Oscillation Theory and Instability of Nonlinear Waves

TL;DR

This paper extends oscillation and instability analysis techniques to nonlinear waves using a generalized Maslov index, broadening their applicability beyond Hamiltonian systems and applying them to spectral instability problems.

Contribution

It introduces a novel implementation of the generalized Maslov index for analyzing spectral instability of nonlinear waves in non-Hamiltonian systems.

Findings

Extended oscillation techniques to non-Hamiltonian systems.

Applied the generalized Maslov index to spectral instability analysis.

Connected the approach to previous Evans function studies.

Abstract

In recent work, Baird et al. have introduced a generalized Maslov index which allows oscillation techniques that have previously been restricted to eigenvalue problems with underlying Hamiltonian structure to be extended to the non-Hamiltonian setting [T. J. Baird, P. Cornwell, G. Cox, C. Jones, and R. Marangell, Generalized Maslov indices for non-Hamiltonian systems, SIAM J. Math. Anal. 54 (2022) 1623-1668]. We show that this approach can be implemented in the analysis of spectral instability for nonlinear waves, taking as our setting a class of equations previously investigated by Pego and Weinstein via the Evans function [R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Phil. Trans. R. Soc. Lond. A 340 (1992) 47-94].