PULSES IN A GIERER-MEINHARDT EQUATION WITH A SLOW NONLINEARITY FRITS VEERMAN, ARJEN DOELMAN Abstract. In this paper, we study in detail the existence and stability o localized pulses in a Gierer- Meinhardt equation with an additional slow nonlinearity. This system is an explicit example o a general class o singularly perturbed, two component reaction-diusion equations that goes signiicantly beyond wellstudied model systems such as Gray-Scott and Gierer-Meinhardt. We investigate the existence o these pulses using the methods o geometric singular pertubation theory. The additional nonlinearity has a proound impact both on the stability analysis o the pulse compared to Gray-Scott/Gierer-Meinhardt type models a distinct extension o the Evans unction approach has to be developed and on the stability properties o the pulse: several (de)stabilization mechanisms turn out to be possible. Moreover, it is shown by numerical simulations that, unlike the Gray-Scott/Gierer-Meinhardt type models, the pulse solutions o the model exhibit a rich and complex behaviour near the Hop biurcations. 1. Introduction. The study o localized pulses in a two-component system o singularly perturbed reaction-diusion equations has been a very active ield o research since the nineties o the previous century. In its most general orm, a system that may exhibit such a pulse reads in one, unbounded, spatial dimension { Ut = U xx + F (U, V ) V t = ε 2 V xx + G(U, V ) (1.1a) (1.1b) with U, V : R R + R, and 0 < ε 1 asymptotically small. The nonlinear reaction terms F, G : R 2 R are assumed to satisy F (Ū, V ) = G(Ū, V ) = 0 or certain (Ū, V ), such that the trivial background state (U, V ) (Ū, V ) is spectrally stable. However, research on pulses in equations o the type (1.1) has been mostly restricted to model equations. In particular two o these models have played a central role in the development o the theory: the (irreversible) Gray-Scott (GS) equation or a class o autocatalytic reactions [18] that became the center o research attention by the intriguing observations in [26, 29] and the Gierer-Meinhardt (GM) equation [17] or (biological) morphogeneses or which the existence problem has already been considered in the mathematical literature or a somewhat longer time [32]. Both or the GS and the GM model, quite precise insight has been obtained in the existence, stability and dynamics o localized (multi-) pulses, also in more than one spatial dimension although one certainly cannot claim that the models are ully understood; see [5, 6, 7, 12, 10, 19, 22, 23, 24, 27, 28, 33, 35] and the reerences therein or the literature on one spatial dimension. The ast V -component o a localized (multi-)pulse solution o a singularly perturbed model (1.1) is asymptotically localized: it decays exponentially to the V -component V o the background state on a spatial scale that is asymptotically shorter than the spatial scale associated to the slow U-component. As a consequence, the two-component (U, V )-low generated by (1.1) is governed by a scalar equation in the slow component U: U t = U xx + F (U, V ) (1.2) except or the asymptotically small spatial regions in which the V -component is not exponentially close to V. Clearly, this is in general a nonlinear equation. However, or the GS and GM models, Universiteit Leiden, Mathematical Insitute, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands 1
this slow reduced scalar equation is linear: (GS) U t = U xx + A(1 U), A > 0 parameter, (Ū, V ) = (1, 0) (GM) U t = U xx αu, α > 0 parameter, (Ū, V ) = (0, 0) (1.3) In act, as ar as we are aware, this the act that the counterpart o (1.2) is linear is the case or all singularly perturbed two-component reaction-diusion equations with exponentially localized pulse solutions considered in the literature (including the Schnakenberg model [34, 31]). There is a number o papers in the literature in which more general classes o equations than the GS or GM models are considered see [4, 6, 11]. In these papers the background state (Ū, V ) is translated to (0, 0) so that F (Ū, V ) = F (0, 0) = 0 in (1.1). Moreover, the nonlinear part o F (U, V ) is assumed to be separable, i.e. F (U, V ) is written as αu + F 1 (U)F 2 (V ). Thereore F 2 ( V ) = F 2 (0) = 0, and these more general systems also reduce to linear slow scalar equations like (1.3) outside the asymptotically small regions where V is not close to V. In this paper, and in its companion paper [14], we consider the potential impact o the nonlinearity o F (U, V ) as unction o U in comparison with the literature on slowly linear model systems such as GS and GM. Here, we consider a very explicit model problem, a Gierer-Meinhardt equation with a slow nonlinearity (see (1.7) below), in ull analytical detail; in [14] we consider the existence and stability o pulses in a general setting, i.e. as solutions o (1.1). We reer to Remark 1.1 or a more speciic motivation o our choice to study equations with slow nonlinearities. In the standard orm (1.1), the classical Gierer-Meinhardt equation [17] is given by U t = U xx α U + σ V 2 V t = ε 2 V xx V + V 2 U (1.4a) (1.4b) in which α > 0 is the main biurcation parameter and σ > 0 is most oten scaled to 1. The pulse type solutions o (1.4) have an amplitude o O ( ) 1 ε [6, 19]. Thereore, we scale U and V and subsequently x and ε, U U ε, V V ε, x ε x, ε ε 2 (1.5) to bring (1.4) in its normal orm [6] ε 2 U t = U xx ε 2 α U + σ V 2 V t = ε 2 V xx V + V 2 U (1.6a) (1.6b) In this paper, we study a slowly nonlinearized version o (1.6), that is obtained rom (1.6) by adding a very simple nonlinear term to its slow U-equation (1.6a): ε 2 U t = U xx ε 2 ( α U γ U d) + σ V 2 (1.7a) V t = ε 2 V xx V + V 2 U (1.7b) with new parameters γ 0, d > 1. Moreover, we now allow σ R\{0}. Systems incorporating such a slow nonlinearity were already encountered in [25] (although no pulse type solutions 2
U h V h x U h V h x (a) (b) Fig. 1.1: The stationary homoclinic pulse (U h (x), V h (x)) o (1.7); (a) σ > 0, (b) σ < 0. were considered in this paper). This equation indeed reduces to a nonlinear slow reduced scalar U-equation away rom the regions in which V is not exponentially close to V = 0: U t = U χχ α U + γ U d (1.8) in which χ = εx is a superslow spatial coordinate see section 2. Note that scaling back the additional slowly nonlinear term γ U d through (1.5) introduces an O ( ε d 1), i.e an asymptotically small, additional term to the Gierer-Meinhardt equation in its classical orm (1.4). We will see in the upcoming analysis that this term has a signiicant impact on the dynamics generated by (1.4). Thus, in a way, our work can also be interpreted as a study o the vulnerability o the classical Gierer-Meinhardt model (1.4) to asymptotically small slowly nonlinear changes to the model. In recent years, the analysis o localized pulses in one-dimensional singularly perturbed reaction-diusion equations has been mostly ocused on pulse dynamics and interactions see [5, 11, 10, 24] and the reerences therein. However like the work on multi-pulse patterns [9, 19, 22, 23, 33, 34] this analysis is based on undamental insights on the existence and stability o stationary, solitary, pulses [6, 7, 12, 32, 35]. On the unbounded domain, i.e. or x R, these pulses correspond to homoclinic solutions o the 4-dimensional spatial dynamical system reduction o the partial dierential equation. Here, we restrict our analysis to the existence and stability o homoclinic stationary pulse solutions (U h (x), V h (x)) to (1.7) that are bi-asymptotic to the background state (0, 0), i.e. lim x ± (U h (x), V h (x)) = (0, 0). Especially the issue o stability requires a signiicant extension o the methods developed in the literature or slowly linear GS/GM-type models. The present results orm the oundation or a subsequent analysis o the multi-pulse patterns see remark 1.1 and pulse interactions. Moreover, already at the level o these most basic pulse solutions, we encounter novel phenomena in the dynamics generated by (1.7) that have not yet been observed in the literature on slowly linear models. The existence problem see section 2 can be studied directly along the lines developed in [6] or slowly linear normal orm models o GM type with a separable nonlinearity. Our main result on the existence o homoclinic pulses (U h (x), V h (x)), Theorem 2.1, can be established by a direct application o the methods o geometric singular perturbation theory [15, 16]. In other words, at the existence level the slow nonlinearity in (1.7) does not require the development o novel theory. However, it is established by Theorem 2.1 that (1.7) does exhibit homoclinic pulse 3
Im Λ 2 Im Λ 15 Im Λ 2 1 10 5 1 3 2 1 1 Re Λ 10 5 5 10 15 20 Re Λ 3 2 1 1 Re Λ 1 5 10 1 (a) 2 15 (b) (c) 2 Fig. 1.2: The orbits through C o the critical eigenvalue λ associated to the spectral stability o (U h (x), V h (x)) as unction o increasing α (γ = 2, σ = 1), to leading order in ε. (a) d = 2 < 3: The same scenario as in the GS and the GM models [6, 7]. Two real positive eigenvalues merge and become a pair o complex conjugate eigenvalues that travels through the imaginary axis: the pulse is stabilized by a Hop biurcation at a critical value o α. (b) d = 5 > 3: A signiicantly dierent scenario. The eigenvalues initially display the same behavior as in the case d < 3: the pulse is again stabilized by a Hop biurcation. However, or α increasing urther, the orbits sharply turn around and ollow the imaginary axis closely in the negative direction see (c), a zoom o (b). Eventually, the orbits branch o, head back to the imaginary axis, and again cross the imaginary axis at a second critical Hop value o α. Finally, the pair meets again at the positive real axis and splits up in 2 positive real valued eigenvalues. patterns that dier signiicantly rom those ound in slowly linear GS/GM-type models. Unlike linear slow reductions as (1.3), the planar stationary problem associated to reduction (1.8) has orbits homoclinic to its saddle point (that corresponds to the background state o (1.7)). As a consequence, unlike the classical GM model (1.4), system (1.7) has homoclinic pulse solutions (U h (x), V h (x)) or σ < 0. At leading order in ε, the slow U-component U h (x) ollows a large part o the homoclinic orbit o (1.8), so that or σ < 0 the slow component o the solitary homoclinic 1-pulse solution has the leading order structure o two combined slow scalar pulses see Figure 1.1b. The spectral stability o (U h (x), V h (x)) is studied in section 3 by the Evans unction D(λ) associated to the linearized stability problem, ollowing the ideas developed in [6, 7]. As is to be expected rom the general theory [3], D(λ) can be decomposed into a slow and a ast component, and all nontrivial eigenvalues are determined by the slow component. In [6, 7], i.e. or the GM and GS models, the zeroes o this slow component are determined analytically by the NLEP method. The linearity o the slow scalar reduction (1.3) plays a central role in this approach as it does in all analytical studies o the spectral stability o pulses in GS/GM-type models (see [5, 19, 22, 23, 24, 33, 35] and the reerences therein). More explicitly, the act that the spectral stability problem is exponentially close to a constant coeicients eigenvalue problem outside the asymptotically small regions in which V is not close to V is a crucial ingredient o the stability analysis o GS/GM-type models. Due to the nonlinearity in the slow scalar reduction (1.8) this is not the case or (1.7): away rom the ast V -pulse, the linear operator associated to stability problem still has coeicients that depend explicitly (and slowly) on x (on χ see (1.8)). Its solution space is thereore not governed by simple, pure exponentials (as or GS/GM-type models). 4
The key to the NLEP approach as developed in [6, 7] is constructing a set o basis unctions or the linear operator/system associated to the stability o the pulse or which the Evans unction D(λ) the determinant o this set can be evaluated, or better: approximated, explicitly. In this paper, and its companion paper [14], we show that the NLEP approach can be based on a set o basis unctions that is determined by the slowly varying problem outside the ast V -pulse region, in such a way that it is still possible to determine an analytical approximation or the zeroes o D(λ). Here, a central role is played by the χ-dependent Sturm-Liouville problem associated to the linearization o (1.8) about its (stationary) homoclinic orbit, deined on a hal-line. This problem has a two-dimensional set o slowly varying solutions. We show that these solutions can take over the role o the slow exponentials coming rom the (slow) stability problem about the trivial state U = 0 o the linear constant coeicient GS/GM-type reductions (1.3). In the present paper, these solutions can be expressed in terms o Legendre unctions, due to the special/simple nature o the nonlinearity in (1.8). In the general setting o [14] the construction o the Evans unctions cannot be this explicit. The main novel analytical result o this paper is given by Theorem 3.10, in which indeed an explicit expression is given or the zeroes o D(λ), that is a generalization o the corresponding slowly linear results in [6, 7]. In section 4, we analyze and interpret the expression obtained in Theorem 3.10. One o our irst and quite straightorward results is Corollary 4.3: the σ < 0 double hump pulses o Figure 1.1b cannot be stable. The σ > 0 pulses o Figure 1.1a however can very well be stable. In Figure 1.2, a graphical description is given o our two main stability results, Theorems 4.6 and 4.7. The stability o the pulse (U h (x), V h (x)) depends strongly on the character o the slow nonlinearity in (1.7). As long as the exponent o the nonlinearity d is smaller than 3, the stability scenario is exactly like that o the slowly linear GS/GM-type models: (U h (x), V h (x)) stabilizes by a Hop biurcation or increasing α even the shape o the orbit o the critical eigenvalues λ(α) through C is very similar to its counterparts in [6, 7]. However, this orbit changes drastically when d becomes larger than 3 see Figure 1.2: or d > 3 there is a second Hop biurcation (as unction o α) that destabilizes (U h (x), V h (x)). Dierent rom the results on GS/GM-type models, or d > 3, there only is a bounded α-region or which (U h (x), V h (x)) can be stable. In Theorem 4.8 this is established rigorously or d > 3 large enough. Finally, in section 5, we present some simulations o (1.7). We have not attempted to perorm a systematic (numerical biurcation) analysis o the dynamics o (1.7). Apart rom checking (and conirming) the outcome o our asymptotic stability analysis, our goal has been to obtain an indication o whether or not the slow nonlinearity o (1.7) generates behavior that is not known rom the (vast) literature on GS/GM-type models. We are not aware o any examples in the literature on GS/GM-type models o stable nonmoving solitary pulses that are not completely stationary. A priori, one would expect that i a pulse is destabilized (or instance by decreasing α in the GM model (1.4)), it may biurcate into a stable standing pulse with a periodically varying amplitude. However, this requires a supercritical Hop biurcation, and all Hop biurcations o stationary pulses in GS/GM-type models reported on in the literature seem to be subcritical: as α decreases through its critical Hop biurcation value, the standing pulse starts to oscillate up and down, but the amplitude o this oscillation grows and ater a certain time the pulse is extinguished see or instance Figure 5.3 (a) in section 5. It should be noted that this statement is based on numerical observations, the nature o the Hop biurcation o solitary, standing pulses in GS/GM-type models has not been analyzed in the literature (or instance by a center maniold reduction). Moreover, it should also 5
2.7 2.7 2.65 2.65 amplitude 2.6 2.55 amplitude 2.6 2.55 2.5 2.5 2.45 0 10 20 30 40 50 60 70 80 90 t 2.45 30 t 40 (a) (b) Fig. 1.3: The dynamics o the maximum o the U-pulse as unction o time in a simulation o (1.7) with γ = 2, σ = 1, ε = 0.002, d = 5 and α = 90.6 or x [ 5000, 5000] with homogeneous Neumann boundary conditions (b) zooms in on a small part (in time) o (a). The position o the maximum does not vary in time. The value o α is close to the second Hop biurcation at which (U h (x), V h (x)) destabilizes see Figure 1.2. be remarked that or instance the GS model does exhibit periodic and even chaotic pulse dynamics see or instance [5, 28]. However, this richer type o behavior only occurs in the context o pulse interactions, it is governed by the interactions between traveling pulses, and/or between pulses and the boundary o the domain. We have not considered this type o dynamics here, as we have completely ocused on the behavior o standing, solitary spatially homoclinic pulses. Nevertheless, we have observed very rich dynamics, much richer than exhibited by linear GS/GM-type models. In section 5 examples are given o periodically oscillating pulses, i.e. standing pulses with an amplitude that varies periodically in time; quasi-periodically oscillating pulses the amplitude o the pulse oscillation is modulated periodically and oscillating pulses o which the amplitude is modulated in an even more complex ashion. A simulation o such a chaotically oscillating pulse is shown in Figure 1.3. It has not been investigated whether the pulse dynamics o Figure 1.3 is chaotic or or instance is quasi-periodic with three or more independent requencies. In other words, we have not studied the details o the associated biurcation scenario and have not computed any measure by which the (possible) chaotic nature o the pulse dynamics can be quantiied. The analytic core o this paper, the analysis o the spectrum associated to the stability o (U h (x), V h (x)) can serve as an ideal starting point or a center maniold analysis o the nature o the Hop (and subsequent) biurcations or pulses and/or multi-pulses occurring in this model (and/or generalizations o (1.7)). This is expected to give analytic insight in (the possible route leading to) the complex/chaotic behavior observed in Figure 1.3. This will be the subject o uture research. Remark 1.1. Our research is strongly motivated by recent indings on the character o the destabilization o spatially periodic multi-pulse patterns with long wavelength L. In [9] it is established or GM-type models that these patterns can only be destabilized by two distinct types o Hop biurcations as L, one in which the linearly growing mode also has wavelength 6
L the most commonly encountered destabilization in the literature and another in which this mode has wavelength 2L. Moreover, these destabilizations alternate countably many times as L. This is called the Hop dance in [9]. This Hop dance also occurs in the GS model, as indicated by the AUTO-simulations in [9]. The GM analysis in [9] shows that this dance is completely driven by the exponential expression E(L) = e L α+λ h associated to the slow reduced eigenvalue equation u xx αu = λ h u originating rom (1.3), in which λ h C is the (complex) eigenvalue o the homoclinic (L ) limit pattern. The rotation o E(L) C as L is the mechanism underpinning the Hop dance. From a generic point o view, it is not at all clear why this linear Hop dance should take place (this is even more obvious or the subsequent belly dance [9]). Hence, to really understand the subtleties involved in the destabilization o long wavelength spatially periodic patterns, one needs to go beyond slowly linear models or which the associated slow reduced eigenvalue problems are not governed by expressions as E(L). In other words, one needs to study systems o the type (1.1) with F (U, V ) not linear as unction o U. 2. Pulse construction. Our goal is to construct a stationary pulse solution which is homoclinic to the trivial background state (U, V ) = (0, 0). To achieve this goal, we use the singularly perturbed nature o the system. The spatial dynamics o the stationary pulse are given by the our-dimensional system u x = p p x = σ v 2 + ε 2 ( α u γ u d) ε v x = q ε q x = v v2 u (2.1a) (2.1b) (2.1c) (2.1d) Along the lines o Fenichel theory, we can perorm a slow-ast decomposition in the spatial variable x: recognizing system (2.1) as the slow system, we can deine the ast variable ξ = x ε to obtain the associated ast system u ξ = ε p p ξ = ε σ v 2 + ε 3 ( α u γ u d) v ξ = q q ξ = v v2 u (2.2a) (2.2b) (2.2c) (2.2d) The trivial background state is in these systems represented by the origin (u, p, v, q) = (0, 0, 0, 0). While the vector ield which generates the low o the system is not deined at the origin due to the singular v2 v2 u term in the v-equation, the ratio u will be well-deined or the constructed pulse. 2.1. Geometric analysis. When ε 0, the slow and ast systems (2.1) and (2.2) reduce to the reduced slow system u xx = σ v 2 q = v v2 u = 0 7 (2.3a) (2.3b)
and the reduced ast system u ξ = p ξ = 0 (2.4a) v ξξ = v v2 u (2.4b) We see that in this limit, the slow and ast dynamics decouple completely. We deine M 0 = {(u, p, v, q) u > 0, v = q = 0} as the two-dimensional normally hyperbolic invariant maniold that consists o hyperbolic equilibria o the reduced ast system (2.4); it has three-dimensional stable and unstable maniolds W s,u (M 0 ) which are the unions o the 2-parameter amilies o one-dimensional stable and unstable maniolds (ibers) at the saddle points (u 0, p 0, 0, 0) M 0. The reduced ast dynamics (2.4) allow a 2-parameter amily o homoclinic solutions v 0,h : v h,0 (ξ; u 0, p 0 ) = 3 u 0 2 sech2 ( 1 2 ξ) (2.5) The union over this amily as a bundle over M 0 orms the intersection W s (M 0 ) W u (M 0 ), see igure 2.1a. Fenichel persistence theory ([15], [16], [20], [21]) states that, or ε suiciently small, the ull system (2.2) has a locally invariant slow maniold M ε which is O(ε) close to M 0. Since M 0 is also invariant under the non-reduced (ast) low o (2.2), we have already ound M ε = M 0. Moreover, Fenichel theory states the existence o three-dimensional stable and unstable maniolds W s,u (M ε ) which are O(ε) close to their unperturbed counterparts W s,u (M 0 ). The intersection W s (M ε ) W u (M ε ) exists, is transversal and thereore determines a two-dimensional maniold. This existence and transversality is based on a Melnikov-type calculation in [6], which can be applied directly to system (2.2). Since the original model equations (1.7) are invariant under relection in the spatial variable x x, this relection is in the our-dimensional system (2.1) equivalent to the momentum relection (p, q) ( p, q). Because the coordinate relection ξ ξ maps W s (M ε ) to W u (M ε ) and vice versa, it ollows that the intersection o these two maniolds is symmetric in the invariant subset o the momentum relection, the two-dimensional hyperplane {(u, p, v, q) p = q = 0}. The transversality o this hyperplane to M ε excludes the possibility that it has the intersection W s (M ε ) W u (M ε ) as a subset, rom which we can conclude that W s (M ε ) W u (M ε ) intersects the hyperplane {(u, p, v, q) p = q = 0} transversally. This determines a 1-parameter amily o orbits bi-asymptotic to M ε. Since both W s (M ε ) and W u (M ε ) are O(ε) close to W s,u (M 0 ) where the 2-parameter amily o homoclinic orbits was parametrized by u 0 and p 0 (see (2.5)), it is convenient to use u 0 to parametrize the 1-parameter amily o orbits bi-asymptotic to M ε determined by W s (M ε ) W u (M ε ). For a sketch o the situation, see igure 2.1b. The next step is to use this structure to construct an orbit homoclinic to (0, 0, 0, 0) in the ull, perturbed system (2.1) / (2.2). For that purpose, it is necessary to consider the dynamics on M ε. The low on M ε can be determined by substituting v = q = 0 in (2.1) and yields Introducing a superslow coordinate χ = εx, this can be written as u xx = ε 2 ( α u γ u d) (2.6) u χχ = α u γ u d (2.7) This equation allows a solution (bi)asymptotic to the trivial background state: since γ > 0, it is 8
M 0 M Ε W u M Ε W s M Ε q q Γ h v v (a) (b) Fig. 2.1: Transversal intersection o the stable and unstable maniolds. (a): The amily o homoclinic orbits v h,0 (ξ; u 0, p 0 ), viewed as a bundle over M 0. Both u- and p-directions are along the vertical axis; M 0 is indicated in blue. (b): For the perturbed system (ε > 0), W s (M ε ) and W u (M ε )intersect transversally: γ h (indicated in red) represents W s (M ε ) W u (M ε ), a oneparameter amily o orbits homoclinic to M ε recall that dim (M ε ) = 2, dim (W s,u (M ε )) = 3 so dim (γ h ) = 2. homoclinic to (0, 0, 0, 0) M ε and explicitly given by [ α(d + 1) u h,0 (χ) = 2γ sech 2 ( 1 2 (d 1) αχ )] 1 d 1 (2.8) The superslow dynamics on M ε allows us to get a grip on picking exactly that orbit bi-asymptotic to M ε rom the intersection W s (M ε ) W u (M ε ) which is also homoclinic to (0, 0, 0, 0) M ε, that is, which is -mostly- asymptotically close to u h,0 M ε. This orbit will make a ast excursion through the V -ield, since this is where the ast dynamics take place (see (2.2), (2.4)). Since our goal is to construct a symmetric pulse, we can choose an interval symmetric around the origin in which the ast jump occurs. The interval needs to be asymptotically small with respect to the slow variable x, but asymptotically large with respect to the ast variable ξ: to be asymptotically close to M ε, the V -component o the pulse needs to be exponentially small. A standard ([6]) 9
v p,q u h,0 v h,0 M Ε u u Fig. 2.2: An asymptotic construction o the orbit γ h (ξ) o Theorem 2.1 drawn in three dimensions. The p- and q-directions are combined, since there is no direct interaction between them. The blue surace represents the persistent slow maniold M ε while the ast dynamics take place on the red surace, which is spanned by the v and q directions. The slow homoclinic orbit u h,0 (χ) is drawn in blue, the ast homoclinic orbit v h,0 (ξ; u, 0) is drawn in red. The jump through the ast ield projected on M ε is indicated by the purple line. choice or this ast spatial region is I = { ξ R ξ < 1 } ε (2.9) Indeed, x 1 and ξ 1 on I. For a sketch o the orbit, see igure 2.2. Now, we deine the take-o and touchdown sets T o,d M ε to be the collection o base points o all Fenichel ibers in W u (M ε ) resp. W s (M ε ) that have points in the transverse intersection W s (M ε ) W u (M ε ). Detailed inormation on T o,d can be obtained by studying the ast system (2.2) on M ε. First, we observe that p ξ = O(ε 3 ) on M ε so the p-coordinate on M ε remains constant to leading order during the ast excursion through the V -ield. Thereore, the change in the p-coordinate o the pulse is completely determined by its accumulated change during its 10
p p u h,0 u h,0 T o T d u * u u * u T d To (a) For σ > 0 the jump is downwards, the resulting pulse is shown in igure 1.1a. (b) For σ < 0 the jump is upwards, the resulting pulse is shown in igure 1.1b. Fig. 2.3: The homoclinic orbit u h,0 (χ) is drawn in blue in the (u, p)-plane. The take-o and touchdown curves T o = { (u, p) p = 3 ε σu 2 } and T d = { (u, p) p = 3 ε σu 2 } are drawn in green. The jump through the ast ield at u = u is indicated by the dashed purple line. excursion through the ast ield, and is given by ξ p = p ξ dξ = I ε σ v 2 + O(ε 3 ) dξ = I ε σ v h,0 (ξ; u 0, p 0 ) 2 dξ + O(ε 2 ) = 6 ε σ u 2 0 + O(ε 2 ) (2.10) where we have used (2.2) and (2.5). Moreover, since u ξ = ε p and p = O(ε) on I, we see that ξ u = O(ε 2 ). This means that during the jump through the ast ield, the u-coordinate o the pulse doesn t change to leading order. Since W s (M ε ) W u (M ε ) intersects the hyperplane {(u, p, v, q) p = q = 0} transversally, we can deine the take-o and touchdown sets as curves T o = { (u, p, 0, 0) M ε p = 3 ε σu 2 } and T d = { (u, p, 0, 0) M ε p = 3 ε σu 2 } (2.11) at leading order. Note that i σ changes sign, the take-o and touchdown curves are interchanged: or σ > 0 the take-o curve has positive p-values, while or σ < 0 the take-o curve has negative p-values. This also means the direction o the ast jump is reversed when σ changes sign, see (2.10) and igure 2.3a. An orbit o the system (2.1) / (2.2) is homoclinic to (0, 0, 0, 0) i its Fenichel iber basepoints in T o,d intersect the superslow homoclinic orbit u h,0 M ε, see igure 2.3a. This intersection can be determined by integrating (2.6) once, ( 1 2 p2 = ε 2 1 2 α u2 γ ) d + 1 ud+1 (2.12) and substituting p = ±3 ε σu 2 rom (2.11) to obtain 2γ d + 1 ud 1 = α 9 σ 2 u 2 (2.13) 11
which or α, γ, σ > 0 and d > 1 always has a unique real positive solution, denoted by u. Furthermore, we deine χ as the (unique) positive χ-value or which u h,0 (χ ) = u, the u-coordinate o the intersection. When σ < 0, we obtain a slightly dierent pulse since part o the slow homoclinic orbit u h,0 is covered twice, see igure 2.3b. This has its consequences or the ormulation o our main existence result: Theorem 2.1. Let ε > 0 be suiciently small. Then, or all values o the parameters α > 0, γ > 0, σ > 0 and d > 1, there exists a unique orbit γ h (ξ) = (u h (ξ), p h (ξ), v h (ξ), q h (ξ)) as a solution o system (2.2) which is homoclinic to (0, 0, 0, 0) and lies in the intersection W s (M ε ) W u (M ε ). Moreover, or all ξ R and v h (ξ) v h,0 (ξ; u, 0) = O(ε), q h (ξ) d dξ v h,0(ξ; u, 0) = O(ε) (2.14) u h (χ) u h,0 (χ sgn(σ) χ ) = O(ε), p h (χ) ε d dχ u h,0(χ sgn(σ) χ ) = O(ε) (2.15) or all χ < 0, while u h (χ) u h,0 (χ + sgn(σ) χ ) = O(ε), p h (χ) ε d dχ u h,0(χ + sgn(σ) χ ) = O(ε) (2.16) or all χ > 0. The orbit γ h corresponds to a homoclinic pulse solution (U h, V h ) o system (1.7). Proo. The missing details in the above geometric construction, especially in the preciese estimates o (2.14), (2.15) and (2.16), can be obtained in a manner identical to the corresponding result on slowly linear systems in [6]. 3. Pulse stability: analysis. The linear stability o the stationary pulse solution (U h, V h ) o (1.7) ound in the previous section is determined by adding a perturbation o the orm (ū(x), v(x)) e λt and linearizing equation (1.7) around the stationary solution, obtaining in the ast variable ξ, ū ξ = ε p p ξ = 2 ε σ V h (ξ) v + ε 3 ( α + λ γ d U h (ξ) d 1) ū v ξ = q ( q ξ = 1 + λ 2 V ) h(ξ) v + V h(ξ) 2 U h (ξ) U h (ξ) 2 ū We write the ast system (3.1) in vector orm where φ(ξ) = ((ū(ξ), p(ξ), v(ξ), q(ξ)) T and (3.1a) (3.1b) (3.1c) (3.1d) d dξ φ = A(ξ; λ, ε)φ (3.2) 0 ε 0 0 ε ( A(ξ; λ, ε) = 3 α + λ γ d U h (ξ) d 1) 0 2 ε σ V h (ξ) 0 0 0 0 1 (3.3) V h (ξ) 2 U h (ξ) 0 1 + λ 2 V h(ξ) 2 U h (ξ) 0 12
Since the V -component o the stationary pulse decays much aster than its U-component, the ratio V h U h is well-deined and converges to zero as ξ ±. This results in the constant coeicient matrix 0 ε 0 0 A (λ, ε) = lim A(ξ; λ, ε) = ε 3 (α + λ) 0 0 0 ξ 0 0 0 1 (3.4) 0 0 1 + λ 0 which has eigenvalues and associated eigenvectors E,± = ± Λ = ± 1 + λ and ± ε 2 Λ s = ±ε 2 α + λ (3.5) ( 0, 0, 1, ± T ( 1 + λ) and E s,± = 1, ±ε T α + λ, 0, 0) (3.6) The essential spectrum o the linear eigenvalue problem (3.1) thereore is σ ess = {λ R λ max( α, 1)}, (3.7) see [30]. Since α > 0, we can conclude that the stability o the pulse (U h, V h ) is determined by its discrete spectrum. 3.1. The Evans unction and its decomposition. The Evans unction, which is complex analytic outside the essential spectrum see [30], [3] and the reerences therein associated to system (3.1) can be deined by D(λ, ε) = det [φ i (ξ; λ, ε)] (3.8) where the unctions φ i, i = 1, 2, 3, 4 satisy boundary conditions at ± (see below) and span the solution space o (3.1). The eigenvalues o (3.2) outside σ ess coincide with the roots o D(λ, ε), including multiplicities. Deinition 3.1. A statement o the orm (x) c g(x) as x is true whenever the 1 limit lim x g(x) (x) = c exists and is well-deined. Lemma 3.2. For all λ C \ σ ess, there are solutions φ,l/r (ξ; λ, ε) and φ s,l/r (ξ; λ, ε) to (3.1) such that the set { φ,l/r (ξ; λ, ε), φ s,l/r (ξ; λ, ε) } spans the solution space o (3.1) and φ,l (ξ; λ, ε) E,+ e Λ ξ φ,r (ξ; λ, ε) E, e Λ ξ φ s,l (ξ; λ, ε) E s,+ e ε2 Λ sξ φ s,r (ξ; λ, ε) E s, e ε2 Λ sξ as ξ (3.9a) as ξ (3.9b) as ξ (3.9c) as ξ (3.9d) Moreover, there exist analytic transmission unctions t,+ (λ, ε) and t s,+ (λ, ε) such that φ,l (ξ; λ, ε) t,+ (λ, ε) E,+ e Λ ξ φ s,l (ξ; λ, ε) t s,+ (λ, ε) E s,+ e ε2 Λ sξ as ξ (3.10a) as ξ (3.10b) 13
where t s,+ (λ, ε) is only deined i t,+ (λ, ε) 0. These choices, when possible, determine φ,l/r and φ s,l uniquely. Proo. Although the linearized system 3.2 is not identical to its counterpart in [6], exactly the same arguments as in [6] can be applied here. Thereore, we reer to [6] or the details o the proo. The Evans unction can be determined by taking the limit ξ o the determinant o the unctions deined in Lemma 3.2, since the Evans unction itsel does not depend on ξ since the trace o A(ξ; λ, ε) vanishes (Abel s theorem). This yields (see [6]) D(λ, ε) = 4ε t,+ (λ, ε) t s,+ (λ, ε) 1 + λ α + λ (3.11) Corollary 3.3. The set o eigenvalues o (3.2) is contained in the union o the sets o roots o t,+ (λ, ε) and t s,+ (λ, ε). Note that, due to the act that t s,+ (λ, ε) only deined when t,+ (λ, ε) 0, the Evans unction D(λ, ε) doesn t necessarily vanish when t,+ (λ, ε) = 0. This is called the resolution to the NLEP paradox in [6] and [7]. The roots o t,+ will be discussed later, in section 3.3. 3.2. The slow solution φ s,l outside I. To obtain more inormation about the roots o t s,+ (λ, ε), it is necessary to determine the leading order behaviour o φ s,l (ξ; λ, ε) in the dierent coordinate regimes. From Lemma 3.2 we know that φ s,l is slowly growing in ξ, since its leading order behaviour or both ξ ± is determined by the exponential growth actor ε 2 Λ s = O(ε 2 ). However, the dynamics governing φ s,l dier signiicantly inside and outside the ast spatial region I. Based on our knowledge o the homoclinic solution stated in Theorem 2.1, we can iner the orm o the matrix A(ξ; λ, ε) both inside and outside I : 0 ε 0 0 ε ( ) A (ξ; λ, ε) = 3 α + λ γ d u d 1 0 2 ε σ v h,0 (ξ; u, 0) 0 0 0 0 1 (3.12) 0 1 + λ 2 v h,0(ξ;u,0) u 0 to leading order or ξ I and v h,0 (ξ;u,0) 2 u 2 0 ε 0 0 A s (ξ; λ, ε) = ε ( 3 α + λ γ d u h,0 ( ε 2 ξ + sgn(σ) χ ) d 1) 0 0 0 0 0 0 1 (3.13) 0 0 1 + λ 0 or ξ / I to leading order. Note that it is the act that this intermediate slow matrix exists, or better: that it is not identical to A (λ, ε) (3.4), that distinguishes the slowly nonlinear Gierer-Meinhardt problem rom slowly linear problems as the classical Gierer-Meinhardt or Gray-Scott systems. Note also that a intermediate matrix as A s (ξ; λ, ε) was already encountered in [8], in the study o a system with non-exponential (algebraic) decay. 14
Lemma 3.4. Consider the system d dξ ψ = A s(ξ; λ, ε)ψ (3.14) with A s (ξ; λ, ε) as given in (3.13). There exist solutions ψ,± (ξ; λ, ε) and ψ s,± (ξ; λ, ε) which span the solution space o (3.14) or ξ < 1 ε and as ξ. ψ,+ (ξ; λ, ε) E,+ e Λ ξ, ψ s,+ (ξ; λ, ε) E s,+ e ε2 Λ sξ, (3.15a) ψ, (ξ; λ, ε) E, e Λ ξ, ψ s, (ξ; λ, ε) E s, e ε2 Λ sξ (3.15b) Proo. The same arguments as in the proo o Lemma 3.2 apply, since lim ξ A s (ξ; λ, ε) = A (λ, ε). Since A(ξ; λ, ε) is to leading order equal to A s (ξ; λ, ε) or ξ < 1 ε and both φ s,l and ψ s,+ E s,+ e ε2 Λ sξ as ξ, combining Lemma 3.2 and Lemma 3.4 yields the ollowing Corollary: Corollary 3.5. For ξ < 1 ε, we can write to leading order. φ s,l (ξ; λ, ε) = ψ s,+ (ξ; λ, ε) The slow evolution o the ū-component o ψ s,± can be written, using again χ = ε 2 ξ, as ū χχ ( α + λ γ d u h ( χ + sgn(σ) χ ) d 1) ū = 0 (3.16) We can introduce the coordinate transormation z = 1 d dχ u h,0(χ sgn(σ) χ ) α u h,0 (χ sgn(σ) χ ) = 1 d α dχ log 1 u h,0 (χ sgn(σ) χ ) = tanh ( 1 2 (d 1) α (χ sgn(σ) χ ) ) (3.17) (by (2.8)) or the region χ < 0 to obtain ) (1 z 2 ) ū zz 2 z ū z + (ν(ν + 1) µ2 1 z 2 ū = 0 (3.18) where ν = d + 1 d 1 µ = + 2 d 1 1 + λ α (3.19a) (3.19b) where we have chosen the branch cut associated to σ ess such that Re µ > 0; note that ν > 1. Equation (3.18) is the Legendre dierential equation: its solutions are the associated Legendre 15
unctions P µ ν (z) and Q µ ν (z) ([2], [1]). Given the symmetry z z o the equation, we choose the basis o the solution space to be P µ ν (±z). The limit χ corresponds to the limit z 1. Taking into account the normalization o ψ s,+ rom Lemma 3.4, the correct expression or the ū-component o ψ s,+ is such that where we deine ū(χ) = Γ(1 + µ) e Λssgn(σ) χ P µ ν ( z(χ)) (3.20) lim ū(χ) = Γ(1 + µ) e Λssgn(σ) χ Pν µ (z ) (3.21) χ 0 z = sgn(σ) tanh ( 1 2 (d 1) α χ ) We can express z in terms o u using equation (2.7): integrating once yields so, by equation (3.17) hence z 2 = 1 α u 2 χ = α u 2 by equation (2.13); rom this, we conclude that (3.22) 2 γ d + 1 ud+1 (3.23) u 2 χ u 2 = 1 2γ α(d + 1) ud 1 (3.24) α ( 1 z 2 ) 2γ = d + 1 ud 1 = α 9σ 2 u 2 (3.25) z = 3 σ α u. (3.26) Note that z inherits the sign o σ since χ is chosen to be positive, see section 2.1. Lemma 3.6. Let ū s (ξ; λ, ε) be the ū-component o φ s,l (ξ; λ, ε) as deined in Lemma 3.2. Then ū s (ξ) = Γ(1 + µ) e Λssgn(σ) χ P µ ν (z ) + O(ε ε) or ξ I. (3.27) Moreover, there are two transmission unctions t s,+ (λ, ε) and t s, (λ, ε) such that φ s,l (ξ; λ, ε) = t s,+ (λ, ε) ψ s, ( ξ; λ, ε) + t s, (λ, ε) ψ s,+ ( ξ; λ, ε) or ξ > 1 ε (3.28) up to exponentially small terms in ξ, where t s,+ was already introduced in Lemma 3.2. Proo. The ū-component o φ s,l is constant on I, since both d dξ ūs and d dξ p s,+ are asymptotically small on I. Thereore, we can determine its leading order value using Corollary 3.5 and (3.21). The matrix A s as deined in (3.13) is symmetric in ξ. For the region ξ > 1 ε we can thereore use the same ψ,± and ψ s,± rom Lemma 3.4 as a basis or the solution space in this region, under the relection ξ ξ. The role o ψ s,+ and ψ s, is reversed compared to 16
the interval ξ < 1 ε : we see that ψ s, ( ξ) grows (slowly) exponentially as ξ, whereas ψ s,+ ( ξ) has an exponential (slow) decay under the same limit. The normalization o φ s,l or ξ, which by Lemma 3.2 introduces t s,+ (λ, ε) in (3.28), does not exclude the possibility that or ξ > 1 ε, φ s,l has components which decay (slowly) as ξ. Thereore, we write the leading order expression o φ s,l in this region as a linear combination o a slowly increasing and a slowly decreasing component, and introduce t s, (λ, ε) to measure the decreasing component. A term containing the ast decreasing component is omitted, since or ξ > 1 ε this would only give an exponentially small correction to the result in (3.28). Based on the results o Lemma 3.6, we have to leading order. lim ū s (χ) = Γ(1 + µ) [ t s,+ (λ, ε) Pν µ ( z ) + t s, (λ, ε) Pν µ (z ) ] (3.29) χ 0 Corollary 3.7. Combining equations (3.27) and (3.29) yields to leading order. t s,+ (λ, ε) P µ ν ( z ) + t s, (λ, ε) P µ ν (z ) = P µ ν (z ) + O(ε ε) (3.30) This gives a (irst) relation between t s,+ (λ, ε) and t s, (λ, ε). 3.3. The ast components o φ s,l inside I. Since ū s (ξ; λ, ε) is constant to leading order or ξ I (see Lemma 3.6), we can represent it by its value at 0 I. Moreover, the equation or the v-component in (3.1) decouples and yields an inhomogeneous Sturm-Liouville problem, ( v ξξ (1 + λ) 2 ) v h,0 (ξ; u, 0) v = 1 u u 2 v h,0 (ξ; u, 0) 2 ū s (0) (3.31) where we used that u h (ξ) = u and v h (ξ) = v h,0 (ξ; u, 0) or ξ I to leading order (see Theorem 2.1). Based on the slow behavior o φ s,l determined in Lemmas 3.2 and 3.6, we observe that the solution v o (3.31) must extinguish as ξ I, which implies that v must decay exponentially ast in ξ. By the nature o the Gierer-Meinhardt equation (1.4) and its slow nonlinearity the problem can be solved exactly along the same lines as done in section 3.2 or the slow problem. First, we introduce a coordinate transormation similar to (3.17), ζ = d dξ v h,0(ξ; u, 0) v h,0 (ξ; u, 0) using (2.5). In this coordinate, v h can be writen as = d dξ log 1 v h,0 (ξ; u, 0) = tanh ( 1 2 ξ) (3.32) v h (ζ; u, 0) = 3 u 2 ( 1 ζ 2 ) (3.33) and equation (3.31) is transormed to ( ) (1 ζ 2 4(1 + λ) ) v ζζ 2 ζ v ζ + 12 1 ζ 2 v = 9 ū s (0) (1 ζ 2 ) (3.34) 17
and I = {ζ R ζ < 1} up to exponentially small terms, compare (2.9). Its homogeneous reduction ( ) (1 ζ 2 4(1 + λ) ) v ζζ 2 ζ v ζ + 12 1 ζ 2 v = 0 (3.35) can again be solved using associated Legendre unctions; it is a special case (α = 1, d = 2) o the slow eigenvalue problem (3.18). It should be noted that there is a crucial dierence between (3.18) and (3.34). The slow equation (3.18) is only deined on part o the ull domain: z ( 1, z ) ( 1, 1). Thereore, the eivenvalues o (3.18) do not yield direct implications or the stability o the pulse (U h, V h ). This is very dierent rom the ast system (3.34). It has three eigenvalues; its corresponding eigenunctions are λ (0) = 5 4, v(0) (ζ) = ( 1 ζ 2) 3 2 (3.36a) λ (1) = 0, v (1) (ζ) = ζ ( 1 ζ 2) = 2 dζ d 3u dξ dζ v h(ζ; u, 0) (3.36b) λ (2) = 3 4, v(2) (ζ) = ( ζ 2 1 5) 1 ζ 2 (3.36c) Reerring to [6], we recall that the roots o t,+ (λ, ε) are to leading order given by the eigenvalues o (3.35), so Lemma 3.8. There are unique λ (i) (ε) R such that lim ε 0 λ (i) (ε) = λ (i) t,+ (λ (i) (ε), ε) = 0 with multiplicity 1 or i = 0, 1, 2. and Proo. See [6]. Hence, the eigenvalues o (3.35) are to leading order zeroes o the ast component o the Evans unction D(λ, ε) given in (3.11) and thus in principle candidates or being a zero o the ull Evans unction. For all λ C \ σ ess, the solution space o (3.35) is spanned by the associated Legendre unctions v ± (ζ; λ) = c ± (λ)p 2 1+λ 3 (±ζ); lim ζ ±1 v ±(ζ; λ) = 0 (3.37) where we normalize v ± (i.e. choose c ± ) such that their Wronskian is given by W ( v, v + )(ζ; λ) = 1 1 ζ 2 (3.38) which implies that c + (λ) c (λ) = 1 (4 2 Γ + 2 ) ( 1 + λ Γ 3 + 2 ) 1 + λ (3.39) Indeed, the expression in (3.39) has poles at λ = λ (i), i = 0, 1, 2. This is due to the act that v ± (ζ; λ) cannot span the two-dimensional solution space or λ = λ (i). Since we have normalized the Wronskian (3.38), this is now encoded in the values o c ± (λ). 18
We know that the inhomogeneous equation (3.34) has a unique bounded solution v in (ξ; λ) or all λ C \ σ ess and λ λ (0,1,2). It can be determined using the Green s unction G(ζ, s; λ) = v (s; λ) v + (ζ; λ) W ( v, v + )(s; λ) (1 s 2 ) v (ζ; λ) v + (s; λ) W ( v, v + )(s; λ) (1 s 2 ) s < ζ s > ζ (3.40) so that v in (ζ; λ) = 1 1 9 ū s (0) (1 s 2 ) G(ζ, s; λ) ds [ = 9 ū s (0) v + (ζ; λ) ζ (1 s 2 ) v (s; λ) ds + v (ζ; λ) 1 1 ζ (1 s 2 ) v + (s; λ) ds ] (3.41) Note that the inhomogeneous term in (3.34) is only orthogonal to the eigenunction corresponding to λ (1) = 0; or the other two eigenvalues the solvability condition 1 1 9 ū s (0) (1 ζ 2 ) v (i) (ζ) dζ = 0 i = 0, 1, 2 (3.42) is not satisied since both v (0,2) (ζ) and 9(1 ζ 2 ) are even unctions in ζ. This means that v in as a unction o λ has a simple pole at λ (0) and λ (2), and is smooth at λ(1) = 0.. To summarize this section, the resulting expression o v in is restated in the ollowing Lemma: Lemma 3.9. The unique solution v in (ζ; λ) to equation (3.34) is given by v in (ζ; λ) = 9 ū s (0) v + (ζ; λ) [ ζ (1 s 2 ) v (s; λ) ds + v (ζ; λ) 1 1 ζ (1 s 2 ) v + (s; λ) ds ] (3.43) with v ± (ζ; λ) as deined in (3.37) and subject to condition (3.39). 3.4. The slow transmission unction t s,+ (λ, ε). In section 3.2 we studied φ s,l outside I and in section 3.3 we considered its ast dynamics inside I. However, we did not yet combine these results. Using (3.1), we see that ū ξξ = 2 ε 2 σ V h (ξ) v + O(ε 4 ) = 2 ε 2 σ v h,0 (ξ; u, 0) v in (ζ(ξ); λ) (3.44) 19
to leading order in I. Thus, the total change o ū ξ over I is given by ξ ū ξ = u ξξ dξ I = 2 ε 2 σ = 2 ε 2 σ = 2 ε 2 σ 1 1 1 1 v h,0 (ξ; u, 0) v in (ζ(ξ); λ) dξ v h,0 (ξ(ζ); u, 0) v in (ζ; λ) 3 u 2 (1 ζ2 ) v in (ζ; λ) 2 dζ 1 ζ 2 2 dζ 1 ζ 2 1 = 6 ε 2 σ u v in (ζ; λ) dζ := (3.45) 1 all to leading order. Using the expression or v in (ζ; λ) rom Lemma 3.9 and the symmetry in ζ between v + and v, this can be rewritten as 1 ζ = 108 ε 2 σ u ū s (0) v + (ζ; λ) v (s; λ)(1 s 2 ) ds dζ (3.46) 1 1 The desired coupling between the slow and ast dynamics can now be obtained by realizing that this change in ū ξ should match with the slow behaviour o φ s,l outside I. Using Corrolary 3.5 and Lemma 3.6, ( ) ( ) = ξ ū ξ = ū ξ 1 ε ū ξ 1 ε = ε 2 Γ(1 + µ) dz d [ ts,+ P µ dχ dz ε 2 Γ(1 + µ) dz d dχ dz = ε 2 Γ(1 + µ) dz d dχ dz ν ( z) + t s, Pν µ [ P µ ν ( z) ] z= z (z) ] z=z [ ts,+ P µ ν ( z) + (t s, + 1) P µ ν (z) ] z=z (3.47) to leading order. Together, expressions (3.45) and (3.47) can be used to obtain a second relation between the two transmission unctions t s,± (λ, ε), see Corollary 3.7. Thus, we can eliminate t s, and obtain a leading order expression or t s,+ : t s,+ ε 2 dz d dχ dz [ P µ ν ( z) P ν µ ( z ) Pν µ (z ) P ν µ ] (z) z=z = so that, using the Wronskian W ( P µ ν (z), P µ ν ( z)) (z ), dz d 2 ε2 Γ(1 + µ) dχ dz [ P µ ν (z) ] z=z dz d t s,+ = Pν µ Γ(1+µ) 2 ε2 dχ dz (z ) [ P ν µ (z)] z=z ε 2 dz dχ W ( Pν µ (z), Pν µ ( z) ) (z ) (3.48) which, using (3.45) and (3.26), leads to the ollowing Theorem: Theorem 3.10. Let ε > 0 be suiciently small. The unction t s,+ (λ, ε) is meromorphic as a unction o λ outside σ ess. It has simple poles at λ (0) (ε) and λ (2) (ε) and is analytic elsewhere. 20
The leading order behaviour o t s,+ is given by α z 1 t s,+ (λ, 0) = Pν µ Γ(1+µ) 1 (z ) v in (ζ; λ) dζ + dz d dχ dz [ P ν µ 1 dz 2 dχ W ( Pν µ (z), Pν µ ( z) ) (z ) (z)] z=z (3.49) The nontrivial roots o the Evans unction D(λ, ε) coincide with the roots o t s,+ (λ, ε). These roots determine the stability o the pulse (U h (ξ), V h (ξ)). Note that it is clear rom (3.49) that t s,+ inherits the poles o v in at λ = λ (0,2). The roots o the Evans unction D(λ, ε) outside σ ess are given by the roots o the product t,+ (λ, ε) t s,+ (λ, ε). Based on orthogonality arguments, we have established that t s,+ (λ, 0) has simple poles at λ = λ (0,2), see the solvability condition (3.42). These coincide with the (simple) roots o t,+ (λ, 0) (see Lemma 3.8), so the Evans unction will not necessarily be zero at these values o λ. Moreover, since the Evans unction is analytic, this statement continues to hold or ε > 0. Note that λ = 0 is always a trivial eigenvalue or system (3.1), with eigenunction d dξ (U h(ξ), V h (ξ)); it does not appear as a zero o t s,+ (λ, 0). 4. Pulse stability: results. The purpose o this section is to analyze the roots o t s,+ (λ, 0) as given in Theorem 3.10. The Wronskian in the denominator is always inite or 1 < z < 1 because the underlying dierential equation (3.18) is only singular at z = 1, 1. We can thereore ocus at the numerator, which is zero whenever P µ ν Using and α z Γ(1 + µ) 1 1 v in (ζ; λ) dζ + dz d dχ dz (z ) = 0 or [ P µ ν (z) ] z=z = 0 (4.1) [ ] dz = 1 dχ z=z 2 (d 1) α(1 z ) 2 (4.2) d [ P µ ν (z) ] dz z=z = 1 ( ) 1 z 2 (ν µ)p µ ν 1 (z ) z νpν µ (z ) equation (4.1) can be rewritten into 18 z P µ ν (z ) 1 ζ 1 1 v + (ζ; λ) v (s; λ)(1 s 2 ) ds dζ + 1 ( ) 2 (d 1) (ν µ)p µ ν 1 (z ) z νpν µ (z ) (4.3) = 0, (4.4) using (3.46) and recalling that ū s (0) = Γ(1 + µ) Pν µ (z ) to leading order by Lemma 3.6. Since this equation is only relevant i Pν µ (z ) 0, we divide by z Pν µ (z ) (note that z 0 since u 0, see (3.26)) to obtain the ollowing: Corollary 4.1. I Pν µ (z ) 0, the nontrivial roots o the Evans unction D(λ, ε) as deined in (3.11) are given to leading order by the solutions o the equation 1 ζ 18 v + (ζ; λ) v (s; λ)(1 s 2 ) ds dζ = 1 1 1 ν 1 21 ( ν (ν µ) P µ ν 1 (z ) z P µ ν (z ) ), (4.5)
LHS 5 LHS 5 1 1 2 RHS 5 1 1 2 RHS 5 Fig. 4.1: Here, LHS(λ) is plotted in blue or λ ( 1, 2); the red line is the graph o RHS( λ α ; ν, z ). In the let plot α = 0.05, ν = 2 and z = 0.75. In the right plot α = 1.5, ν = 2 and z = 0.60; the let igure is a representation o the statement in Theorem 4.2. with µ, ν, z as given in (3.19) and (3.26). The lethand side o this equation is a unction o λ only; all parameters are contained in the righthand side. Moreover, we have restricted our parameter space (α, γ, σ, d) R >0 R >0 R \ {0} (1, ), a union o two orthants in R 4 to (α, ν, z ) R >0 (1, ) ( 1, 0) (0, 1), the union o two (semi-compact) slabs in R 3. It is useul to deine the let- and righthand sides o equation (4.5) separately: LHS(λ) = 18 1 ζ 1 RHS( λ α ; ν, z ) = 1 ν 1 1 v + (ζ; λ) v (s; λ)(1 s 2 ) ds dζ (4.6) λ µ( P α ;ν) ν 1 (z ) ; ν)) ν (ν µ( λ α z P µ( λ α ;ν) ν (z ) (4.7) In Figure 4.1, the graphs o LHS(λ) and RHS( λ α ; ν, z ) are plotted or real values o λ. It is worthwhile to note that LHS(λ) = 288 R(P = 1 + λ; 2, 2) as used in [6]. 4.1. Immediate results: σ < 0 and γ 0. In this subsection we present the irst immediate implications o the developed theory or the stability o the pulse (U h, V h ). Theorem 4.2. Let ε > 0 be suiciently small. For all σ < 0, there is a real zero λ pos > λ (0) > 0 o the Evans unction associated with the stability problem (3.1). Proo. As λ, rom (3.19) we iner that µ ν such that the ratio P µ ν 1 (z ) P µ ν (z ) 1. Thereore, RHS( λ α ; ν, z ) µ 1 ν 1 z 1 3σu λ as λ. Using an equivalent argument as in [13], Lemma 4.1 (ii), one can show that LHS(λ) increases monotonically (rom ) to zero or λ > λ (0). Thereore, there is a λ > 5 4 or which LHS and RHS intersect and which thereore solves (4.5) or all parameter values when σ < 0, see Figure 4.1. Corollary 4.3. A pulse with a double hump in the U-component, as shown in Figure 1.1b, is always unstable. 22
A direct consequence o the above Corollary is that in order to obtain any stability result, we have to conine ourselves to the interval 0 < z < 1 since sgn(z ) = sgn(σ), see (3.26). It would be beneicial to a complete understanding o the linear stability o the constructed pulse (z ). However, while some inormation can be obtained regarding the number o zeroes o Pν µ (z ) or real values o µ (see [1], the general case will be treated in [14]), the authors are not aware o any general analytic expressions concerning zeroes o Pν µ (z ) or complex µ. Notwithstanding, direct numerical evaluation o Pν µ (z ) or a broad parameter range has led to the ollowing Conjecture: i more would be known about the zeroes o P µ ν Conjecture 4.4. For all λ C or which Re λ > 0, Pν µ (z ) 0 or all 0 < z < 1. Moreover, or Im λ 0, Pν µ (z ) 0 or all 0 < z < 1. Based on this observation, the study o linear stability o the pulse can be conined to the study o solutions o (4.5). Moreover, any additional eigenvalues originating rom zeroes o Pν µ (z ) would occur on the real line and be negative. Note that in the ollowing results, this Conjecture is not needed. Equation (4.7) can be studied or dierent parameter values (and limits thereo) to obtain inormation about the pulse spectrum. Another direct result can be obtained by taking the limit γ 0 to remove the inluence o the slow nonlinearity in (1.7) and obtain the classical Gierer- Meinhardt equations. As γ 0, u α 3 σ (see (2.13)) so z sgn(σ) using (3.26). Note that, while the limit γ 0 reduces equation (1.7) to the classical Gierer-Meinhardt equation where the slow evolution in U is linear, yielding a simple exponential instead o an associated Legendre unction the coordinate z is ill-deined or γ = 0, see (3.17) in relation to (3.16). Thereore, some o the expressions in the ollowing will still depend on ν, while ν disappears rom (1.7) as γ 0 Since P µ ν (z ) 1 Γ(1 + µ) P µ ν (see [2], [1]), this means that ( ) µ 1 z 2 as z 1 (4.8a) 2 ( ) µ Γ(µ) 1 + z 2 (z ) as z 1 (4.8b) Γ(µ ν)γ(1 + µ + ν) 2 lim RHS( λ γ 0 α ; ν, z (α, γ, σ, d)) = lim RHS( λ z α ; ν, z ) = sgn(σ) sgn(σ) = sgn(σ) µ ν 1 1 + λ α, (4.9) Moreover, Pν µ (z ) can be written as Pν µ (z ) = ( 1 z ) µ 2 2 F (z ) where F (z) has a regular expansion (see [2], [1]). Near z = 1, F (z) can be expanded as F (z) = k=0 a ( 1 z ) k, k 2 with ( µ ) 2 k j a k = ( ν) j(ν + 1) j (4.10) Γ(1 + j + µ)(k j)!j! j=0 1 Since a 0 = Γ(1+µ) 0 or all µ considered since Re µ > 0 and the limit γ 0 only inluences the value o z, it ollows that Pν µ (z ) does not have any zeroes asymptotically close to, but 23