Asymptotic behavior near transition fronts for equations of generalized Cahn Hilliard form

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1 Asymptotic behavior near transition fronts for equations of generalized Cahn Hilliard form Peter HOWARD February 13, 7 Abstract We consider the asymptotic behavior of perturbations of standing wave solutions arising in evolutionary PDE of generalized Cahn Hilliard form in one space dimension. Such equations are well known to arise in the study of spinodal decomposition, a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into its two components with their concentrations separated by sharp transition layers. Motivated by work of Bricmont, Kupiainen, and Taskinen [5], we regard the study of standing waves as an interesting step toward understanding the dynamics of these transitions. A critical feature of the Cahn Hilliard equation is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis, a feature that is known to complicate the step from spectral to nonlinear stability. Under the assumption of spectral stability, described in terms of an appropriate Evans function, we develop detailed asymptotics for perturbations from standing wave solutions, establishing phase-asymptotic orbital stability for initial perturbations decaying with appropriate algebraic rate. 1 Introduction We consider the asymptotic behavior of perturbations of standing-wave solutions ū(x, ū(± = u ± arising as equilibrium solutions in equations of generalized Cahn Hilliard form for which we assume (H b, c C (R. (H1 b(u ± >, c(ū(x c >, x R. u t = (b(uu x x (c(uu xxx x, x, u, b, c R, t >, (1.1 Our analysis is motivated by the distinguished role the Cahn Hilliard model plays in the study of spinodal decomposition, a phenomenon in which the rapid cooling of a homogeneously mixed binary alloy causes separation to occur, resolving the mixture into its two components with their concentrations separated by sharp transition layers [9, 11]. In this context, and for the case of incompressible fluids, the Cahn Hilliard equation is u t = {M(u (F (u κ u} u ν = u ν = ; x Ω, (1. where Ω denotes a bounded open subset of R n (n typically or 3, ν denotes the unit normal vector to Ω, u denotes the concentration of one component of the binary alloy (or possibly a difference between this concentration and the concentration at homogenous mixing, F(u denotes the bulk free energy of the alloy, and the typically small parameter κ is a measure of the strength of interfacial energy. (In the case that κ depends on u, a new term appears in (1. of the form 1 κ (u u ; see [9]. We take this natural form of the equation from [1].. 1

2 An intesting feature of equations of form (1.1 is that the second and fourth order regularizations can balance, and a wide variety of stationary solutions can arise, including standing waves. For example, in the case of (1.1 with b(u = 3 u 1 ; c(u 1 (1.3 (the case studied in [5], corresponding with F(u = (1/8u 4 (1/4u, M(u 1, and κ = 1 in (1., one readily verifies that the standing wave ū(x = tanh( x (1.4 is such a solution, often referred to as the kink solution. Motivated by the elegant result of Bricmont, Kupiainen, and Taskinen [5] (which we state below for comparison with the current analysis, we regard the study of standing waves as an interesting first step toward understanding the dynamics of these transitions. (See also [36] and [38] for related results in the case of multiple space dimensions. We stress that our focus is on standing waves only, and remark that linearization of (1.1 about a traveling wave leads to a linearized problem with non-vanishing convection and consequently an entirely different flavor than the problem that arises upon linearization about a standing wave. Indeed, upon shifting to a coordinate system moving along with the shock, a flux function is introduced f(u = su, and the wave can be regarded as a regularized undercompressive shock profile. Such problems have been considered in [16], and are quite similar to the problems considered in [5, 6]. A critical feature of equations of form (1.1 is that the linear operator that arises upon linearization of the equation about a standing wave solution has essential spectrum extending onto the imaginary axis, a feature that is known to complicate the step from spectral to nonlinear stability. The purpose of this paper is to study precisely this step from spectral to nonlinear stability. In particular, under the assumption of spectral stability (described below in terms of an appropriate Evans function and verified in the particular case (1.1 (1.3 with wave (1.4, we develop detailed asymptotics for perturbations from standing wave solutions, establishing phase-asymptotic orbital stability for initial perturbations decaying with algebraic rate (1 + x 3/. Our approach to this problem will be to extend to this setting methods developed previously in the context of conservation laws with diffusive and/or dispersive regularity, u t + f(u x = (b(uu x x + (c(uu xx x +..., (1.5 which also have no spectral gap. More precisely, we proceed by computing pointwise estimates on the Green s function for the linear equation that arises upon linearization of (1.1 about the wave ū(x (employing a contour-shifting approach introduced in [, 5]; see also [3, 45]. Such estimates are dependent upon the spectrum of the linear operator, which we understand here in terms of an appropriate Evans function (see, for example, [14, 1, 35, 5] and the discussion and references below. Finally, we employ the local tracking method developed in [34], an approach through which Green s function estimates on the linearized operator can be used to approximately locate shifts from the standing wave ū. Our general approach is similar to the analysis of [5], in which case the authors also employ Green s function estimates on the linear operator in order to close an iteration on the perturbation in some appropriately weighted space. A fundamental difference between the two analyses is that in [5], this iteration is carried out by the renormalization group method, a theory that has its origins in particle physics (see [8] and was introduced in the context of time-asymptotic behavior for nonlinear PDE in [17, 18, ], and further developed in [3, 4] (it pre-dates the current approach by about eight years. Briefly, the renormalization group method is an approach toward understanding asymptotic behavior of PDE that makes use of certain natural scalings in the PDE. This method is ideally suited for cases such as (1.1, for which the equation that arises upon linearization about the wave ū(x has the same natural scaling as one would use for the heat equation. Indeed, though the analysis of [5] is carried out in the context of a self-adjoint linearized equation (the integrated equation, the method could in principle be extended to the current setting for which the linearized operator is not generally self-adjoint even after integration. Nonetheless, the Green s function estimates required for any such extension would almost certainly be obtained by methods very similar to those employed here, which have been developed particularly for the case of non-self-adjoint problems. In this way, the current approach seems to represent a change of thinking from the point of view of classical mathematical physics to that of modern Evans function techniques. We emphasize, however, that the spectral analysis of both [5] and

3 the current analysis rely on the self-adjoint nature of the equation, and in the non-self-adjoint case spectral stability is left as a separate analysis (see the discussion following Remark 1.1. A more fundamental difficulty with the use of such a scaling technique as the renormalization group method in the context of the Cahn Hilliard equation arises in the extension to multiple space dimensions, in which case it is known that the leading eigenvalue of the linearized operator about a planar wave ū(x 1 (leading in the case of stability scales as λ ξ 3 (see [5], and consequently introduces a new cubic scale into the problem, which competes with the natural heat-equation scaling of (1.1. (Here, ξ R d1 denotes a Fourier variable associated with spatial components transverse to the planar wave. A different approach is taken in this setting in [36, 38], quite similar to the method employed here, and the authors conclude stability in space dimensions d 3 for the planar wave ū(x 1 = tanh x 1, arising in (1. with F(u = (1/8u 4 (1/4u, M(u 1 and κ = 1. The primary difference between the approach of [36, 38] and the current analysis is the local tracking function δ(t employed here, which can be generalized to the case of multiple space dimensions as a function of t and the transverse variable x = (x, x 3,..., x d (see [5, 6, 7, 8]. It is precisely this local tracking, which does not seem to fit in any straightforward fashion into the renormalization group framework, that allows us here to obtain stability for more slowly decaying data than considered in [5], and that allows for the extension of this method to the case of space dimensions d (an analysis we carry out in a separate paper. Finally, we note that in adddition to the standing waves considered in the current analysis and the planar waves discussed above, the Cahn Hillard equation admits a wide range of periodic solutions, and the methods here can be extended to that setting in a manner similar to the analyses of Oh and Zumbrun in the case of periodic solutions arising in the context of viscous conservation laws [46, 47]. Before setting up the analysis, we mention that equations of form (1.1 have been shown to arise in the area of mathematical biology in the context of cell formation and aggregation [44], and also that equations similar to (1.1 arise naturally in the modeling of thin film flows, for which under certain circumstances, the height h(t, x of a film moving along an inclined plane can be modeled by fourth order equations h t + (h h 3 x = β(u 3 u x x γ(u 3 u xxx x (see [6] and the references therein. As in the case of conservation laws, it is readily seen that solutions u(t, x initially near ū(x, will not generally approach ū(x, but rather will approach a translate of ū(x determined uniquely by the amount of perturbation mass (measured as R (u(, x ū(xdx carried into the shock layer. We proceed, then, by defining the perturbation v(t, x = u(t, x + δ(t ū(x, (1.6 for which δ(t will be chosen by the analysis to track the location of the perturbed solution in time. In this way, we compare the shapes of u and ū rather than their locations. (The type of stability we will conclude is orbital. Substituting v into (1.1, we obtain the perturbation equation with v t + (a(xv x = (b(xv x x (c(xv xxx x + δ(t(ū x (x + v x + Q(x, v, v x, v xxx x, (1.7 a(x = b (ūū x + c (ūū xxx ; b(x = b(ū(x; c(x := c(ū(x Q(x, v, v x, v xxx = O( vv x + O( vv xxx + O(e η x v, where η > and Q is a continuously differentiable function of its arguments. According to hypotheses (H (H1, we have that a C 1 (R, b, c C (R and for k =, 1 k xa(x = O(e α x ; k x(b(x b ± = O(e α x ; k x(c(x c ± = O(e α x, (1.9 where α > and ± represent the asymptotic limits as x ±. Regarding the convection coefficient a(x, we note that it decays to at both ± (see Section for more details. In comparison with conservation laws u t + f(u x = (b(uu x x, (1.8 3

4 this corresponds with the degenerate case, for which one of the asymptotic convection coefficients a ± = f (u ± for a traveling wave ū(x st is equal to the wave speed s, and both values can be taken without loss of generality as (see [3, 4]. In the case (1.1 (1.3 the wave (1.4 satisfies a(x = 3ū(xū x (x, from which we see that for x <, we have a(x >, while for x >, we have a(x <. In this way, the wave (1.4 is similar to Lax waves in the sense that characteristic speeds (values of the convection coefficient a(x impinge on the transition layer from both sides. The critical difference, again, is that in the case of equations of form (1.1 these characteristic speeds approach asymptotically. Finally, we mention that whereas this decay of a(x is algebraic in the case of degenerate viscous shock waves, it is exponential in the current setting, a difference critical in the Evans function analysis of Section. Another important feature of equations of form (1.1 is that the diffusion coefficient b(ū(x may not always be positive. In the case (1.1 (1.3 with wave (1.4, we have b(ū = 3 ū 1, (1.1 which is asymptotically positive at both ±, but negative near the transition layer at x =. Our assumptions (H (H1 clearly do not give us control over the possible destabilizing effect of this feature of the equations. The effect is, however, encoded in the spectrum of the linear operator Lv = (c(xv xxx x + (b(xv x x (a(xv x, (1.11 which we discuss further below. For now, we suffice to remark that the essential spectrum of L consists of the negative real axis, up to the imaginary axis, and so as in the case of regularized conservation laws, the step from spectral stability to nonlinear stability is problematic. Integrating (1.7, we have (after integration by parts and upon observing that e Lt ū x = ū x the integral equation v(t, x = t G(t, x; yv (ydy + δ(tū x (x [ G y (t s, x; y Q(y, v, v y, v yyy + δ(sv ] dyds, where G(t, x; y denotes a Green s function for the linear part of (1.7: (1.1 G t = LG; G(, x; y = δ y (x. (1.13 We divide G(t, x; y into terms for which the x dependence is exactly ū x (x (the excited terms and the remainder, G(t, x; y. In general, the excited terms do not decay for large time and correspond with mass accumulating at the origin, shifting the asymptotic location of the waves. Writing we have v(t, x = t ū x (x G(t, x; y = G(t, x; y + ū x (xe(t, y, G(t, x; yv (ydy + ū x (x t [ G y (t s, x; y Q(y, v, v y, v yyy + δ(sv Choosing δ(t to eliminate precisely the excited term, we have δ(t = e(t, yv (ydy + e(t, yv (ydy + δ(tū x (x ] dyds [ e y (t s, y Q(y, v, v y, v yyy + δ(sv ] dyds. t [ e y (t s, y Q(y, v, v y, v yyy + δ(sv ] dyds, (1.14 4

5 after which, we have v(t, x = t G(t, x; yv (ydy [ G y (t s, x; y Q(y, v, v y, v yyy + δ(sv ] dyds. (1.15 Integral equations for v x and δ can immediately be obtained by differentiation of both sides of (1.14 and (1.15. Our approach will be to determine estimates on G(t, x; y and e(t, y sufficient for closing simultaneous iterations on v(t, x, v x (t, x, δ(t, and δ(t. To this end, we analyze the Green s function G(t, x; y through its Laplace transform G λ (x, y, which satisfies the ODE (t transformed to λ LG λ λg λ = δ y (x, (1.16 and can be estimated by standard methods. Letting φ + 1 (x; λ and φ+ (x; λ denote the (necessarily two linearly independent asymptotically decaying solutions at + of the eigenvalue ODE Lφ = λφ, (1.17 and φ 1 (x, λ and φ (x, λ similarly the two linearly independent asymptotically decaying solutions at, we write G λ (x, y as a linear combination { φ 1 G λ (x, y = (x; λn+ 1 (y; λ + φ (x; λn+ (y; λ, x < y φ + 1 (x; λn 1 (y; λ + φ+ (x; λn (y; λ, x > y. (1.18 Insisting, as usual, on the continuity of G λ (x, y and its first two x-derivatives in x, and on the jump in 3 x G λ(x, y at x = y, we have N 1 + (y; λ = W(φ+ 1, φ+, φ ; N + c(yw λ (y (y; λ = W(φ+ 1, φ+, φ 1 c(yw λ (y N 1 (y; λ = W(φ 1, φ, φ+ c(yw λ (y N (y; λ = W(φ 1, φ, φ+ 1, c(yw λ (y (1.19 where W(φ 1, φ,..., φ n denotes a square determinant of column vectors created by augmentation with an appropriate number of x-derivatives and W λ (y := W(φ 1, φ, φ+ 1, φ+. We see immediately from (1.18 and (1.19 that, away from essential spectrum, G λ (x, y is bounded so long as W λ (y is bounded away from. Since W λ (y is a Wronskian for (1.17, for each fixed λ its sign does not change as y varies. In light of this, we define the Evans function as D(λ = W λ (. (1. Introduced by Evans in the context of nerve impulse equations [14] (see also the early analysis of Jones for pulse solutions to the FitzHugh Nagumo equation [35], the Evans function serves as a characteristic function for the operator L. More precisely, away from essential spectrum, zeros of the Evans function correspond in location and multiplicity with eigenvalues of the operator L, an observation that has been made precise in [1] in the case pertaining to reaction diffusion equations of isolated eigenvalues, and in [1, 5] and [4] in the cases pertaining respectively to conservation laws and the nonlinear Schrödinger equation of nonstandard effective eigenvalues embedded in essential spectrum. (The latter correspond with resonant poles of L, as also examined in [48]. Briefly, we remark on the connection between our form of the Evans function (1. and the definition of Evans et al. [1, 14, 35]. Letting Φ ± k := (φ± k, φ± k, φ ± k, φ ± k tr, define Y ± (x; λ = Φ ± 1 (x; λ Φ± (x; λ, (1.1 where represents a wedge product on the space of vectors in R 4 (i.e., denotes an associative, bilinear operation on vectors v R 4, characterized by the relation between standard R 4 basis vectors e j and e k, e j e k = e k e j. In this case, we obtain through straightforward calculation that D(λ = e R x tra(y;λdy Y (x; λ Y + (x; λ, (1. 5

6 which is the form of [1, 14, 35]. (As described below, the matrix A is simply the matrix that arises when (1.17 is written as a first order system. One notable advantage of the formulation (1. is that the -forms Y ± (x; λ are the asymptotically decaying solutions of the equations where A is defined by Y ± = A ±(x; λy ±, Y ± = (Φ± 1 Φ± = Φ ± 1 Φ ± + Φ ± 1 Φ± = (AΦ ± 1 Φ± + Φ± 1 AΦ± = A ±Φ ± 1 Φ± = A ±Y ±. In this way, the wedge formulation can remain analytically valid in cases for which the wedge products Y ± cannot be uniquely resolved into components. In the current setting, the Φ ± k remain sufficiently distinct (away from essential spectrum that the two formulations are equivalent. We will observe in Section that D(λ is not analytic at λ =, but can be extended analytically on the Riemann surface ρ = λ. (This is quite similar to the analysis of Kapitula and Rubin in the case of the complex Ginzburg Landau equation [39], and the analyses of Guès et al. and of Zumbrun in the case of multidimensional shock stability in the vicinity of glancing points [19, 51], but differs markedly from the case of degenerate conservation laws, for which the Evans function has a dependence of the form λ lnλ [3, 4]. In light of this, we define the function D a (ρ := D(λ. The primary concern of this analysis is to show that under assumptions (H (H1, nonlinear stability of standing wave solutions ū(x of (1.1 is implied by the spectral condition (D: (D: The Evans function D(λ has precisely one zero in {Reλ }, necessarily at, and ρρ D a (. Remark 1.1. Under the assumption of condition (D, the point spectrum of L, aside from the distinguished point at λ =, lies to the left of a parabolic contour, which passes through the negative real axis and opens into the negative real half plane, λ s (k = c k + i c 1 k + c. (1.3 We will denote this contour Γ s. The essential spectrum of L consists of the negative real axis. While condition (D is generally quite difficult to verify analytically (see, for example, [13, 15, 35], it can be analyzed numerically [, 33, 31]. In the case of equations (1.1 (1.3 and the standing wave (1.4, (D is straightforward to verify (see [5] or Lemma.6 of the current analysis, in which we review the analysis of [5] for the sake of completeness. We are now in a position to state the first theorem of the paper. Theorem 1.1. Suppose ū(x is a standing wave solution to (1.1 and suppose (H (H1 hold, as well as stability criterion (D. Then for some fixed C and C E, and for positive contants M and K sufficiently large, and for η >, depending only on the spectrum and coefficients of L, the Green s function G(t, x; y described in (1.13 satisfies the following estimates for y (with symmetric estimates in the case y. (I For either x y Kt or t 1, and for α a multi-index in the variables x and y, α G(t, x; y Ct 1+ α 4 e xy 4/3 Mt 1/3, α 3. (II For x y Kt and t 1 where G(t, x; y = G(t, x; y + E(t, x; y, 6

7 (i y x (ii x y G(t, x; y = G y (t, x; y = G x (t, x; y = O( C [e (xy 4b t 4b t C 4b t y [e (xy 4b t + O(t 3/ e (xy Mt + O( x y (xy e Mt t3/ G xy (t, x; y = O(t 3/ e (xy Mt G(t, x; y = C [e (xy 4b t 4b t ] e (x+y 4b t + O(t 1 e (xy Mt ] e (x+y 4b t x + y (x+y e Mt t3/ + O(t 1/ e η x e y Mt + O(t 3/ e (xy Mt + O( y t 3/ eη x e y Mt. ] e (x+y 4b t + O(t 1 e (xy Mt x y G y (t, x; y = O( G x (t, x; y = t 3/ e (xy Mt C 4b t x [e (xy 4b t + O(t 3/ e (xy Mt ] e (x+y 4b t + O(t 1/ e η x e y x + y Mt + O( t G xy (t, x; y = O(t 3/ e (xy Mt 3/ e (x+y Mt + O(t 3/ e (xy Mt (iii y x and in all cases G(t, x; y = O(t 1/ e (x t 3/ s b+ b y Mt x b+ b y (x G y (t, x; y = O( e x b+ b y (x G x (t, x; y = O( e t 3/ G xy (t, x; y = O(t 3/ e (x s b+ b y Mt s b+ y b Mt s b+ y b Mt E(t, x; y = C E ū x (x C E + O( y t 3/ eη x e y (x Mt + O(t 3/ e + O(t 1/ e η x e y Mt + O(t 3/ e (x + O( y t 3/ eη x e y Mt y 4b t E y (t, x; y = y 4b tūx(xe ( e ζ dζ 1 + O(e η y 4b t ( 1 + O(e η y. s b+ b y Mt s b+ b y The estimates of Theorem 1.1 are quite similar to those of Lemma.1 in [38], in which the authors are considering a multidimensional generalization of (1.1 (1.3. Similar estimates are also obtained in [5] during the course of the analysis, though in that case the authors work with the integrated equation and consequently the term E(t, x; y, which does not decay as t grows, does not appear. In both of these cases, the analysis is restricted to the case of (1., with F(u = (1/8u 4 (1/4u, M(u 1, and κ = 1, and the wave (1.4. The estimates of Theorem 1.1 are sufficient for establishing the following theorem on the perturbations v(t, x. Mt 7

8 Theorem 1.. Suppose ū(x is a standing wave solution to (1.1, and suppose (H (H1 hold, as well as stability criterion (D. Then for Hölder continuous initial perturbations (u(, x ū(x C +γ (R, γ >, with (u(, x ū(xdx = (1.4 u(, x ū(x E (1 + x 3/, for some E sufficiently small, and for δ(t as defined in (1.14, there holds u(t, x + δ(t ū(x CE (1 + x + t 3/ x (u(t, x + δ(t ū(x CE [ t 1/4 (1 + t 1/4 (1 + x + t 3/ + (1 + t 3/4 e η x ] δ(t CE (1 + t 1/4 δ(t CE (1 + t 5/4. (1.5 In the estimates of Theorem 1., we have included v x (t, x in order to take advantage of the particular form of the nonlinearity Q, in which the term O( vv x is better than O(v, which arises in a similar analysis in the case of regularized conservation laws. Remark 1.. We note that in the current setting the assumption of zero mass initial data can be taken without loss of generality. As in the case of Lax waves arising in regularized conservation laws, the asymptotic shift of the wave can be located here entirely from the initial perturbation. In this way, the shift to a zero mass perturbation is simply a change of perspective, in which we consider the stability of the asymptotically selected wave rather than of the original wave. The validity of this a priori selection of the asymptotic translate is justified precisely by our decay estimate on δ(t, from which we verify that perturbations of the zero-mass wave do indeed approach the zero-mass wave. Of course, this means the stability we conclude is orbital. In light of Remark 1., we can state the following theorem regarding initial perturbations with non-zero mass. Corollary 1.1. Suppose ū(x is a standing wave solution to (1.1, and suppose (H (H1 hold, as well as stability criterion (D. Then for Hölder continuous initial perturbations (u(, x ū(x C +γ (R, γ >, with u(, x ū(x E (1 + x 3/, (1.6 for some E sufficiently small, there holds where u(t, x + x ū(x CE [(1 + x + t 3/ + (1 + t 1/4 e η x ], (1.7 1 x = u + u u(, x ū(xdx. We note specifically the relationship between Corollary 1.1 and the result of [5]. In [5], the authors employ the renormalization group method to establish the following result on (1.1 (1.3: For any p >, there exists δ > such that for v Xp δ, { } X p := v C (R : v X := sup(1 + x p+1 v (x < x R, the Cauchy problem (1.1 (1.3 with u(1, x = ū(x + v (x, where ū(x is as in (1.4, has a unique smooth solution u(t, x, t (1, satisfying for some x R depending only on v. u(t, ū( x L (R as t, 8

9 Moreover, for some constants A and B depending only on v, there holds u(t, x + x = ū(x + 1 d (e x 4t (Aū(x + B + o(t 1. 4πt dx The difference between Corollary 1.1 and the result of [5] (for the equation (1.1 (1.3, with wave (1.4 lies predominately in the difference in assumptions on the initial perturbation. In taking more rapidly decaying initial perturbations, we could increase the decay rate of our perturbation in time. More precisely, in the estimate of Theorem 1., for 1 < r < initial perturbations of size (1 + x r give temporal decay on u(t, x + δ(t ū(x at rate (1 + t r/ and δ(t decay at rate (1 + t (1r/. For r >, δ(t decays at the maximal rate (1 + t 1/, and the estimate of Corollary 1.1 becomes u(t, x + x ū(x CE [(1 + t 1/ e η x + (1 + t 1], (1.8 which almost recovers the detail of [5] Theorem 1.1. Finally, we note that the second order term in [5] Theorem 1.1 is the analogue here of the diffusion waves of Liu [41], and could be recovered in the current setting by an analysis similar to that of [9, 3, 49]. Outline of the paper. In Section, we carry out a general analysis of the Evans function for the eigenvalue problem (1.17, providing full details in the case (1.1 (1.3 with wave (1.4. In addition, we use this analysis to develop estimates on the Laplace-transformed Green s function G λ (x, y. In Section 3, we prove Theorem 1.1, while in Section 4 we prove Theorem 1.. Finally, some representative estimates on the integrations arising from our integral equations are provided in Section 5. Analysis of the Evans Function In this section we analyze the Evans function, as defined in (1., for our linear eigenvalue problem (1.17. We begin with a remark on the structure of standing waves ū(x arising in the context of (1.1. Upon substitution of such a wave into (1.1, we have the ODE (c(ūū xxx x (b(ūū x x =. (.1 Observing that ū x (x as x ±, we can integrate (.1 to obtain the third order equation c(ūū xxx b(ūū x =. (. Asymptotically, solutions to (. behave like solutions to the constant coefficient equations c ± ū xxx b ± ū x =, (.3 whose solutions grow and decay with rates ± b ± /c ± and. Since the solutions with rate correspond with constant solutions to (.1, we see that any solution that does not blow up must approach its endstates at exponential rate. We now begin our analysis of the Evans function by writing the eigenvalue problem (1.17 as a first order system W = A(x; λw, (.4 where A(x; λ = λ c(x a (x c(x a(x c(x + b (x c(x b(x c(x Under assumptions (H (H1, A(x; λ has the asymptotic behavior { A (λ + E(x; λ, x < A(x; λ = A + (λ + E(x; λ, x >, c (x c(x. 9

10 where lim A(x; λ = A ±(λ = x ± λ c ± 1 1 1, (.5 b ± c ± and for λ bounded E(x; λ = O(e α x. The eigenvalues of A ± (λ satisfy the relation, b ± ± b µ ± 4λc ± =. c ± Ordering these so that j < k Reµ j (λ Reµ k (λ, we have µ ± 1 (λ = b ± c ± + O( λ ; µ ± (λ = 1 b± λ + O( λ 3/ µ ± 3 (λ = + 1 b± λ + O( λ 3/ ; µ ± 4 (λ = + b ± c ± + O( λ, with associated eigenvectors V ± k = (1, µ ± k, (µ± k, (µ ± k 3 tr. A straightforward way in which to understand these expressions is through the relationship b ± b 4c± λ ± 4λc ± =, b ± + b ± 4λc ± where the right-hand side is now in the form of λ multiplied by a function that for λ sufficiently small is analytic in λ. We note that for λ sufficiently small, and away from essential spectrum (the negative real axis, these eigenvalues remain distinct. Also, we clearly have µ ± 1 = µ± 4 and µ± = µ± 3. Lemma.1. For the eigenvalue problem (1.17, assume a C 1 (R, b, c C (R, with b ± > and c ± >, and additionally that (1.9 holds. Then for some ᾱ > and k =, 1,, 3, we have the following estimates on a choice of linearly independent solutions of (1.17. For λ r, some r > sufficiently small, there holds: (i For x (ii For x k x φ 1 (x; λ = eµ 3 (λx (µ 3 (λk + O(e ᾱ x k x φ (x; λ = eµ 4 (λx (µ 4 (λk + O(e ᾱ x x k ψ 1 (x; λ = eµ 1 (λx (µ 1 (λk + O(e ᾱ x x k 1 ( ψ (x; λ = µ (λ µ (λk e µ (λx µ 3 (λk e µ(λx 3 + O(e ᾱ x. k x φ+ 1 (x; λ = eµ+ 1 (λx (µ + 1 (λk + O(e ᾱ x x k φ+ (x; λ = eµ+ (λx (µ + (λk + O(e ᾱ x x k 1 ( ψ+ 1 (x; λ = µ + 3 (λ µ + 3 (λk e µ+ 3 (λx µ + (λk e µ+ (λx + O(e ᾱ x k xψ + (x; λ = eµ+ 4 (λx (µ + 4 (λk + O(e ᾱ x. Proof. Lemma.1 can be proved by standard methods such as those of [5], pp We remark here only on the choice of our slow growth solutions ψ (x; λ and ψ+ 1 (x; λ. We could alternatively make the choice x k ψ (x; λ = eµ (λx (µ (λk + O(e ᾱ x k x ψ + 1 (x; λ = eµ+ 3 (λx (µ + 3 (λk + O(e ᾱ x. 1

11 The difficulty with this choice is that these growth solutions coalesce with the slow decay solutions φ 1 and φ + as λ. (See [3], in which this latter convention is taken. Following the analysis of [5], we take the linear combination of solutions stated in order to have a set of solutions of (1.17 that remains linearly independent as λ. In order to see that ψ (x; λ and ψ+ 1 (x; λ are valid choices, we note that their derivation depends precisely on this coalescence with the slow decay solutions φ 1 and φ+ as λ. In the case, for example, of ψ (x; λ, we proceed by letting φ (x be any λ = solution of (1.17 that neither grows nor decays as x. (The existence of such a solution follows immmediately from the slow decay solution φ 1. We search for solutions of the form ψ (x; λ = φ (xw(x; λ, for which direct substitution reveals c(xφ w (c(x(3φ w + 3φ w + φ w + b(xφ w + (b(xφ w a(xφ w = λφ w, (.6 and we make the critical observation that the only coefficient of undifferentiated w is λφ. In this way, the first order system associated with (.6 has the form (W k = k w W = A (λw + E(x; λw, where A (λ is precisely as in (.5, while E(x; λ has the particular form E(x; λ =, (.7 λo(e α x O(e α x O(e α x O(e α x for some α >. Searching for solutions W = e µ 3 x Z, we have the equation which can be integrated as Z(x = V 3 (λ + x Z + µ 3 IZ = A (λz + EZ, e (Aµ 3 (xy P E(y; λz(y; λdy M + e (Aµ 3 (xy QE(y; λz(y; λdy, x where similarly as in [5], P is a projection operator projecting onto the eigenspace associated with µ 1, µ, and µ 3, while Q is the projection operator projecting onto the eigenspace associated with µ 4. In this way, the first integral decays at exponential rate due to the exponential decay of E and the second integral decays at exponential rate due to the exponential decay of the combination e (µ 4 µ 3 (xy E(y; λ. Consequently, a standard contraction mapping closes for all x sufficiently large. This in turn justifies a direct iteration, in which V 3 (λ is taken as a first approximation for Z(x. Upon substitution for a second iterate, we observe 1 E(y; λv 3 + (λ = µ 3 (λ µ λo(e α y O(e α y O(e α y O(e α y 3 (λ = O( λ1/ e α y. µ 3 (λ3 Continuing the iteration, we conclude φ 1 (x; λ = eµ 3 (λx φ (x(1 + O( λ1/ e α x. Similarly, a slow growth solution can be constructed from φ (x, as ψ (x; λ = eµ (λx φ (x(1 + O( λ1/ e α x. 11

12 Any linear combination of φ 1 (x; λ and ψ (x; λ is also a solution of (1.17, and so we are justified in defining ψ 1 ( (x; λ := ψ µ (x; λ φ 1 (x; λ. We have, then, 1 ( ψ (x; λ φ 1 (x; λ µ = 1 ( µ e µ (λx φ (x(1 + O( λ1/ e α x e µ 3 (λx φ (x(1 + O( λ1/ e α x. = φ (x µ ( e µ (λx e µ 3 (λx + O( λ 1/ e α x. The estimate we state is obtained by scaling φ (x as φ (x = 1 + O(eα x. The case of ψ 1 + (x; λ can be analyzed similarly. Lemma.. Under the assumptions of Lemma.1, and for φ ± k, ψ± k as in Lemma.1, with k =, 1 and for D(λ as in (1. we have the following estimates. For some α >, (i For x x k W(φ 1, φ, ψ 1 = c(xw λ (x x k W(φ 1, φ, ψ = c(xw λ (x 1 (λx( c(d(λ eµ (µ k (µ 1 µ 4 (µ 1 µ 3 (µ 4 µ 3 + O(e α x 1 (λx( c(d(λ eµ 1 (µ 1 k (µ µ 4 (µ µ 3 x k W(φ 1, ψ 1, ψ 1 = (λx( c(xw λ (x c(d(λ eµ 4 (µ 4 (µ µ 1 (µ µ 3 µ x k W(φ, ψ 1, ψ 1 ( (µ = µ 1 (µ µ 4 (µ 1 µ 4 c(xw λ (x c(d(λ µ + O( λ 1/ e α x. µ (µ 4 µ 3 + O(e α x (µ 1 µ 3 + O(e α x ( e µ 3 (λx (µ 3 k e µ (λx (µ k (ii For x x k W(φ + 1, φ+, ψ+ 1 = c(xw λ (x x k W(φ + 1, φ+, ψ+ = c(xw λ (x x k W(φ + 1, ψ+ 1, ψ+ = c(xw λ (x x k W(φ +, ψ+ 1, ψ+ = c(xw λ (x 1 + (λx( c(d(λ eµ 4 (µ + 4 k ( µ+ 3 µ+ µ + (µ + 3 µ+ 1 (µ+ µ+ 1 + O(e α x (λx( c(d(λ eµ 3 (µ + 3 k (µ + 4 µ+ (µ+ 4 µ+ 1 (µ+ µ+ 1 + O(e α x 1 ( (µ + 4 µ + 3 (µ+ 4 µ+ 1 (µ+ 3 µ+ 1 ( e µ+ (λx (µ + c(d(λ k e µ+ 3 (λx (µ + 3 k + O( λ 1/ e α x µ (λx( c(d(λ eµ 1 (µ + 1 k (µ + 4 µ+ 3 (µ+ 4 µ+ (µ+ 3 µ+ + O(e α x. Proof. The estimates of Lemma. are each carried out in a similar straightforward manner, so we proceed only in the case of W(φ, ψ 1, ψ x. c(xw λ (x µ + 3 1

13 Computing directly, and observing (c(xw λ (x =, we have x W(φ, ψ 1, ψ c(xw λ (x For the denominator, we note Abel s relation, or = xw(φ, ψ 1, ψ. c(xw λ (x W λ(x = W λ (e R x tra(y;λdy = D(λe R x c (y c(y dy = D(λ c( c(x, For the numerator, we compute φ x W(φ ψ1 ψ, ψ 1, ψ = det = 1 µ 1 µ c(xw λ (x = c(d(λ. (.8 φ φ ψ1 ψ1 ψ ψ det µ 4 µ 1 µ e (µ 1 +µ +µ 4 x (µ 4 3 (µ 1 3 (µ det µ 4 µ 1 µ 3 (µ 4 3 (µ 1 3 (µ 3 3 e (µ 1 +µ 3 +µ 4 x + O(e α x. We mention particularly that in the exponentially decaying terms of this last calculation, we have observed the estimate 1 µ (λx e µ 3 (λx O(e ᾱ x 1 C 1 (λ(eµ µ (λ λx O(e ᾱ x for some < α < ᾱ. Computing the determinants, we find x W(φ, ψ 1, ψ = 1 µ = O(e α x, [ (µ 3 (µ µ 1 (µ µ 4 (µ 3 µ 4 e(µ 1 +µ +µ 4 x (µ (µ 3 µ 1 (µ 3 µ 4 (µ 1 µ 4 ]e (µ 1 +µ 3 +µ 4 x + O(e α x, for which we observe that with µ 1 = µ 4 and µ = µ 3, we have and (µ µ 1 (µ µ 4 (µ 3 µ 4 = (µ 3 µ 1 (µ 3 µ 4 (µ 1 µ 4, e (µ 1 +µ +µ 4 x = e µ 3 x e (µ 1 +µ 3 +µ 4 x = e µ x. Combining these last observations with (.8, we conclude our argument for the case of interest. The remaining estimates follow similarly. In addition to Lemmas.1 and., we have the following estimates on the N ± k. Lemma.3. Under the assumptions of Theorem 1.1, we have the following estimates on the N ± k (with similar estimates for x. For x, k =, 1, and for λ r, some r sufficiently small, k x N 1 (x; λ = O( λ1+ k e µ (λx k x N (x; λ = O( λ 1 + k e µ (λx k xn + 1 (x; λ = O( λ 1 ( e µ (λx µ (λk e µ 3 (λx µ 3 (λk + O( λ 1 + k e µ 3 (λx k xn + (x; λ = O( λ1+ k e µ (λx + O(1e µ 4 (λx. (.9 of (

14 Proof. We note at the outset that for x, we will take the expansions φ + k (x; λ = A+ k (λφ 1 (x; λ + B+ k (λφ (x; λ + C+ k (λψ 1 (x; λ + D+ k (λψ (x; λ. (.1 Without loss of generality, we take the convention according to which there must hold φ + 1 (x; = ū x(x = φ (x;, (.11 A + 1 (λ = O( λ1/ ; B + 1 (λ = 1 + O( λ1/ ; C + 1 (λ = O( λ1/ ; D + 1 (λ = O( λ1/ ; A + (λ = O(1; B+ (λ = O(1; C+ (λ = O(1; D+ (λ = O(1. (.1 As each estimate of Lemma.3 is proven similarly, we provide details only in the case of x N1. We have, similarly as in the proof of Lemma. x N1 (x; λ = xw(φ 1, φ, φ+, c(xw λ (x for which we compute φ x W(φ 1 φ φ + 1, φ, φ+ = det. φ 1 φ 1 φ φ φ + φ + In order to derive expressions for the terms φ ± k, we consider the eigenvalue problem (1.17. In general, we have the relation (c(xφ ± k + (b(xφ ± k (a(xφ ± k = λφ ± k. (.13 For φ 1 and φ, upon integration from up to x, and keeping in mind that a(x decays at exponential rate as x ±, we have c(xφ k x + b(xφ k a(xφ k = λ φ k (y; λdy. The most straightforward case is φ (x; λ, which for x < decays at exponential rate even for λ =. In this case, we define x W (x; λ = φ (y; λdy = O(1eµ 4 (λx. (.14 Proceeding similarly for φ 1 (x; λ, which fails to decay for λ =, we define W1 (x; λ = x λ φ 1 (y; λdy = x λ e µ 3 (λy (1 + O(e ᾱ y dy = 1 λ µ 3 (λx + λo(e α x, 3 (λeµ so that with c(xφ 1 + b(xφ 1 a(xφ 1 = λw1 (x; λ, W1 (x; λ = eµ 3 λ (λx ( µ 3 (λ + O( λ1/ e α x. Finally, for φ + (x; λ in the case x <, we must expand in terms of the linearly independent solutions φ k and ψ k, φ + (x; λ = A+ (λφ 1 (x; λ + B+ (λφ (x; λ + C+ (λψ 1 (x; λ + D+ (λψ (x, λ. 14

15 In this case, we integrate (.13 for y [x, + to obtain c(xφ + (x; λ b(xφ + (x; λ + a(xφ + (x; λ = λ where we divide this final integration up as M φ + (y; λdy = φ + (y; λdy + x x M x φ + (y; λdy =: λw + (x; λ, φ + (y; λdy, where M > is chosen sufficiently large so that the estimates of Lemma.1 hold for x M. For the integration over y > M, we have φ + (y; λdy = O(1dy + M M On the other hand, for integration over y < M, we have M x Combining these last observations, we conclude e µ+ (λy (1 + O(e ᾱ y dy = O( λ 1/. M M φ + (y; λdy = A+ (λ φ 1 (y; λdy + B+ (λ x x M M + C + (λ ψ1 (y; λdy + D+ (λ x x φ ψ (y; λdy (y, λdy. W + (x; λ = O(1 + M M λ [A + (λ φ 1 (y; λdy + B+ (λ φ (y; λdy x x M M ] + C + (λ ψ1 (y; λdy + D+ (λ ψ (y, λdy = O(1e µ 1 (λx x Omitting coefficient dependence on y for brevity of notation, we have φ 1 φ φ + det φ 1 φ 1 = det φ φ φ + φ + φ 1 φ φ + φ 1 b c φ 1 a c φ 1 λ c W 1 1 = det 1 = det a c φ 1 b c 1 b c φ φ x a c φ λ c W φ 1 φ φ + φ 1 λ φ φ + c W 1 λ c W λ φ 1 φ φ +. φ φ + λ c W 1 λ c W λ c W+ c W+ φ + b c φ+ a c φ+ λ c W+ We proceed by expanding this last determinant along its bottom row, giving (.15 λ c(x W 1 (x; λw(φ, φ+ + λ λ c(x W (x; λw(φ 1, φ+ + c(x W+ (x; λw(φ 1, φ. (.16 For the first expression in (.16, we observe W(φ, φ+ = A+ (λw(φ, φ 1 + C+ (λw(φ, ψ 1 + D+ (λw(φ, ψ = O(1e (µ 1 +µ 4 y = O(1, 15

16 while similarly In this way, the terms in (.16 give respectively W(φ 1, φ+ = O(1e(µ 1 +µ 3 y. O( λ 1/ e (µ 1 +µ 3 +µ 4 y + O( λ e (µ 1 +µ 3 +µ 4 y + O( λ 1/ e (µ 1 +µ 3 +µ 4 y. We now recall the relationship µ 3 + µ =, and note that according to spectral criterion (D, there holds D(λ = c 1 λ + O( λ 3/. Upon division, then, by c(yw λ (y = c(d(λ, we conclude the claimed estimate. The remaining estimates of Lemma.3 follow similarly. Lemma.4. Under the assumptions of Theorem 1.1, and for C + k (λ and D+ k (λ as defined in (.1, there holds C 1 + (λd+ (λ C+ (λd+ 1 (λ = O( λ. Proof. We first observe that by augmenting (.1 with y-derivatives up to third order, we obtain the matrix equations φ 1 φ ψ1 ψ A + φ + φ 1 φ ψ1 ψ k k φ 1 φ ψ1 ψ B + k C + = φ + k k φ + D + k. (.17 k Proceeding by Cramer s rule, we have, for example, φ 1 φ ψ 1 ψ C + 1 (λ = W(φ 1, φ, φ+ 1, ψ W(φ 1, φ, ψ 1, ψ, where the W notation is defined immediately following (1.19. By linear independence, the denominator of this last expression is non-zero, while for the numerator, we compute similarly as in (.15 W(φ 1, φ, φ+ 1, ψ = det = ( ψ φ + k φ 1 φ φ + 1 ψ φ 1 φ 1 λ c(x W 1 b(x c(x ψ φ φ λ c(x W+ + a(x b(x ψ ψ 1 + ψ 1 + λ c(x W+ 1 ψ ψ ψ b(x c(x ψ W(φ 1, φ, φ+ 1 + O( λ, + a(x b(x ψ where ψ is treated in a distinguished manner because it cannot be integrated to an asymptotic limit. Proceeding similarly, we also have W(φ 1, φ, φ 1, ψ C+ (λ = W(φ 1, φ, φ+, ψ = ( ψ b(x c(x ψ a(x + b(x ψ W(φ 1, φ, φ 1, ψ D+ 1 (λ = W(φ 1, φ, ψ 1, φ+ 1 ( = ψ 1 b(x c(x ψ a(x 1 + b(x ψ 1 W(φ 1, φ, φ 1, ψ D+ (λ = W(φ 1, φ, ψ 1, φ+ ( = ψ 1 b(x c(x ψ a(x 1 + b(x ψ 1 W(φ 1, φ, φ+ + O( λ1/ W(φ 1, φ, φ+ 1 + O( λ W(φ 1, φ, φ+ + O( λ1/. 16

17 Combining these observations, we compute W(φ 1, φ, φ 1, ψ ( C + 1 (λd+ (λ C+ (λd+ 1 (λ = [( ψ b(x [ ( ψ1 [( ψ c(x ψ + a(x b(x ψ b(x c(x ψ a(x 1 + b(x c(x ψ [ ( ψ1 b(x = ( ψ b(x c(x ψ ( ψ1 b(x b(x ψ 1 + a(x b(x ψ a(x + b(x ψ W(φ 1, φ, φ+ 1 + O( λ ] ] W(φ 1, φ, φ+ + O( λ1/ W(φ 1, φ, φ+ + O( λ1/ c(x ψ a(x ] 1 + b(x ψ 1 W(φ 1, φ, φ+ 1 + O( λ W(φ 1, φ, φ+ 1 O( λ1/ c(x ψ 1 ( ψ b(x ( + ψ1 + O( λ 3/. c(x ψ + a(x b(x ψ 1 + a(x b(x ψ b(x c(x ψ a(x 1 + b(x ψ 1 W(φ 1, φ, φ+ O( λ W(φ 1, φ, φ+ O( λ W(φ 1, φ, φ+ 1 O( λ1/ Recalling that φ (x; = ū x(x = φ + 1 (x,, we have that determinants involving both of these functions must vanish as λ, necessarily at rate λ 1/ or better by analyticity in ρ. We conclude the claimed estimate. We next state our main theorem on behavior of the Evans function for small values of λ, which we note is simply the extension to the current setting of Proposition.7 of [7]. Lemma.5. Under the assumptions of Theorem 1.1, and for D(λ as defined in (1., we have D(λ = γ(u + u λ + O( λ 3/, ] with γ = 1 φ 1 (; λ ū x( φ + (; λ c( det φ 1 (; λ ūxx ( φ + (; λ. φ 1 (; λ ūxxx ( φ + (; λ Proof. Observing that the solutions φ ± k (x; λ are analytic in the variable ρ := λ, we set D a (ρ := D(λ. We have, then, ρ D a ( = W( ρ φ 1, φ, φ+ 1, φ+ + W(φ 1, ρφ, φ+ 1, φ+ + W(φ 1, φ, ρφ + 1, φ+ + W(φ 1, φ, φ+ 1, ρφ + = W(φ 1, ρ(φ φ+ 1, ū x(, φ +, where the terms for which neither φ + 1 nor φ are differentiated are due to the choice of scaling (.11. We observe here that the fast-decaying solutions φ and φ+ 1 have been derived analytically in λ = ρ, and thus ρ φ (x; = ρ φ + 1 (x; =. (.18 In this way, we see immediately that ρ D a ( =. 17

18 Proceeding similarly for ρρ D a (ρ, we find ρρ D a ( = W( ρ φ 1 (;, ρ(φ φ+ 1 (;, ū x(, φ + (; + W(φ 1 (;, ρ(φ φ+ 1 (;, ū x(, ρ φ + (; + W(φ 1 (;, ρφ (;, ρφ + 1 (;, φ+ (; + W(φ 1 (;, ρρ(φ φ+ 1 (;, ū x(, φ + (;. Upon substitution of (.18, all terms with only single partials are eliminated, and we have ρρ D a ( = W(φ 1 (;, ρρ(φ φ+ 1 (;, ū x(, φ + (;. In order to understand ρ derivatives of φ and φ+ 1, we proceed similarly as in (.13, setting (c(xφ ± k + (b(xφ ± k (a(xφ ± k = ρ φ ± k, (.19 where prime here denotes differentiation with respect to x. Keeping in mind that φ and φ+ 1 exponential rate even for λ =, we have, upon integration on (, x], both decay at c(xφ + b(xφ a(xφ = ρ W, where W is as in (.14, and similarly for φ+ 1. Upon differentiation with respect to ρ, we find c(x( ρ φ + b(x( ρ φ a(x( ρ φ = ρw + ρ ( ρ W. with again a similar relationship for φ + 1. We next take a second ρ derivative of (.19 to obtain, upon integration and evaluation at ρ =, c(x( ρρ φ + b(x( ρρ φ a(x( ρρ φ = W (x;. Recalling that we have and similarly We conclude W (x; ρ := x φ (y; ρdy, x W (x; = ū y (ydy = ū(x u, x W 1 + (x; = ū y (ydy = ū(x u +. c(x( ρρ (φ φ+ 1 + b(x( ρρ (φ φ+ 1 a(x( ρρ (φ φ+ 1 = (u + u. Finally, we have φ 1 (; ρρ(φ φ+ 1 (; ū x( φ + (; φ 1 (; ρρ (φ ρρ D a ( = det φ+ 1 (; ū xx ( φ + (; φ 1 (; ρρ (φ φ+ 1 (; ū xxx ( φ + (; φ 1 (; ρρ (φ φ+ 1 (; ū xxxx ( φ + (; φ 1 (; ρρ(φ φ+ 1 (; ū x( φ + (; φ 1 (; ρρ (φ = det φ+ 1 (; ū xx ( φ + (; φ 1 (; ρρ (φ φ+ 1 (; ū xxx ( φ + (;, c( (u + u from which the claimed relationship is immediate. We now state for completeness a lemma asserting that (D is known to hold in the case (1.1 (1.3 with wave (

19 Lemma.6. For the case (1.1 (1.3, and for the standing front (1.4, spectral condition (D holds. Proof. In order to verify the first assertion of (D that the only zero that D(λ has with non-negative real part is λ = we first note that the integrated variable w(x = x φ(y; λdy satisfies the integrated eigenvalue problem [ Lw = λw; Lw := x ( xx V (xw x ], where V (x = (3/cosh (x/, and L is clearly a self-adjoint operator. Away from essential spectrum, the eigenvalues of L correspond with those of L, and so aside from the point λ = (which is an eigenvalue of L but not of L, our study reduces to that of L, which has been considered in [5, 36, 38] (see additionally the alternative analysis of [1]. For completeness, we briefly review the observations of [5, 36, 38]. As pointed out in [36, 38], the middle operator Mw := w xx + (1 + V (xw has been shown in [43] to have two isolated eigenvalues at and 3 4, and essential spectrum on the line [1,. It is an immediate consequence of the Spectral Theorem (see e.g. [37], p. 36 that M is a (nonstrictly positive operator. In the event that λ is an eigenvalue of L (necessarily real, we have that for some eigenfunction w associated with λ that (Mw x x = λw. Upon multiplication by w and integration over R, we have w x Mw x dx = λ w L. We can conclude by the positivity of M that λ. According to Lemma.5, the second assertion in stability criterion (D is verified if γ. This can be checked by a careful study of solutions to the eigenvalue problem (1.17. In the case of (1.1 (1.3, and profile (1.4, (1.17 becomes φ xxxx + (b(xφ xx = λφ, (. with b(x = 3 ū(x 1 ; ū(x = tanhx. In the case of φ 1 (x; λ, (. can be integrated on x (, x], and we obtain where as in the proof of Lemma.3 Setting λ =, we have φ 1 xxx + (b(xφ 1 x = λw1 (x; λ, W 1 (x; λ = eµ 3 (λx( λ µ 3 (λ + O( λ1/ e α x. φ 1 (x; xxx + (b(xφ 1 (x; x =. (.1 Proceeding similarly, we see that φ + (x; solves precisely the same equation. We can solve (.1 exactly by integrating once, and solving the equation φ xx + b(xφ = c 1. Taking c 1 = 1, we find the three linearly independent solutions φ 1(x = ū x (x [ φ (x = ū x(x φ 3 (x = cosh x. sinh x cosh3 x + 3 sinh x cosh x + 3 x ] It follows that φ 1 (x; and φ+ (x; must both be linear combinations of φ and φ 3, where without loss of generality we can choose to remove a constant multiple of the exponentially decaying function ū x without 19

20 affecting the estimates of Lemma.1. More precisely, we obtain a scaling that agrees with that of Lemma.1 by choosing φ 1 (x; = (φ (x + φ 3 (x φ + (x; = φ (x φ 3(x. Computing directly, we find γ =. This concludes the proof of Lemma.6. We conclude this section by combining the estimates of Lemmas.1.3 to obtain estimates on G λ (x, y for values of λ sufficiently small. We have the following lemma. Lemma.7. Under the assumptions of Theorem 1.1, and for G λ (x, y as defined in (1.18, we have the following estimates. For G λ (x, y = G λ (x, y + E λ (x, y, and λ < r, for some suitably small constant r, there holds (i For y x (ii For x y (iii For y x G λ (x, y = c 1 λ 1/ (e µ (λx e µ 3 (λx e µ (λy + O( λ 1/ e µ 3 (λ(x+y + O(1e µ (λ(xy µ y Gλ (x, y = c (λ 1 (e µ λ 1/ (λx e µ 3 (λx e µ (λy + O(1e µ 3 (λ(x+y + O( λ 1/ e µ (λ(xy + O( λ 1/ e η x e µ (λy + O(e η xy x Gλ (x, y = O(1(e µ (λx + e µ 3 (λx e µ (λy + O( λ 1/ e η x e µ (λy + O(e η xy xy Gλ (x, y = O( λ 1/ (e µ (λx + e µ 3 (λx e µ (λy + O(e η x e µ (λy + O(e η xy G λ (x, y = c λ 1/ e µ (λx (e µ (λy e µ 3 (λy + O( λ 1/ e µ 3 (λ(x+y + O(1e µ 3 (λ(xy y Gλ (x, y = O(1e µ (λx (e µ (λy + e µ 3 (λy + O( λ 1/ e η x e µ (λy + O(e η xy µ x Gλ (x, y = c (λ e µ λ 1/ (λx (e µ (λy e µ 3 (λy + O(1e µ 3 (λ(x+y + O( λ 1/ e µ 3 (λ(xy + O( λ 1/ e η x e µ (λy + O(e η xy xy Gλ (x, y = O( λ 1/ e µ (λx (e µ (λy + e µ 3 (λy + O( λ 1/ e η x e µ (λy + O(e η xy. G λ (x, y = O( λ 1/ e µ+ (λxµ (λy y Gλ (x, y = O(1e µ+ (λxµ (λy + O(e η x e µ (λy x Gλ (x, y = O(1e µ+ (λxµ (λy + O( λ 1/ e η x e µ (λy xy Gλ (x, y = O( λ 1/ e µ+ (λxµ (λy + O(e η x e µ (λy

21 and in all cases E λ (x, y = c E λ Remark.1. We note that the difference terms µ ūx(xe (λy (1 + O(e η y µ y E λ (x, y = c (λ E ū x (xe µ (λy (1 + O(e η y. λ (. c 1 λ 1/ (e µ (λx e µ 3 (λx, which clearly vanish at x =, arise naturally from our choice of ψ (x; λ and moreover are expected since the mass acculating near x = is recorded in the excited terms. However, since we additionally have a term O( λ 1/ e µ 3 (λ(x+y, there is no real advantage to be taken from the cancellation. In this way, our estimates do not seem quite as sharp as those of [38], obtained for a multidimensional generalization of (1.1 (1.3. We would also mention that we will point out in the proof of Lemma.7 precisely why it is that y-derivatives of E λ (x, y do not have a term that would correspond with differentiation of e ηy. Proof. We observe at the outset the relations, y N1 (y; λ = C+ (λ yw(φ 1, φ, ψ 1 (y; λ D + c(yw λ (y (λ yw(φ 1, φ, ψ (y; λ c(yw λ (y y N (y; λ = C+ 1 (λ yw(φ 1, φ, ψ 1 (y; λ + D 1 + c(yw λ (y (λ yw(φ 1, φ, ψ (y; λ. c(yw λ (y (.3 Since proofs of the four estimates in each case are similar, we provide details only for the estimates on y-derivatives. The case of y-derivatives is chosen because there is a subtle point in this case in which the estimates of Lemma. do not quite suffice, and we must use (.3 and the estimates of Lemma.3. Case (i: y x. For y x, we have y G λ (x, y = φ + 1 (x; λ yn1 (y; λ + φ+ (x; λ yn (y; λ [ = (A + C+ 1 A+ 1 C+ φ 1 (x + (B+ C+ 1 B+ 1 C+ φ (x ] + (D + C+ 1 D+ 1 C+ ψ (x y W(φ 1, φ, ψ 1 (y c(yw λ (y [ + (A + D+ 1 A+ 1 D+ φ 1 (x + (B+ D+ 1 B+ 1 D+ φ (x ] + (D 1 + C+ D+ C+ 1 ψ 1 (x y W(φ 1, φ, ψ (y. c(yw λ (y (.4 We group these into three sets of terms, beginning with the leading order G λ expression (D + C+ 1 D+ 1 C+ ψ (x yw(φ 1, φ, ψ 1 (y c(yw λ (y = (c 1 λ 1/ + O(1(e µ (λx e µ 3 (λx + O( λ 1/ e α x e µ (λy (µ (λ + O(e α y µ = c (λ 1 (e µ λ 1/ (λx e µ 3 (λx e µ (λ + O( λ 1/ e µ (λ(xy + O( λ 1/ e η x e µ (λy O(e η( x + y. We note in particular that we have made critical use here of Lemma.4; otherwise, we have employed the estimates of Lemma. in straightfoward fashion. 1

22 Next, we consider contributions to the excited term y E λ (x, y. In this case, we take [ B 1 + (λφ (x; λ C + (λ yw(φ 1, φ, ψ 1 (y; λ + D + c(yw λ (y (λ yw(φ 1, φ, ψ (y; λ ] c(yw λ (y = B 1 + (λφ (x; λ yn1 (y; λ ( ( = c 1 λ 1/ + O(1 ū x (x + O( λ 1/ e η x e µ (λy (1 + O(e η y = c 1 λ 1/ ū x (xe µ (λy (1 + O(e η y + O(e η x e µ (λx. It is of particular importance to note that the undifferentiated excited term is precisely the same thing with y-derivatives omitted. In this way, we see that y-differentiation improves the singularity of E λ (x, y as λ. For the remaining terms, we group according to (.3 in order to obtain A + (λφ 1 (x; λ yn (y; λ + A+ 1 (λφ 1 (x; λ yn1 (y; λ + B + (λφ (x; λ yn (y; λ + (D+ (λc+ 1 (λ D+ 1 (λc+ (λψ 1 (x; λ yw(φ 1, φ, ψ c(yw λ (y (.5 For the first three terms on the right hand side of (.5, we have an estimate by ( c 1 + O( λ 1/ e µ 3 (λxµ (λy (1 + O(e η x = O(1e µ 3 (λxµ (λy. For the third term on the right hand side of (.5, keeping in mind Lemma.4, we have (D + (λc+ 1 (λ D+ 1 (λc+ (λψ 1 (x; λ yw(φ 1, φ, ψ c(yw λ (y Case (ii: x y. For x y, we have = O(1e µ 1 (λ(xy. (.6 y G λ (x, y = φ 1 (x; λ yn 1 + (y; λ + φ (x; λ yn + (y; λ. (.7 Beginning again with the leading order G λ estimate, we have while for the excited and correction terms, we compute φ 1 (x; λ yn + 1 (y; λ = O(1eµ 3 (λ(xy, (.8 φ (x; λ yn + (y; λ ( = ū x (x + O( λ 1/ e η x [(k 1 λ 1/ + O(1e µ (λy (1 + O(e α y + O(1e µ(λy] 4 = k 1 λ 1/ ū x (xe µ (λy (1 + O(e α y + O(e η x e µ (λy + O(e η xy. For the excited term, we take Case (iii: y x. For y x, we have For the leading order G λ term, we take y E λ (x, y = k 1 λ 1/ ū x (xe µ (λy (1 + O(e α y. y G λ (x, y = φ + 1 (x; λ yn1 (y; λ + φ+ (x; λ yn (y; λ. (.9 φ + (x; λ yn (y; λ = O(1eµ+ (λxµ (λy, while for the excited and correction terms, we compute φ + 1 (x; λ yn1 (ū (y; λ = x (x + O( λ 1/ e η x (n 1 λ 1/ + O(1e µ (λy (1 + O(e α y.