Bounds on unstable spectrum for dispersive Hamiltonian PDEs

TL;DR

This paper provides bounds on the location and number of off-axis eigenvalues in the complex plane for stability analysis of periodic traveling waves in Hamiltonian PDEs, applicable to various nonlinear dispersive equations.

Contribution

It establishes new bounds and estimates for unstable eigenvalues in Hamiltonian PDEs, utilizing Gershgorin disks and spectral symmetry, extending stability analysis tools.

Findings

Bounds on eigenvalues away from the imaginary axis.

Estimates for the number of off-axis eigenvalues.

Applicable to a broad class of nonlinear dispersive equations.

Abstract

We study quasi-periodic eigenvalue problems that arise in the stability analysis of periodic traveling wave solutions to Hamiltonian PDEs. We establish bounds on regions in the complex plane when the eigenvalues may deviate from the imaginary axis, and estimates for the number of such off-axis eigenvalues. These relations hold when the dispersion relation grows sufficiently rapidly in the wavenumber. The proofs involve a Gershgorin disk argument together with the Hamiltonian symmetry of the spectrum. The results are applicable to a broad class of nonlinear dispersive equations including the generalized Korteweg--de Vries, Benjamin--Bona--Mahoney, and Kawanhara equations.