Stability of travelling waves

Transcription

1 Stability of travelling waves Björn Sanstee Department of Mathematics Ohio State University 231 West 18th Avenue Columbus, OH 43210, USA Abstract An overview of various aspects relate to the spectral an nonlinear stability of travelling-wave solutions to partial ifferential equations is given. The point an the essential spectrum of the linearization about a travelling wave are iscusse as is the relation between these spectra, Freholm properties, an the existence of exponential ichotomies (or Green s functions) for the linear operator. Among the other topics reviewe in this survey are the nonlinear stability of waves, the stability an interaction of wellseparate multi-bump pulses, the numerical computation of spectra, an the Evans function, which is a tool to locate isolate eigenvalues in the point spectrum an near the essential spectrum. Furthermore, methos for the stability of waves in Hamiltonian an monotone equations as well as for singularly perturbe problems are mentione. Moulate waves, rotating waves on the plane, an travelling waves on cylinrical omains are also iscusse briefly. This work was partially supporte by the NSF uner grant DMS

2 Contents 1 Introuction 4 2 Set-up an examples Set-up Examples Spectral stability Reformulation Exponential ichotomies Spectrum an Freholm properties Fronts, pulses an wave trains Homogeneous rest states Perioic wave trains Fronts Pulses Fronts connecting perioic waves Absolute an convective instability The Evans function Definition an properties The computation of the Evans function, an applications The erivative D (0) The asymptotic behaviour of D(λ) as λ Extension across the essential spectrum Spectral stability of multi-bump pulses Spatially-perioic wave trains with long wavelength Outline of the proof of Theorem Discussion of Theorem Multi-bump pulses Strategies for using Theorem An alternative approach using the Evans function Fronts an backs A review of existence an stability results of multi-bump pulses an applications Weak interaction of pulses

3 6 Numerical computation of spectra Continuation of travelling waves Computation of spectra of spatially-perioic wave trains Computation of spectra of pulses an fronts Perioic bounary conitions Separate bounary conitions Nonlinear stability 47 8 Equations with aitional structure 49 9 Moulate, rotating, an travelling waves 52 References 55 3

4 1 Introuction This survey is evote to the stability of travelling waves. Travelling waves are solutions to partial ifferential equations that move with constant spee c while maintaining their shape. In other wors, if the solution is written as U(x, t) where x an t enote the spatial an time variable, respectively, then we have U(x, t) = Q(x ct) for some appropriate function Q(ξ). Note that c = 0 escribes staning waves that o not move at all. In homogeneous meia, travelling waves arise as one-parameter families: any translate Q(x + τ ct) of the wave Q(x ct), with τ R fixe, is also a travelling wave. We can istinguish between various ifferent shapes of travelling waves (see Figure 1): Wave trains are spatially-perioic travelling waves so that Q(ξ + L) = Q(ξ) for all ξ for some L > 0. Homogeneous waves are steay states that o not epen on ξ so that Q(ξ) = Q 0 for all ξ. Fronts, backs an pulses are travelling waves that are asymptotically constant, i.e. that converge to homogeneous rest states: lim ξ ± Q(ξ) = Q ±. For fronts an backs, we have Q + Q, whereas pulses converge towars the same rest state as ξ ± so that Q + = Q. Travelling waves arise in many applie problems. Such waves play an important role in mathematical biology (see e.g. [121]) where they escribe, for instance, the propagation of impulses in nerve fibers. Various ifferent kins of waves can often be observe in chemical reactions [99, 182]; one example are flame fronts that arise in problems in combustion [182]. Another fiel where waves arise prominently is nonlinear optics (see e.g. [1]): of interest there are moels for the transmission an propagation of beams an pulses through optical fibers or waveguies. We refer to [38, 43, 89] for applications to water waves. Travelling waves also arise as viscous shock profiles in conservation laws that moel, for instance, problems in flui an gas ynamics or magneto-hyroynamics [171]. Localize structures in soli mechanics can be moelle by staning waves (see [172, 173, 174]). We refer to [59] for the existence an stability of patterns on boune omains. In this article, we focus on the stability of a given travelling wave. That is, we are intereste in the fate of solutions whose initial conitions are small perturbations of the travelling wave uner consieration. If any such solution stays close to the set of all translates of the travelling wave Q( ) for all positive times, then we say that the travelling wave Q( ) is stable. If there are initial conitions arbitrarily close to the wave such that the associate solutions leave a small neighbourhoo of the wave an its translates, then the wave is sai to be unstable. In other wors, we are intereste in orbital stability of travelling waves. There exists an enormous number of ifferent approaches to investigate the stability of waves: which of these is the most appropriate epens, for instance, on whether the partial ifferential equation is issipative or conservative, or whether one can exploit a special structure such as monotonicity or singular perturbations. Given this variety, writing a comprehensive survey is quite ifficult: thus, the selection of topics appearing in this survey is a very personal one an, of course, by no means complete. We refer to the recent review [184] an to the monograph [182] for many other references relate to the existence an stability of waves. A natural approach to the stuy of stability of a given travelling wave Q is to linearize the partial ifferential equation about the wave. The spectrum of the resulting linear operator L shoul then provie clues as to the stability of the wave with respect to the full nonlinear equation. As we shall see in Section 3, the spectrum (a) (b) (c) () Figure 1 Travelling waves with various ifferent shapes are plotte: pulses in (a), spatially-perioic wave trains in (b), fronts in (c), an backs in (). Note that the istinction between fronts an backs is, in general, rather artificial. 4

5 of L is the union of the point spectrum, efine as the set of isolate eigenvalues with finite multiplicity, an its complement, the essential spectrum. Point an essential spectra are also relate to Freholm properties of the operator L λ. Most of the results presente here are formulate using the first-orer operator T (λ) that is obtaine by casting the eigenvalue operator L λ as a first-orer ifferential operator. In Section 4, we review the efinition an properties of the Evans function, which is a tool to locate an track the point spectrum of L. In Section 7, we iscuss uner what conitions spectral stability of the linearization L implies nonlinear stability, i.e. stability of the wave with respect to the full partial ifferential equation. The stability analysis of a given wave is often facilitate by exploiting the structure of the unerlying equation. In Section 8, we provie some pointers to the literature for Hamiltonian an monotone equations as well as for singularly perturbe problems. In many applications, it appears to be ifficult to analyze the stability of travelling waves analytically. For this reason, we comment in Section 6 on the numerical computation of the spectra of linearizations about travelling waves. An interesting problem that is relevant for a number of applications is the stability of multi-bump pulses that accompany primary pulses. Recent results in this irection are reviewe in Section 5. Most of the results presente in this survey are also applicable to other waves, for instance, rotating waves such as spiral waves in two space imensions, moulate waves (waves that are time-perioic in an appropriate moving frame), an travelling waves on cylinrical omains. Some of these extensions are iscusse in Section 9. Acknowlegments I am grateful to Bernol Fieler, Arn Scheel an Alice Yew for helpful comments an suggestions on the manuscript. 2 Set-up an examples 2.1 Set-up We consier partial ifferential equations (PDEs) of the form U t = A( x )U + N (U), x R, U X. (2.1) Here, A(z) is a vector-value polynomial in z, an X is an appropriate Banach space consisting of functions U(x) with x R, so that A( x ) : X X is a close, ensely efine operator. Lastly, N : X X enotes a nonlinearity, perhaps not efine on the entire space X, that is efine via pointwise evaluation of U an, possibly, erivatives of U. We refer to [85, 133] for more backgroun. Travelling waves are solutions to (2.1) of the form U(x, t) = Q(x ct). Introucing the coorinate ξ = x ct, we seek functions U(ξ, t) = U(x ct, t) that satisfy (2.1). In the (ξ, t)-coorinates, the PDE (2.1) reas U t = A( ξ )U + c ξ U + N (U), ξ R, U X, (2.2) an the travelling wave is then a stationary solution Q(ξ) that satisfies The linearization of (2.2) about the steay state Q(ξ) is given by The right-han sie efines the linear operator 0 = A( ξ )U + c ξ U + N (U). (2.3) U t = A( ξ )U + c ξ U + U N (Q)U. (2.4) L := A( ξ ) + c ξ + U N (Q). 5

6 Spectral stability of the wave Q is etermine by the spectrum of the operator L, i.e. by the eigenvalue problem λu = A( ξ )U + c ξ U + U N (Q)U = LU (2.5) that etermines whether (2.4) supports solutions of the form U(ξ, t) = e λt U(ξ). Note that the steay-state equation (2.3) an the eigenvalue problem (2.5) are both orinary ifferential equations (ODEs). As such, they can be cast as first-orer systems. The steay-state equation, for instance, can be written as u = f(u, c), u R n, = ξ. (2.6) The travelling wave Q(ξ) correspons to a boune solution q(ξ) of (2.6). The PDE eigenvalue problem (2.5) becomes u = ( u f(q(ξ), c) + λb)u, (2.7) where B is an appropriate n n matrix that encoes the PDE structure (see Section 2.2 below for examples). An important relation is given by c f(u(ξ), c) = Bu (ξ) (2.8) which follows from inspecting (2.5). In this survey, we focus on the ODE formulation (2.6) an (2.7). In particular, travelling waves can be sought as boune solutions of (2.6), an we refer to the textbooks [33, 80, 107] for a ynamical-systems approach to constructing such solutions. 2.2 Examples We give a few examples that fit into the framework outline above. Example 1 (Reaction-iffusion systems) Let D be a iagonal N N matrix with positive entries an F : R N R N be a smooth function. Consier the reaction-iffusion equation U t = DU xx + F (U), x R, U R N, (2.9) pose on the space X = C 0 unif (R, RN ) of boune, uniformly continuous functions. In the moving frame ξ = x ct, the system (2.9) is given by U t = DU ξξ + cu ξ + F (U), ξ R, U R N. (2.10) Suppose that U(ξ, t) = Q(ξ) is a stationary solution of (2.10) such that DQ ξξ (ξ) + cq ξ (ξ) + F (Q(ξ)) = 0, ξ R. (2.11) The eigenvalue problem associate with the linearization of (2.10) about Q(ξ) is given by This eigenvalue problem can be cast as ( ) ( ) ( Uξ V = D 1 = (λu U F (Q(ξ))U cv ) V ξ λu = DU ξξ + cu ξ + U F (Q)U =: LU. (2.12) 0 i D 1 (λ U F (Q(ξ))) cd 1 ) (U ) V which we write as u ξ = A(ξ; λ)u = (Ã(ξ) + λb)u, u Rn = R 2N (2.13) 6

7 with u = (U, V ) an Ã(ξ) = ( 0 i D 1 U F (Q(ξ)) cd 1 ), B = ( 0 0 D 1 0 Boune solutions to (2.12), namely (L λ)u = 0, an (2.13) are then in one-to-one corresponence. In particular, if Q( ) is not a constant function, then λ = 0 is an eigenvalue of L with eigenfunction Q ξ (ξ). This can be seen by taking the erivative of (2.11) with respect to ξ which gives D(Q ξ ) ξξ + c(q ξ ) ξ + U F (Q)Q ξ = 0 so that LQ ξ = 0. Hence, u(ξ) = (Q ξ (ξ), Q ξξ (ξ)) satisfies (2.13) for λ = 0. One important example is the FitzHugh Nagumo equation (FHN) u t = u xx + f(u) w w t = δ 2 w xx + ɛ(u γw), for instance with f(u) = u(1 u)(u a). It amits various travelling waves such as pulses, fronts an backs (see e.g. [91, 105, 176] for references). The stability of pulses has been stuie in [90, 185]. Stability results for spatially-perioic wave trains can be foun in [53, 156], whereas the stability of concatenate fronts an backs has been stuie in [124, 147] an [125, 154]. Many other results on the stability of waves to reaction-iffusion equations can be foun in the literature (see e.g. [47, 60]). One class of such equations that has been stuie extensively are monotone systems (see [37, 141, 182] an Section 8). We also refer to [85, Section 5.4] for instructive examples. Example 2 (Phase-sensitive amplification) The issipative fourth-orer equation U t + ( ξξ + 2U 2 (2κ η 2 ))( ξξ + 2U 2 η 2 )U + 4σ(3U( ξ U) 2 + U 2 ξξ U) = 0 (2.14) moels the transmission of pulses in optical storage loops uner phase-sensitive amplification (see [106]). This equation with σ = 0 amits the explicit solution Q(ξ) = η sech(ηξ) for every κ 0. Its stability has been investigate in [3, 106]. For σ > 0, (2.14) has multi-bump pulses whose existence an stability has been analyze in [150]. Example 3 (Korteweg e Vries equation) The generalize Korteweg e Vries equation (KV) is given by U t + U xxx + U p U x = 0, x R, where p is a positive parameter. Formulate in the moving frame ξ = x ct, the generalize Korteweg e Vries equation reas U t + U ξξξ cu ξ + U p U ξ = 0, ξ R, where c enotes the wave spee. This equation amits a family of pulses given by [ ] 1 ( ) c(p + 1)(p + 2) p 2 Q(ξ) = sech p p cξ 2 2 for any positive values of c an p. The stability of these solitons was investigate in [10], whereas asymptotic stability has been stuie in [134, 135]. The KV equation is Hamiltonian for all p > 0 an known to be completely integrable for p = 1, 2. ). 7

8 Example 4 (Nonlinear Schröinger equation) The nonlinear Schröinger equation (NLS) reas iφ t + Φ ξξ + 4 Φ 2 Φ = 0, ξ R, with Φ C. If we seek solutions of the form Φ(ξ, t) = e iωt U(ξ, t), then we obtain the equation iu t + U ξξ ωu + 4 U 2 U = 0, ξ R, (2.15) where U C an ω > 0. It is known [183] to support stable pulses given by ω Q(ξ) = 2 sech( ω ξ). The NLS equation is Hamiltonian an, in fact, completely integrable. Of interest is the persistence an stability of these waves upon aing perturbations that moel various physical imperfections. An important example is the perturbation to the issipative complex cubic-quintic Ginzburg Lanau equation (CGL) iu t + U ξξ ωu + 4 U 2 U + 3α U 4 U = iɛ( 1 U ξξ + 2 U + 3 U 2 U + 4 U 4 U) (2.16) for small α R an ɛ > 0. We refer to [177, 97, 98, 1] for various stability an instability results for solitary waves to this equation. Mathematically, the transition from (2.15) to (2.16) is interesting since the perturbation estroys the Hamiltonian nature of (2.15). We refer to [99, 121, 171, 176, 182] for problems where reaction-iffusion equations arise naturally. Formal erivations of the KV, NLS an CGL equation can be foun in [1] an [38, 89] for problems in nonlinear optics an for water waves, respectively. 3 Spectral stability In this section, we review results on the structure of the spectrum of the linearization of a nonlinear PDE about a travelling wave. Notation Throughout this survey, we enote the range an the null space of an operator L by R(L) an N(L), respectively. Eigenvalues of operators an matrices are always counte with algebraic multiplicity. Consier a matrix A C n n. We often refer to the eigenvalues of the matrix A as the spatial eigenvalues. We say that A is hyperbolic if all eigenvalues of A have non-zero real part, i.e. if spec(a) ir =. We refer to eigenvalues of A with positive (negative) real part as unstable (stable) eigenvalues. Similarly, the generalize eigenspace of A associate with all eigenvalues with positive (negative) real part is calle the unstable (stable) eigenspace of A. The δ-neighbourhoo of an element or a subset z of a vector space is enote by U δ (z). 3.1 Reformulation As mentione above, it is often avantageous to write the eigenvalue problem associate with the linearization as a first-orer ODE. Therefore, we consier the family T of linear operators efine by T (λ) : D H, u u A( ; λ)u ξ for λ C. We take either D = C 1 unif(r, C n ), H = C 0 unif(r, C n ) 8

9 or D = H 1 (R, C n ), H = L 2 (R, C n ). (3.1) Throughout this survey, we assume that the following hypothesis is met. Hypothesis 3.1 The matrix-value function A(ξ; λ) C n n is of the form A(ξ; λ) = Ã(ξ) + λb(ξ) where Ã( ) an B( ) are in C (R, R n n ). The operators T (λ) are close, ensely efine operators in H with omain D. We are intereste in the set of λ for which T (λ) is not invertible. 3.2 Exponential ichotomies Spectral properties of T can be classifie by using properties of the associate ODE u = A(ξ; λ)u (3.2) ξ with u C n. We enote by Φ(ξ, ζ) the evolution operator 1 associate with (3.2). Note that Φ(ξ, ζ) = Φ(ξ, ζ; λ) epens on λ, but we often suppress this epenence in our notation. A particularly useful notion associate with linear ODEs such as (3.2) is exponential ichotomies. Suppose that we consier a linear constant-coefficient equation u = A(λ)u, (3.3) ξ so that A(λ) oes not epen on ξ. We want to classify solutions to (3.3) accoring to their asymptotic behaviour as ξ. Suppose that the matrix A(λ) is hyperbolic so that the spatial spectrum spec(a(λ)) has no points on the imaginary axis. Consequently, C n = E s 0(λ) E u 0 (λ) (3.4) where the two spaces on the right-han sie are the generalize stable an unstable eigenspaces of the matrix A(λ). We enote by P s 0(λ) the spectral projection of A(λ), so that R(P s 0(λ)) = E s 0(λ), N(P s 0(λ)) = E u 0 (λ). (3.5) These subspaces are invariant uner the evolution Φ(ξ, ζ) = e A(λ)(ξ ζ) of (3.3). Furthermore, solutions u(ξ) with initial conitions u(ζ) in E s 0(λ) ecay exponentially for ξ > ζ, while solutions with initial conitions u(ζ) in E u 0 (λ) ecay exponentially for ξ < ζ. We are intereste in a similar characterization of solutions to the more general equation (3.2): Definition 3.1 (Exponential ichotomies) Let I = R +, R or R, an fix λ C. We say that (3.2), with λ = λ fixe, has an exponential ichotomy on I if constants K > 0 an κ s < 0 < κ u exist as well as a family of projections P (ξ), efine an continuous for ξ I, such that the following is true for ξ, ζ I. With Φ s (ξ, ζ) := Φ(ξ, ζ)p (ζ), we have Φ s (ξ, ζ) Ke κs (ξ ζ), ξ ζ, ξ, ζ I. 1 i.e. Φ(ξ, ξ) = i, Φ(ξ, τ)φ(τ, ζ) = Φ(ξ, ζ) for all ξ, τ, ζ R an u(ξ) = Φ(ξ, ζ)u 0 satisfies (3.2) for every u 0 C n 9

10 R(P (ξ)) R(P (ζ)) Φ s (ξ, ζ) N(P (ξ)) Φ u (ζ, ξ) N(P (ζ)) Figure 2 A plot of the stable an unstable spaces associate with an exponential ichotomy. Vectors in the stable space R(P (ζ)) are contracte exponentially uner the linear evolution Φ s (ξ, ζ) for ξ > ζ. Similarly, vectors in the unstable space N(P (ξ)) are contracte uner the linear evolution Φ u (ζ, ξ) for ξ > ζ. Define Φ u (ξ, ζ) := Φ(ξ, ζ)(i P (ζ)), then Φ u (ξ, ζ) Ke κu (ξ ζ), ξ ζ, ξ, ζ I. The projections commute with the evolution, Φ(ξ, ζ)p (ζ) = P (ξ)φ(ξ, ζ), so that Φ s (ξ, ζ)u 0 R(P (ξ)), ξ ζ, ξ, ζ I Φ u (ξ, ζ)u 0 N(P (ξ)), ξ ζ, ξ, ζ I. The ξ-inepenent imension of N(P (ξ)) is referre to as the Morse inex of the exponential ichotomy on I. If (3.2) has exponential ichotomies on R + an on R, the associate Morse inices are enote by i + (λ ) an i (λ ), respectively. Roughly speaking, (3.2) has an exponential ichotomy on an unboune interval I if each solution to (3.2) on I ecays exponentially either in forwar time or else in backwar time. The set of initial conitions u(ζ) leaing to solutions u(ξ) that ecay for ξ > ζ, with ξ, ζ I, is given by the range R(P (ζ)) of the projection P (ζ). Similarly, the set of initial conitions u(ζ) leaing to solutions u(ξ) that ecay for ξ < ζ, with ξ, ζ I, is given by the null space N(P (ζ)). The spaces R(P (ξ)) are mappe into each other by the evolution associate with (3.2); this is also true for the spaces N(P (ξ)); see Figure 2 for an illustration. For the constant-coefficient equation (3.3), we have P (ξ) = P s 0(λ) ue to (3.5). Note that, for constant-coefficient equations, the Morse inex of the exponential ichotomy is simply the imension of the generalize unstable eigenspace. Exponential ichotomies persist uner small perturbations of the equation. This result is often referre to as the roughness theorem for exponential ichotomies. If we, for instance, perturb the coefficient matrix A(λ) of the constant-coefficient equation (3.3) by aing a small ξ-epenent matrix, we expect that the two subspaces E s 0(λ) an E u 0 (λ) that appear in (3.4) perturb slightly to two new ξ-epenent subspaces that contain all initial conitions that lea to exponentially ecaying solutions for forwar or backwar times. Theorem 3.1 ([36, Chapter 4]) Firstly, let I be R + or R. Suppose that A( ) C 0 (I, C n n ) an that the equation u = A(ξ)u (3.6) ξ has an exponential ichotomy on I with constants K, κ s an κ u as in Definition 3.1. There are then positive constants δ an C such that the following is true. If B( ) C 0 (I, C n n ) such that sup ξ I, ξ L B(ξ) < δ/c 10

11 for some δ < δ an some L 0, then a constant K > 0 exists such that the equation u = (A(ξ) + B(ξ))u (3.7) ξ has an exponential ichotomy on I with constants K, κ s + δ an κ u δ. Moreover, the projections P (ξ) an evolutions Φ s (ξ, ζ) an Φ u (ξ, ζ) associate with (3.7) are δ-close to those associate with (3.6) for all ξ, ζ I with ξ, ζ L. Seconly, if I = R, then the above statement is true with L = 0. Thus, to get persistence of exponential ichotomies on R + or R, the coefficient matrices of the perturbe equation nee to be close to those of the unperturbe equation only for all sufficiently large values of ξ. For I = R, the coefficient matrices nee to be close for all ξ R to get persistence. Theorem 3.1 can be prove by applying Banach s fixe-point theorem to an appropriate integral equation whose solutions are precisely the evolution operators that appear in Definition 3.1 (see [143, 137]). Inee, suppose that Φ s (ξ, ζ) an Φ u (ξ, ζ) enote exponential ichotomies of (3.6) on I = R +, say. If sup ξ 0 B(ξ) is sufficiently small, then the ichotomies Φ s (ξ, ζ) an Φ u (ξ, ζ) associate with (3.7) can be foun as the unique solution of the integral equation 0 = Φ s (ξ, ζ) Φ s (ξ, ζ) + ξ ζ Φ s (ξ, τ)b(τ) Φ s (τ, ζ) τ + 0 = Φ u (ξ, ζ) Φ u (ξ, ζ) ξ 0 ξ ξ Φ s (ξ, τ)b(τ) Φ u (τ, ζ) τ ζ Φ u (ξ, τ)b(τ) Φ s (τ, ζ) τ (3.8) ζ 0 Φ s (ξ, τ)b(τ) Φ u (τ, ζ) τ, Φ u (ξ, τ)b(τ) Φ u (τ, ζ) τ ζ Φ u (ξ, τ)b(τ) Φ s (τ, ζ) τ, 0 ζ ξ 0 ξ ζ (see [143, 137] for etails). We emphasize that exponential ichotomies are not unique: on R +, for instance, the range of P (ξ) is uniquely etermine, whereas the null space of P (0) can be chosen to be any complement of R(P (0)); any such choice then etermines the null space of P (ξ) for any ξ > 0 by the requirement that the projections an the evolution operators commute. The above integral equation fixes such a complement. Remark 3.1 If the perturbation B(ξ) in (3.7) converges to zero as ξ with ξ I, then the projections an evolutions of (3.7) converge to those of (3.6) (see e.g. [143, 137]). It is also true that, if (3.2) has an exponential ichotomy for λ = λ, then the evolutions an projections that appear in Definition 3.1 can be chosen to epen analytically on λ for λ close to λ (see again [143, 137]). It is often of interest to istinguish solutions accoring to the strength of the ecay or growth. For instance, for the constant-coefficient system (3.3), we might be intereste in istinguishing solutions u 1 (ξ) that satisfy u 1 (ξ) e (η δ)ξ u 1 (0), ξ > 0 from solutions u 2 (ξ) that satisfy u 2 (ξ) e (η+δ)ξ u 2 (0), ξ < 0 for some chosen η an some small δ > 0. In other wors, rather than separating stable an unstable eigenvalues of A(λ), we ivie the spectrum spec(a(λ)) into two isjoint sets accoring to the presence of a spectral gap at Re ν = η; see Figure 3b. This may soun more general than the situation consiere above, but, in fact, it is not: the scaling v(ξ) = u(ξ)e ηξ (3.9) 11

12 (a) ir (b) ir Re ν = η Figure 3 Two ifferent spectral ecompositions of spec(a(λ)) are plotte: in (a), stable an unstable eigenvalues are separate by the imaginary axis, whereas the spectrum in (b) is ivie into the two isjoint sets {ν spec(a(λ)); Re ν < η} an {ν spec(a(λ)); Re ν > η}, exploiting a spectral gap. transforms (3.3) into the equation v = (A(λ) η)v, ξ an the two spectral sets associate with the spectral gap at Re ν = η for the matrix A(λ) become the stable an unstable spectral sets for the matrix A(λ) η. Thus, the results state above are applicable to any two spectral sets of A(λ) of the form {ν spec(a(λ)); Re ν < η} an {ν spec(a(λ)); Re ν > η} (assuming, of course, that Re ν η for every ν spec(a(λ))). Note that the transformation (3.9) changes only the length of vectors but not their irection. In particular, subspaces of solutions are not change. In summary, we may wish to replace the conition κ s < 0 < κ u that appears in Definition 3.1 by the weaker conition κ s < κ u. Using the transformation (3.9) for an appropriate η, we see that all the results mentione above are also true uner this weaker conition, i.e. for arbitrary spectral gaps. 3.3 Spectrum an Freholm properties We consier the family of operators T (λ) : D H, u u A( ; λ)u ξ with parameter λ. We are intereste in characterizing those λ for which the operator T (λ) : D H is not invertible. The set of all such λ is the spectrum of the linearization L about the travelling wave. We emphasize that the spectrum of the iniviual operators T (λ) : D H, for fixe λ, is of no interest to us. Definition 3.2 (Spectrum) We say that λ is in the spectrum Σ of T if T (λ) is not invertible, i.e. if the inverse operator oes not exist or is not boune. We say that λ Σ is in the point spectrum Σ pt of T or, alternatively, that λ Σ is an eigenvalue of T if T (λ) is a Freholm operator with inex zero. The complement Σ \ Σ pt =: Σ ess is calle the essential spectrum. The complement of Σ in C is the resolvent set of T. Recall that an operator L : X Y is sai to be a Freholm operator if R(L) is close in Y, an the imension of N(L) an the coimension of R(L) are both finite. The ifference im N(L) coim R(L) is calle the Freholm inex of L. It is a measure for the solvability of Lx = y for a given y Y. Freholm operators are amenable to a stanar perturbation theory using Liapunov Schmit reuction. If L ɛ : X Y enotes a Freholm operator that epens continuously on ɛ R in the operator norm, then Liapunov Schmit reuction replaces the equation L ɛ u = 0 by a reuce equation of the form L ɛ u = 0, Lɛ : N(L 0 ) R(L 0 ) 12

13 that is vali for ɛ close to zero, where R(L 0 ) is a complement of R(L 0 ). Note that both spaces appearing in the above equation are finite-imensional. We refer to [180, Chapter 3] an [75, Chapters I.3 an VII] for introuctions to Liapunov Schmit reuction. For any λ in the point spectrum of T, we efine the multiplicity of λ as follows. Recall that A(ξ; λ) is of the form A(ξ; λ) = Ã(ξ) + λb(ξ). Suppose that λ is in the point spectrum of T, with T (λ) = Ã(ξ) λb(ξ), ξ such that N(T (λ)) = span{u 1 ( )}. We say that λ has multiplicity l if functions u j D can be foun for j = 2,..., l such that ξ u j(ξ) = (Ã(ξ) + λb(ξ))u j(ξ) + B(ξ)u j 1 (ξ), ξ R for j = 2,..., l, but so that there is no solution u D to ξ u = (Ã(ξ) + λb(ξ))u + B(ξ)u l(ξ), ξ R. Lastly, we say that an arbitrary eigenvalue λ of T has multiplicity l if the sum of the multiplicities of a maximal set of linearly inepenent elements in N(T (λ)) is equal to l. Example 1 (continue) Recall the operator L = D ξξ + c ξ + U F (Q) an the associate family T (λ) with Ã(ξ) = ( T (λ) = Ã(ξ) λb ξ 0 i D 1 U F (Q(ξ)) cd 1 ), B = ( 0 0 D 1 0 Suppose that λ is in the spectrum of L an T. The Joran-block structures of the operators L λ an T (λ) are then the same, i.e. geometric an algebraic multiplicities an the length of each maximal Joran chain are the same whether compute for L λ or for T (λ). This justifies our efinition of multiplicity for eigenvalues of T. It is also true that the Freholm properties, an the Freholm inices, of L λ an T (λ) are the same (see e.g. [153, 157]). ). Remark 3.2 The point spectrum is often efine as the set of all isolate eigenvalues with finite multiplicity, i.e. as the set Σ pt of those λ for which T (λ) is Freholm with inex zero, the null space of T (λ) is non-trivial, an T ( λ) is invertible for all λ in a small neighbourhoo of λ (except, of course, for λ = λ). The sets Σ pt an Σ pt iffer in the following way. The set of λ for which T (λ) is Freholm with inex zero is open. Take a connecte component C of this set, then the following alternative hols. Either T (λ) is invertible for all but a iscrete set of elements in C, or else T (λ) has a non-trivial null space for all λ C. This follows, for instance, from using the Evans function (see Section 4.1). The following theorem prove by Palmer relates Freholm properties of the operator T (λ) to properties pertaining to the existence of ichotomies of (3.2) u = A(ξ; λ)u. ξ 13

14 Theorem 3.2 ([131, 132]) Fix λ C. The following statements are true. λ is in the resolvent set of T if, an only if, (3.2) has an exponential ichotomy on R. λ is in the point spectrum Σ pt of T if, an only if, (3.2) has exponential ichotomies on R + an on R with the same Morse inex, i + (λ) = i (λ), an im N(T (λ)) > 0. In this case, enote by P ± (ξ; λ) the projections of the exponential ichotomies of (3.2) on R ±, then the spaces N(P (0; λ)) R(P + (0; λ)) an N(T (λ)) are isomorphic via u(0) u( ). λ is in the essential spectrum Σ ess if (3.2) either oes not have exponential ichotomies on R + or on R, or else if it oes, but the Morse inices on R + an on R iffer. As a consequence, eigenfunctions associate with elements in the point spectrum of T ecay necessarily exponentially as ξ. Remark 3.3 To summarize the relation between Freholm properties of T an exponential ichotomies of (3.2), we remark that T is Freholm if, an only if, (3.2) has exponential ichotomies on R + an on R. The Freholm inex of T is then equal to the ifference i (λ) i + (λ) of the Morse inices of the ichotomies on R an R + (see [131, 132]). If T (λ) is not Freholm, then typically the range R(T (λ)) of T (λ) is not close in H. Suppose that T (λ) is invertible, an enote by Φ s (ξ, ζ; λ) an Φ u (ξ, ζ; λ) the exponential ichotomy of (3.2) on R. The inverse of T (λ) is then given by u(ξ) = [T (λ) 1 h](ξ) = ξ Φ s (ξ, ζ; λ)h(ζ) ζ + ξ Φ u (ξ, ζ; λ)h(ζ) ζ. If T (λ) is Freholm with inex i, then its range is given as follows. Consier the ajoint equation an the associate ajoint operator ξ v = A(ξ; λ) v (3.10) T (λ) : D H, v v ξ A( ; λ) v (3.11) (note that T (λ) is the genuine Hilbert-space ajoint of T (λ) only when pose on the spaces (3.1)). The ajoint operator T (λ) is Freholm with inex i. We have that h R(T (λ)) if, an only if, ψ(ξ), h(ξ) ξ = 0 (3.12) for each ψ N(T (λ) ), i.e. for each boune solution ψ(ξ) of (3.10). In fact, the following remark is true. Remark 3.4 Suppose that the equation u = A(ξ; λ)u (3.13) ξ has an exponential ichotomy on I with projections P (ξ; λ) an evolutions Φ s (ξ, ζ; λ) an Φ u (ξ, ζ; λ), then the equation ξ v = A(ξ; λ) v (3.14) also has an exponential ichotomy on I with projections P (ξ; λ) an evolutions Φ s (ξ, ζ; λ) an Φ u (ξ, ζ; λ). The projections an evolutions of (3.13) an (3.14) are relate via P (ξ; λ) = i P (ξ; λ), Φs (ξ, ζ; λ) = Φ u (ζ, ξ; λ), Φu (ξ, ζ; λ) = Φ s (ζ, ξ; λ). 14

15 This is a consequence of Definition 3.1 together with the following observation (see also [157, Lemma 5.1]): if Φ(ξ, ζ) enotes the evolution of (3.13), then, upon ifferentiating the ientity Φ(ξ, ζ)φ(ζ, ξ) = i with respect to ξ, we see that Φ(ξ, ζ) = Φ(ζ, ξ) is the evolution of (3.14). In particular, ψ N(T (λ) ) if, an only if, ψ(0) N( P (0; λ)) R( P + (0; λ)) = ( N(P (0; λ)) + R(P + (0; λ))), where P ± (ξ; λ) an P ± (ξ; λ) are the projections for (3.13) an (3.14), respectively, on I = R ±. Remark 3.5 Note that u(ξ), v(ξ) = 0, ξ ξ R for any two solutions u(ξ) an v(ξ) of (3.13) an (3.14), respectively. In particular, if u(ξ) an v(ξ) are both boune, an one of them converges to zero as ξ or ξ, then u(ξ), v(ξ) = 0 for all ξ. 3.4 Fronts, pulses an wave trains In this section, we iscuss the consequences of the above results for fronts, pulses an wave trains Homogeneous rest states Suppose that the travelling wave Q(ξ) is a homogeneous stationary solution, so that Q(ξ) = Q 0 R n oes not epen on ξ. The coefficients of the PDE linearization about Q 0 are constant an o not epen on ξ. Thus, assume that A(ξ; λ) = A 0 (λ) = Ã0 + λb 0 oes not epen on ξ an consier (3.2), now given by ξ u = A 0(λ)u. This equation has an exponential ichotomy on R if, an only if, A 0 (λ) is hyperbolic. In fact, if A 0 (λ) is hyperbolic, then Φ s (ξ, ζ; λ) = e A0(λ)(ξ ζ) P0(λ), s Φ u (ξ, ζ; λ) = e A0(λ)(ξ ζ) P0 u (λ), where P s 0(λ) an P u 0 (λ) are the spectral projections of A 0 (λ) associate with the stable an unstable spectral sets, respectively. We have the following alternative: λ is in the resolvent set of T if, an only if, A 0 (λ) is hyperbolic. λ is in the essential spectrum Σ ess if, an only if, A 0 (λ) has at least one purely imaginary eigenvalue, i.e. Σ ess = {λ C; spec(a 0 (λ)) ir }. In particular, the point spectrum is empty. Example 1 (continue) Suppose that Q 0 is a homogeneous rest state. Hence, ( ) 0 i A 0 (λ) = D 1 (λ U F (Q 0 )) cd 1, an λ is in the essential spectrum of T if, an only if, 0 (λ, k) = et[a 0 (λ) ik] = 0 15

16 has a solution k R. The function 0 (λ, k) is often referre to as the (linear) ispersion relation. Typically, the essential spectrum consists of the union of curves λ (k) in the complex plane, where λ (k) is such that 0 (λ (k), k) = 0 for k R. Alternatively, the essential spectrum can be calculate by substituting U(ξ, t) = e λt+ikξ U 0 into the linear equation U t = L 0 U. An interesting quantity is the group velocity c group = k Im λ (k) which is the velocity with which wave packets with Fourier spectrum centere near the frequency k evolve with respect to the equation U t = L 0 U. We refer to [24, Section 2] for more etails regaring the physical interpretation of the group velocity Perioic wave trains If we consier the linearization about a spatially-perioic travelling wave Q(ξ) with spatial perio L, i.e. about a wave train Q(ξ) with Q(ξ + L) = Q(ξ) for all ξ, then the coefficients of the PDE linearization have perio L in ξ. Thus, we assume that the matrix A(ξ; λ) is perioic in ξ with perio L > 0, so that A(ξ + L; λ) = A(ξ; λ), ξ R, u = A(ξ; λ)u = (Ã(ξ) + λb(ξ))u (3.15) ξ has perioic coefficients. By Floquet theory (see e.g. [83, Chapter IV.6]), the evolution Φ(ξ, ζ; λ) of (3.15) is of the form Φ(ξ, 0; λ) = Φ per (ξ; λ)e R(λ)ξ where R(λ) C n n an Φ per (ξ + L; λ) = Φ per (ξ; λ) for all ξ R with Φ per (0; λ) = i. Note that it is not clear whether we can choose R(λ) to be analytic in λ (though this is always possible locally in λ). We have the following alternatives: The point spectrum Σ pt is empty, an λ is in the resolvent set of T if, an only if, spec(r(λ)) ir =, i.e. if Φ(L, 0; λ) has no purely imaginary Floquet exponent (or, equivalently, if Φ(L, 0; λ) has no spectrum on the unit circle). Σ ess = {λ C; spec(r(λ)) ir } = {λ C; spec(φ(l, 0; λ)) S 1 }. Consequently, λ is in the essential spectrum if, an only if, the bounary-value problem u = A(ξ; λ)u, 0 < ξ < L (3.16) ξ u(l) = e iγ u(0) has a solution u(ξ) for some γ R. This is the case precisely if iγ is a purely imaginary Floquet exponent of Φ(L, 0; λ). The approach via Floquet theory is also applicable in higher space imensions [163, 164], then often referre to as ecomposition into Bloch waves, an we refer to [23, 118, 119] for generalizations an applications to Turing patterns. Suppose that the wave train is foun as a perioic solution q(ξ) to u = f(u, c) ξ 16

17 an that (3.15) is given by with ξ u = ( uf(q(ξ), c) + λb)u c f(u(ξ), c) = Bu (ξ). As a consequence, λ = 0 is containe in the essential spectrum, since q (ξ) satisfies (3.16) for γ = 0. Furthermore, spatially-perioic wave trains with perio L typically exist for any perio L, in a certain range, for a wave spee c(l) that epens on L (see e.g. [33, 80, 107]). It is not har to verify, using the equations above, that L c(l) = c group = γ Im λ(γ) γ=0 where λ(γ) enotes the solution to (3.16) that satisfies λ(0) = 0. The group velocity at λ = 0 is therefore relate to the nonlinear ispersion relation c = c(l) that relates wave spee an wavelength of the wave trains. We refer to [24, 117] for the physical interpretation of the group velocity. We remark that, for each fixe γ R, the multiplicity of an eigenvalue λ to (3.16) can again be efine as in Section 3.3 by using Joran chains ξ u j = A(ξ; λ)u j + B(ξ)u j 1, u j (L) = e iγ u j (0) (see [68]). These eigenvalues, counte with their multiplicity, can be sought as zeros of the Evans function D per (γ, λ) = et[e iγ Φ(L, 0; λ)]. (3.17) It has been prove in [68] that, for fixe γ R, λ is a solution to (3.16) with multiplicity l if, an only if, λ is a zero of D per (γ, λ) of orer l Fronts Suppose that the travelling wave Q(ξ) is a front, so that the limits lim Q(ξ) = Q ± R N ξ ± exist. The vectors Q ± are homogeneous stationary solutions to the unerlying PDE, an we refer to Q ± as the asymptotic rest states. Thus, the coefficients of the unerlying PDE linearization have limits as ξ ±. We assume that there are n n matrices ñ an B ± such that lim Ã(ξ) = ñ, ξ ± lim ξ ± B(ξ) = B ± an efine A ± (λ) = ñ + λb ±. The existence of exponential ichotomies for the equation u = A(ξ; λ)u = (Ã(ξ) + λb(ξ))u (3.18) ξ on R ± is relate to the hyperbolicity of the asymptotic matrices A ± (λ). The next theorem rephrases the statement of Theorem

18 Theorem 3.3 ([36, Chapter 6]) Fix λ C. Equation (3.18) has an exponential ichotomy on R + if, an only if, the matrix A + (λ) is hyperbolic. In this case, the Morse inex i + (λ) is equal to the imension im E+(λ) u of the generalize unstable eigenspace E+(λ) u of A + (λ). This statement is also true on R with A + (λ) replace by A (λ). Lastly, (3.18) has an exponential ichotomy on R if, an only if, it has exponential ichotomies on R + an on R with projections P ± (ξ; λ) such that N(P (0; λ)) R(P + (0; λ)) = C n ; this requires in particular that the Morse inices i + (λ) an i (λ) are equal. With a slight abuse of notation, we will refer to the number of unstable eigenvalues of a hyperbolic n n matrix A, counte with multiplicity, as its Morse inex. We observe that, using this notation, the Morse inices of the asymptotic matrices A ± (λ) are equal to the Morse inices i ± (λ) of the exponential ichotomies on R ± by the above theorem. Note that T (λ) is Freholm with inex zero if, an only if, the number of linearly inepenent solutions to (3.18) that ecay as ξ an the number of solutions that ecay as ξ a up to the imension n of C n. As a consequence of Theorems 3.2 an 3.3, we have the following options: λ is in the resolvent set of T if, an only if, A ± (λ) are both hyperbolic with the same Morse inex i + (λ) = i (λ) such that the projections P ± (ξ; λ) of the exponential ichotomies of (3.18) on I = R ± satisfy N(P (0; λ)) R(P + (0; λ)) = C n. λ is in the point spectrum Σ pt if, an only if, the asymptotic matrices A ± (λ) are both hyperbolic with ientical Morse inex i + (λ) = i (λ) such that the projections P ± (ξ; λ) of the exponential ichotomies of (3.18) on I = R ± satisfy N(P (0; λ)) R(P + (0; λ)) {0}. λ is in the essential spectrum Σ ess if either at least one of the two asymptotic matrices A ± (λ) is not hyperbolic (so that λ is in the essential spectrum of one or both rest states Q ± ) or else if A + (λ) an A (λ) are both hyperbolic but their Morse inices iffer, so that i + (λ) i (λ). The reason that the bounary of the essential spectrum epens only on the asymptotic rest states Q ± is relate to the fact that the operators T (λ) an ˆT (λ) = Â( ; λ) ξ with Â(ξ; λ) = { A (λ) for ξ < 0 A + (λ) for ξ 0 iffer only by a relatively compact operator (see [85, Appenix to Section 5: Theorem A.1 an Exercise 2]). Typically, the essential spectrum of fronts contains open sets in the complex plane, namely regions where T (λ) is Freholm with non-zero inex i (λ) i + (λ) 0. Note that λ = 0 is always containe in the spectrum with eigenfunction Q (ξ). Example 1 (continue) Suppose that Q(ξ) is a front connecting the asymptotic rest states Q ±, so that ( ) 0 i A ± (λ) = D 1 (λ U F (Q ± )) cd 1. Thus, λ is in the essential spectrum of T if either λ is in the essential spectrum of Q + or Q (see Section 3.4.1) or else if the Morse inices i (λ) an i + (λ), i.e. the number of unstable eigenvalues of A ± (λ), iffer. 18

19 3.4.4 Pulses Suppose that the travelling wave Q(ξ) is a pulse such that lim Q(ξ) = Q 0 R N. ξ In other wors, a pulse is a front that connects to the same rest state Q 0 as ξ ±. Thus, we assume that there are n n matrices Ã0 an B 0 such that lim Ã(ξ) = Ã0, ξ lim ξ B(ξ) = B 0 an efine A 0 (λ) = Ã0 + λb 0. This is a special case of the situation for fronts consiere above. The main ifference is that the Morse inices at ξ = + an ξ = are always the same. As a consequence, the operator T (λ) is either not Freholm or is Freholm with inex zero. We have the following statement. λ is in the resolvent set of T if, an only if, the asymptotic matrix A 0 (λ) is hyperbolic, an the projections P ± (ξ; λ) of the exponential ichotomies of (3.18) on R ± satisfy N(P (0; λ)) R(P + (0; λ)) = C n. λ is in the point spectrum Σ pt if, an only if, A 0 (λ) is hyperbolic, an the projections P ± (ξ; λ) of the exponential ichotomies of (3.18) on R ± satisfy N(P (0; λ)) R(P + (0; λ)) {0}. λ is in the essential spectrum Σ ess if the asymptotic matrix A 0 (λ) is not hyperbolic, i.e. if λ is in the essential spectrum of the asymptotic rest state Q 0. Again, λ = 0 is always containe in the spectrum with eigenfunction Q (ξ) (see Example 1) Fronts connecting perioic waves Similar results are true for fronts that connect spatially-perioic waves to each other or to homogeneous rest states. The essential spectrum of such fronts is etermine by the essential spectra of the asymptotic wave trains or rest states an their Morse inices: The essential spectra of the asymptotic wave trains or rest states has been compute in Sections an above. Exponential ichotomies for the asymptotic linearizations generate exponential ichotomies for the linearization about the front, an vice versa, by Theorem 3.1. We omit the etails. 3.5 Absolute an convective instability In this section, we report on absolute an convective instabilities that are relate to the essential spectrum of a travelling wave. We refer to [16, 24, 135, 162, 155] for further etails an more backgroun, an also to [140] for instructive examples. Relate phenomena for matrices (via iscretizations of PDE operators) are reviewe in [175]. Suppose that we consier a travelling wave that has essential spectrum in the right half-plane, so that the wave is unstable. Such an instability can manifest itself in ifferent ways. The physics literature istinguishes between two ifferent kins of instability, namely absolute an convective instabilities. An absolute instability occurs if perturbations grow in time at every fixe point in the omain (see Figure 4a). Convective instabilities are characterize by the fact that, even though the overall norm of the perturbation grows in time, perturbations ecay locally at every fixe point in the unboune omain; in other wors, the growing perturbation is transporte, or convecte, towars infinity (see Figure 4b). 2 2 Note that the ifference between absolute an convective instabilities epens crucially on the choice of the spatial coorinate system: changing to a moving frame can turn a convective instability into an absolute instability 19

20 (a) (b) U ξ Figure 4 The otte waves are the initial ata U 0 (ξ) to the linearize equation U t = LU, whereas the soli waves represent the solution U(ξ, t) at a fixe positive time t. In (a), an absolute instability is shown: the solution grows without bouns at each given point ξ in space as t. In (b), a convective instability is shown: the solution U(ξ, t) grows but also travels towars ξ = + ; U(ξ, t) actually ecays for each fixe value of ξ as t. The operator L woul then have stable spectrum in the norm η for a certain η > 0. We outline how absolute an convective instabilities can be capture mathematically on the unboune omain R. Suppose that the linearization of the PDE about a pulse, say, is given by the operator L, acting on the space L 2 (R) with norm. To escribe convective instabilities, it is convenient to introuce exponential weights [162]: for a given real number η, efine the norm η by U 2 η = e ηξ U(ξ) 2 ξ, (3.19) an enote by L 2 η(r), equippe with the norm η, the space of functions U(ξ) for which U η <. Note that the norms η for ifferent values of η are not equivalent to each other. We consier L as an operator on L 2 η(r) an compute its spectrum using the new norm η for appropriate values of η. The key is that, for η > 0, the norm η penalizes perturbations at, while it tolerates perturbations (which may in fact grow exponentially with any rate less than η) at +. Thus, if an instability is of transient nature, so that it manifests itself by moes that travel towars +, then the essential spectrum calculate in the norm η shoul move to the left as η > 0 increases. Inee, as the perturbations travel towars +, they are multiplie by the weight e ηξ which is small as ξ an therefore reuces their growth or even causes them to ecay. Exponential weights have been use to stuy a variety of problems pose on the real line such as reaction-iffusion operators [162], conservative systems such as the KV equation [135, 136], an generalize Kuramoto Sivashinsky equations that escribe thin films [30, 31]. They have also been applie to spiral waves [159] on the plane. Absolute instabilities occur if the spectrum cannot be stabilize by any choice of η. Conitions for the presence of an absolute instability were erive for homogeneous rest states in [24, Section 2] an for wave trains in [16]. We also refer to [155] for relate phenomena. Introucing the weight (3.19) has the effect that the operator T given by is replace by the operator T (λ) : D H, u u A( ; λ)u ξ T η (λ) : D H, u u [A( ; λ) η]u ξ upon using the transformation (3.9). In particular, the essential spectrum of the operator L in the weighte space can be compute by applying the theory outline in the previous sections to the operator T η (λ), rather than to the operator T (λ). These arguments also apply to fronts instea of pulses: it is then, however, often necessary to consier ifferent exponents for ξ < 0 an for ξ > 0 in the weight function to accommoate the ifferent asymptotic matrices A ± (λ). We refer to [155] for more etails an references. 20

21 4 The Evans function We have seen that the spectrum of T is the union of the essential spectrum Σ ess an the point spectrum Σ pt. For pulses an fronts, the essential spectrum can be calculate by solving the linear ispersion relation of the asymptotic rest states (see Section 3.4). In this section, we review the Evans function which provies a tool for locating the point spectrum [2, 134]. The Evans function can also be use to locate the essential spectrum of wave trains [68] (see Section 3.4.2). 4.1 Definition an properties Consier the eigenvalue problem ξ u = A(ξ; λ)u, u Cn, ξ R. (4.1) Since we are intereste in locating the point spectrum, we assume that λ is not in the essential spectrum Σ ess of T (see, however, Section 4.3). Owing to Theorem 3.2, equation (4.1) therefore has exponential ichotomies on R + an R with projections P + (ξ; λ) an P (ξ; λ), respectively, an the Morse inices im N(P + (0; λ)) = im N(P (0; λ)) are the same. Recall from Definition 3.1 that R(P + (0; λ)) contains all initial conitions u(0) whose associate solutions u(ξ) of (4.1) ecay exponentially as ξ. Analogously, N(P (0; λ)) consists of all initial conitions u(0) whose associate solutions u(ξ) ecay exponentially as ξ. In particular, owing to Theorem 3.2, we have that λ Σ pt if, an only if, N(T (λ)) = N(P (0; λ)) R(P + (0; λ)) {0}. Any eigenfunction u(ξ) is a boune solution to the eigenvalue problem (4.1): u(0) shoul therefore lie in R(P + (0; λ)), so that u(ξ) is boune for ξ > 0, an in N(P (0; λ)), so that u(ξ) is boune for ξ < 0. The Evans function D(λ) is esigne to locate non-trivial intersections of R(P + (0; λ)) an N(P (0; λ)). Let Ω be a simply-connecte subset of C \ Σ ess. Note that, in most applications, the essential spectrum Σ ess will be containe in the left half-plane; otherwise the wave is alreay unstable. The set Ω of interest is then the connecte component Ω of C \ Σ ess that contains the right half-plane. The Morse inex im N(P (0; λ)) = im N(P + (0; λ)) is constant for λ Ω, see Remark 3.1, an we enote it by k. We choose orere bases [u 1 (λ),..., u k (λ)] an [u k+1 (λ),..., u n (λ)] of N(P (0; λ)) an R(P + (0; λ)), respectively. On account of [100, Chapter II.4.2], we can choose these basis vectors in an analytic fashion, so that u j (λ) epens analytically on λ Ω for j = 1,..., n. We can also choose these basis vectors to be real whenever λ is real (recall that we assume that the matrices Ã(ξ) an B(ξ) are real). Definition 4.1 (The Evans function) The Evans function is efine by D(λ) = et[u 1 (λ),..., u n (λ)]. An immeiate consequence of this efinition is that D(λ) = 0 if, an only if, λ is an eigenvalue of T. Note that the Evans function epens on the choice of the basis vectors u j (λ). Any two Evans functions, however, iffer only by a prouct with an analytic function that never vanishes; this factor is given by the eterminants of the transformation matrices that escribe the change of bases. Since this ambiguity in the construction is of no consequence, we sometimes use, with an abuse of notation, the shortcut D(λ) = N(P (0; λ)) R(P + (0; λ)) 21