Large stable pulse solutions in reaction-diffusion equations

Abstract

ABSTRACT. In this paper we study the existence and stability of asymptotically large stationary multi-pulse solutions in a family of singularly perturbed reaction-diffusion equations. This family includes the generalized Gierer-Meinhardt equation. The existence of N-pulse homoclinic orbits (N ≥ 1) is established by the methods of geometric singular perturbation theory. A theory, called the NLEP (=NonLocal Eigenvalue Problem) approach, is developed, by which the stability of these patterns can be studied explicitly. This theory is based on the ideas developed in our earlier work on the Gray-Scott model. It is known that the Evans function of the linear eigenvalue problem associated to the stability of the pattern can be decomposed into the product of a slow and a fast transmission function. The NLEP approach determines explicit leading order approximations of these transmission functions. It is shown that the zero/pole cancellation in the decomposition of the Evans function, called the NLEP paradox, is a phenomenon that occurs naturally in singularly perturbed eigenvalue problems. It follows that the zeroes of the Evans function, and thus the spectrum of the stability problem, can be studied by the slow transmission function. The key ingredient of the analysis of this expression is a transformation of the associated nonlocal eigenvalue problem into an inhomogeneous hypergeometric differential equation. By this transformation it is possible to determine both the number and the position of all elements in the discrete spectrum of the linear eigenvalue problem. The method is applied to a special case that corresponds to the classical model proposed by Gierer and Meinhardt. It is shown that the one-pulse pattern can gain (or lose) stability through a Hopf bifurcation at a certain value µ Hopf of the main parameter µ. The NLEP approach not only yields a leading order approximation of µ Hopf , but it also shows that there is another bifurcation value, µ edge , at which a new (stable) eigenvalue bifurcates from the edge of the essential spectrum. Finally, it is shown that the N-pulse patterns are always unstable when N ≥ 2. 443

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“…Before making a case distinction, we do some preparatory work. As noted before, from expression (14), it can be seen that the stability of the homogeneous steady state can be manipulated by increasing c. Namely, there is only one term that depends on c, and for k 2 small, this term is approximated by 4c 2 a 1 a 4 < 0. By choosing the right value of c, it can be inferred that ReðkÞ ¼ 0 for some k 2 but nowhere ReðkÞ > 0.…”

“…Because Ð p0 0 1 p dp diverges, for p small enough, the sign of (27) will again be positive and Eq. (14) has no solutions, again implying stability by Lemma 4 (Appendix A). ٗ Uniqueness of the destabilization locus on I for fixed p or fixed c is an immediate consequence of the strict monotonicity.…”

“…First note that all terms in (14) are positive except a 1 a 4 < 0 on the second line and possibly d 1 d 2 k 2 À C, which appears on both lines. For c ¼ 0, this confirms the well-known fact that C > 0 is a necessary condition for a Turing instability, see (11).…”

“…We continue with some useful estimates that can be derived from (14). We first note an upper bound for the wavenumbers that can become critical given by…”

“…Now, all terms of (14) are positive for k 2 2 a 4 d 2 ; 1 Â Á , so these wavenumbers cannot be critical.…”