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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Construction and asymptotic stability of structurally stable internal layer solutions


Author: Xiao-Biao Lin
Journal: Trans. Amer. Math. Soc. 353 (2001), 2983-3043
MSC (2000): Primary 34D15, 34E05, 35B25; Secondary 35C20, 37C29, 34C37
Published electronically: March 22, 2001
MathSciNet review: 1828598
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Abstract:

We introduce a geometric/asymptotic method to treat structurally stable internal layer solutions. We consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability. Proofs of the existence of exact solutions and eigenvalue-eigenfunctions are outlined.

Multi-layered solutions are constructed by a new shooting method through a sequence of pseudo Poincaré mappings that do not require the transversality of the flow to cross sections. The critical eigenvalues are determined by a coupling matrix that generates the SLEP matrix. The transversality of the shooting method is related to the nonzeroness of the critical eigenvalues.

An equivalent approach is given to mono-layer solutions. They can be determined by the intersection of a fast jump surface and a slow switching curve, which reduces Fenichel's transversality condition to the slow manifold. The critical eigenvalue is determined by the angle of the intersection.

We present three examples. The first treats the critical eigenvalues of the system studied by Angenent, Mallet-Paret & Peletier. The second shows that a key lemma in the SLEP method may not hold. The third is a perturbed activator-inhibitor system that can have any number of mono-layer solutions. Some of the solutions can only be found with the new shooting method.


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Additional Information

Xiao-Biao Lin
Affiliation: Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27695-8205
Email: xblin@xblsun.math.ncsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-01-02769-6
PII: S 0002-9947(01)02769-6
Keywords: Singular perturbation, matched asymptotic expansion, internal layers, critical eigenvalues, stability, structural stability
Received by editor(s): March 18, 1998
Received by editor(s) in revised form: May 5, 2000
Published electronically: March 22, 2001
Additional Notes: Research partially supported by NSF grant 9501255
Article copyright: © Copyright 2001 American Mathematical Society