambda\in\mathbb{C}:\lambda=\lambda_{0}-r^{2},\
\ 0<r<r_{0}\right\\}.$

Case 2: The results for Case 2 follow from the orientations of the essential
spectrum curves. Upon crossing $\Sigma_{\text{ess}}^{0}$ from right to left a
left spatial eigenvalue crosses the imaginary axis. Crossing
$\Sigma^{c,1,2}_{\text{ess}}$ from right to left results in a right spatial
eigenvalue crossing the imaginary axis. Since spatial eigenvalues are analytic
in $\lambda$, there must be values of $\lambda\in\Omega$ for which the real
parts of the right and left spatial eigenvalues coincide to give an absolute
spectrum curve.

In Case 1, we prove the existence of an absolute spectrum branch locally near
$\lambda_{0}$ which extends leftward in the complex plane. Theorem 3
guarantees the existence of a spectral curve within $\Omega$ for Case 2,
signifying that these models can undergo an absolute instability if $\Omega$
reaches into the positive half plane.

##  6 Application to Barkley and Karma models

In this section, the results of Theorems 1-3 are applied to planar spiral
waves formed in the Barkley and Karma models, which are well-studied nonlinear
reaction-diffusion systems [3, 4, 24, 23, 12, 13, 6, 10, 11]. As we will see,
these two models also furnish examples of the two absolute spectrum cases.

###  6.1 Barkley Model

The Barkley model is given by

$\displaystyle u_{t}$ $\displaystyle=\ \Delta
u+\frac{1}{\epsilon}u(1-u)\left(u-\frac{v+b}{a}\right)$ $\displaystyle v_{t}$
$\displaystyle=\ \delta\Delta v+u-v,$

where we fix parameters $a=0.7$, $b=0.001$, and $\epsilon=0.02$. For these
parameters, there is numerical evidence that a stable spiral wave exists [2].
The effects of removing diffusion are studied by modifying the diffusion
coefficient $\delta$ for the slowly diffusing species $v$ in the interval
$[0,0.2]$. The angular frequency changes with $\delta$ and falls between
$\omega_{0}=1.87\ (\delta=0.2)$ and $\omega_{0}=2.09\ (\delta=0)$. Note that
$\bar{g}=-1$.

Figure 4: Essential spectra for the Barkley model for $\delta=0$ and
$\delta>0$

Essential spectrum curves are calculated using numerical continuation methods
on system (2.6) as described in [18]. Computations are performed with Matlab
on a 256 grid point periodic domain with Fourier spectral methods used for
spatial derivatives. Below, these numerically computed spectral curves will be
compared to the analytical predictions.

Figure 4 shows essential spectrum curves of the spiral wave calculated for
$\delta=0$ and $\delta=0.2$. Vertically periodicity of the branches arises due
to the Floquet symmetry of the spiral eigenfunctions. As noted above and as
indicated in the theorems, the curves look very different for these two cases.
For $\delta=0.2$, the curves are unbounded. In the case $\delta=0$, adjacent
branches meet at the limit points $\lambda_{0}$ and form cusps. The limit
points are predicted to be $\lambda_{0}=-1+\mathrm{i}\omega_{0}n$.

Numerically computed curves for decreasing $\delta$ are shown in Figure 5 with
the inset focusing on the region near $\lambda_{0}$. For small, non-zero
values of $\delta$, curves begin to turn toward the $\lambda_{0}$ limit point
before diverging.

Figure 5: Essential spectra for the Barkley model under changing diffusion.
Inset at top right focuses on area near $\lambda_{0}$.

We verify the form of $\lambda_{*}^{0}(\alpha)$ computed in Theorem 1 by
analyzing the convergence of numerically computed spectral curves; denote
these curves by $\lambda_{c}$. From $\lambda_{*}^{0}(\alpha)$, we expect
$\mathop{\mathrm{missing}}{Re}\nolimits\left(|\lambda_{c}-\lambda_{0}|\right)=\mathrm{O}(\alpha^{2})$
and
$\mathop{\mathrm{missing}}{Im}\nolimits\left(|\lambda_{c}-\lambda_{0}|\right)=\mathrm{O}(\alpha^{3})$.
Log-log plots of the convergence are shown in Figure 7 and show the
$\mathrm{O}(\alpha^{2})$ dependence of the real terms. Note that
$\mathop{\mathrm{missing}}{Im}\nolimits\left(|\lambda_{c}-\lambda_{0}|\right)=\mathrm{O}(\alpha^{5})$.
This leads to the following observation.

Figure 6:  Convergence of real and imaginary parts of the essential spectrum
computed with continuation in the Barkley model for $\delta=0$.

###### Corollary 1

If the linearized term $g_{1}(x)$ is a constant, then $\lambda_{3}=0$ in
$\lambda_{*}^{0}(\alpha)$.

This corollary is immediately proved by integrating $\lambda_{3}$ and using
the periodicity of $f_{2}(x)$. The linearization $g_{1}(x)=1$ in the Barkley
model, and Corollary 1 therefore applies.

Numerical computation of absolute spectra following methods in [18] for
$\delta=0$ and $\delta=0.2$ are shown in Figure 1. There is an exceptional
difference between the two: the Y-shaped branches present for $\delta=0.2$ are
replaced by short segments that lie to the left of the essential spectrum
branches for $\delta=0$.

Under the nonlinearities and parameters used here for the Barkley model, the
orientation of the essential spectrum curves means the system falls into
absolute spectrum case 1 described in Theorem 3. The numerical computations
confirm the existence of an absolute spectrum branch that emerges leftward
from $\lambda_{0}$ (Figure 1a). Moreover, no absolute spectrum is numerically
found within $\Omega$.

###  6.2 Karma Model

The Karma model is given by

$\displaystyle u_{t}=$ $\displaystyle\ 1.1\Delta
u+400\left(-u+\left(1.5414-v^{4}\right)\left(1-\tanh(u-3)\right)\frac{u^{2}}{2}\right)$
$\displaystyle v_{t}=$ $\displaystyle\ \delta\Delta
v+4\left(\frac{1}{1-\mathrm{e}^{-\mu_{K}}}\theta_{s}(u-1)-v\right)$

where $\mu_{K}=1.2$. As in [1, 12, 13, 6], we consider a smoothed version of
the Heaviside function given by $\theta_{s}(u)=(1+\tanh(su))/2$ for $s=4$.
Alternans are numerically observed for this set of parameter values [12, 13,
6] and spirals have angular frequencies of $\omega_{0}=49.81$ ($\delta=0.1$)
and $\omega_{0}=51.66$ ($\delta=0$). Again, there are distinct differences
between the essential and absolute spectra for $\delta>0$ and $\delta=0$ in
Figure 7. Here, $\bar{g}=-4$.

Figure 7: Shown are the essential spectra in the Karma model for $\delta=0$
and $\delta=0.1$. When $\delta=0$, cusps form in the essential spectrum at
$\lambda_{0}=-4+\mathrm{i}\omega_{0}n$.

From Theorem 1 the cusps are predicted to emerge from
$\lambda_{0}=-4+\mathrm{i}\omega_{0}n$, which is confirmed numerically (Figure
8). For $\delta=0$, essential spectrum curves in the Karma model form loops
that reverse the branch orientations before forming the cusp point, and a
computation shows that $\lambda_{3}<0$. Thus, the Karma model falls into
absolute spectrum case 2, and Theorem 3 predicts absolute spectrum to the
right of $\lambda_{0}$ in region $\Omega$. The numerical computation of the
absolute spectrum shown in Figure 8 confirms this prediction.

Furthermore, the existence of leading absolute spectrum branches in $\Omega$
grants the possibility of the system undergoing an absolute instability. In
fact, regardless of $\delta$ these branches do destabilize when the system
parameter $\mu_{k}$ is increased above 1.4.

Figure 8: Formation of cusps in Karma model for $\delta=0$. Loops in the
essential spectrum change the local orientation (indicated by arrows) at the
cusp points. Spectra computed by numerical continuation.

##  7 Discussion

#### Summary

The central question we pursued is how the continuous spectra of planar spiral
waves behave in the limit of vanishing diffusion in one of the components of a
reaction-diffusion system. We found that these spectra behave discontinuously
near cusp points of the essential spectrum of the zero-diffusion limit. We
also priovide explicit expansions of these spectra in the spatial Floquet
exponent $1/\alpha$ and the small diffusion coefficient $\delta$ that hold for
$0\leq\alpha\ll 1$ and $0\leq\delta\ll 1$ and whose coefficients are
determined by the diffusionless variable. We verified these predictions
numerically in the Barkley and Karma models.

We also discussed the absolute spectra of spiral waves on bounded disks for
$\delta=0$. The sign of $\lambda_{3}$ from $\lambda_{*}^{0}(\alpha)$ divides
models into the two distinct cases described in Theorem 3. Absolute spectra
lie strictly outside of the region $\Omega$ if $\lambda_{3}>0$. If instead
$\lambda_{3}<0$, then there is absolute spectrum inside $\Omega$. These two
cases occur in the Barkley and Karma models as demonstrated in §6.

#### Wave-train spectra

As mentioned in the introduction, the essential spectra of the asymptotic wave
trains depend smoothly on $\delta$ in the zero-diffusion limit [17].
Interestingly, cusp points are not observed for the essential spectra of these
wave trains when $\delta=0$. Figure 9 shows essential spectra for the wave
train in the Barkley model for decreasing values of $\delta$. The limiting
behavior of the wave train essential spectrum can be obtained directly from
the spectral relationship (2.4) and is given by

$\displaystyle\lambda^{0}_{\infty}(\alpha)=\lambda_{0}+\lambda_{2}\alpha^{2}+\mathrm{i}\lambda_{3}\alpha^{3}+\omega\left(\frac{\mathrm{i}}{\alpha}\right)+\mathrm{o}(\alpha^{4}).$

Only the imaginary parts of the spectra differ between
$\lambda^{0}_{\infty}(\alpha)$ and $\lambda^{0}_{*}(\alpha)$. In the limit
$\alpha\rightarrow 0$, we have
$\mathop{\mathrm{missing}}{Re}\nolimits\lambda_{*}^{0}(\alpha)\rightarrow\bar{g}$,
implying
$\mathop{\mathrm{missing}}{Re}\nolimits\lambda^{0}_{\infty}(\alpha)\rightarrow\bar{g}$,
but $\mathop{\mathrm{missing}}{Im}\nolimits\lambda^{0}_{\infty}(\alpha)$
becomes infinite due to the addition of the imaginary $1/\alpha$ term.

For the spiral wave at $\delta=0$, the limit points $\lambda_{0}$ arise
because the $v$-equation decouples at small values of $\alpha$. To further
understand the origins of the differences between wave-train and spiral
spectra, consider the linear dispersion relation for the wave train in the co-
moving frame

$\displaystyle
D\left(\partial_{x}+\nu\right)^{2}\bar{V}+\omega\left(\partial_{x}+\nu\right)^{2}\bar{V}+F_{U}(U_{\infty}(x))\bar{V}-\lambda_{\infty}\bar{V}=0.$

For a two-variable system as studied above with $\delta=0$ and
$\nu=\mathrm{i}\gamma$, the $v$-equation takes the form

$\displaystyle\lambda_{\infty}v=\omega\left(\partial_{x}+\mathrm{i}\gamma\right)v+g_{1}(x)u+\bar{g}v.$

Here, the $v$-equation still retains direct input from $\gamma$, unlike the
corresponding equation for the spiral (4.18). The $\gamma$ provides an
unbounded term for which $\lambda_{\infty}$ must compensate in order to retain
bounded eigenfunctions.

Figure 9: Essential spectrum of wave trains in the Barkley model limits to a
vertical branch under $\delta\rightarrow 0$.

#### Outlook

There are a number of questions that we did not address. One such question is
whether persistence of planar spiral waves that are assumed to exist for
$\delta=0$ into the region $0<\delta\ll 1$ could be established using singular
perturbation theory. Similarly, we discussed only the essential spectra of
planar spiral waves in the vanishing-diffusion limit, and the behavior of
point spectrum was not addressed here. Moreover, we focused on two-component
system, and it is natural to ask whether our results hold for general
$n$-component systems, potentially with several non-diffusing variables: we
expect that each non-diffusing variable creates its own cusps but have not
attempted to prove this. Finally, while Theorem 1 remains true if we allow
$\bar{g}$ to be a $2\pi/\kappa$-periodic function, we did not prove the same
for Theorem 2, though we expect this to hold as well.

On bounded disks, the spectrum consists entirely of discrete eigenvalues,
which fall into disjoint sets that either align along curves of the absolute
spectrum, are members of the extended point spectrum, or arise from the
imposed boundary conditions [21, 22]. Theorem 3 predicts the locations of
absolute spectrum depending on the sign of $\lambda_{3}$, but the effect of
vanishing diffusion on boundary and extended point spectrum remains an open
question.

We leave the reader with a word of caution: the essential spectrum may be
significantly different in the two cases of small positive and zero diffusion
$\delta$. This fact has potentially significant consequences for the stability
of patterns formed in ion channel models, and care should be taken during
computations.

  * Acknowledgments  Dodson was partially supported by the NSF through DMS-1644760. Sandstede was partially supported by the NSF through grants DMS-1714429 and DMS-2106566.

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