Oscillation Theory and Instability of Nonlinear Waves

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].