Construction and asymptotic stability of structurally stable internal layer solutions

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.