Spectral stability of nonlinear waves in KdV-type evolution equations

TL;DR

This paper investigates the spectral stability of nonlinear waves in KdV-type equations, providing an instability index theorem applicable to solitary and periodic waves, and reviews recent related results.

Contribution

It introduces a new instability index theorem for spectral stability analysis of nonlinear waves in KdV-type equations under generic conditions.

Findings

Proves an instability index theorem for solitary and periodic waves.

Analyzes the eigenvalue problem involving a composition of operators.

Reviews recent developments in spectral stability of KdV-type waves.

Abstract

This paper concerns spectral stability of nonlinear waves in KdV-type evolution equations. The relevant eigenvalue problem is defined by the composition of an unbounded self-adjoint operator with a finite number of negative eigenvalues and an unbounded non-invertible symplectic operator $\partial_x$. The instability index theorem is proven under a generic assumption on the self-adjoint operator both in the case of solitary waves and periodic waves. This result is reviewed in the context of other recent results on spectral stability of nonlinear waves in KdV-type evolution equations.