Counting unstable eigenvalues in Hamiltonian spectral problems via commuting operators

TL;DR

This paper introduces a general method to count unstable eigenvalues of certain linear operators using commuting operators, with applications to stability analysis of periodic waves in the KP-II equation.

Contribution

It provides a novel counting result for unstable eigenvalues of operators of the form JL using a commuting operator K, extending stability analysis techniques.

Findings

Unstable eigenvalues of JL are bounded by nonpositive eigenvalues of K.

One-dimensional periodic waves in KP-II are transversely spectrally stable.

These waves are also transversely linearly stable under doubly periodic perturbations.

Abstract

We present a general counting result for the unstable eigenvalues of linear operators of the form $JL$ in which $J$ and $L$ are skew- and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operator $K$ such that the operators $JL$ and $JK$ commute, we prove that the number of unstable eigenvalues of $JL$ is bounded by the number of nonpositive eigenvalues of~$K$. As an application, we discuss the transverse stability of one-dimensional periodic traveling waves in the classical KP-II (Kadomtsev--Petviashvili) equation. We show that these one-dimensional periodic waves are transversely spectrally stable with respect to general two-dimensional bounded perturbations, including periodic and localized perturbations in either the longitudinal or the transverse direction, and that they are transversely linearly stable with respect to doubly periodic perturbations.