Real Lax spectrum implies spectral stability

TL;DR

This paper links the Lax spectrum of integrable equations with the spectral stability of their periodic solutions, showing that real parts of the spectrum correspond to stability, especially in self-adjoint cases.

Contribution

It establishes a connection between the Lax spectrum and spectral stability, providing a method to identify stable eigenvalues via the Lax spectrum for integrable equations.

Findings

Real Lax spectrum contains all stable eigenvalues for self-adjoint cases.

For non-self-adjoint cases, the real line in the Lax spectrum maps to stable eigenvalues.

Method demonstrated on various examples confirms its broad applicability.

Abstract

We consider the dynamical stability of periodic solutions of integrable equations with 2x2 Lax pairs. We construct the eigenfunctions and hence the Floquet discriminant for such Lax pairs. The boundedness of the eigenfunctions determines the Lax spectrum. We use a connection between the Lax spectrum and the stability spectrum to show that the subset of the real line which gives rise to stable eigenvalues is contained in the Lax spectrum. This subset is the full spectrum for self-adjoint members of the AKNS hierarchy. For non-self-adjoint members of the AKNS hierarchy admitting a common reduction, the real line is always part of the Lax spectrum and maps to stable eigenvalues of the stability problem. We demonstrate our methods work for a variety of examples.