The Modultional Instability for a Generalized KdV Equation

TL;DR

This paper analyzes the spectral stability of periodic solutions to the generalized KdV equation using a rigorous modulation theory approach, identifying key indices that determine long-wavelength instability.

Contribution

It introduces a novel spectral stability analysis framework for g-KdV waves, linking stability indices to conserved quantities and providing geometric insights.

Findings

Two instability indices identified for spectral analysis.

A necessary and sufficient condition for long-wavelength instability derived.

Connection established between stability indices and conserved quantities.

Abstract

We study the spectral stability of a family of periodic standing wave solutions to the generalized KdV (g-KdV) in a neighborhood of the origin in the spectral plane using what amounts to a rigorous Whitham modulation theory calculation. In particular we are interested in understanding the role played by the null directions of the linearized operator in the stability of the traveling wave to perturbations of long wavelength. A study of the normal form of the characteristic polynomial of the monodromy map (the periodic Evan's function) in a neighborhood of the origin in the spectral plane leads to two different instability indices. The first index counts modulo 2 the total number of periodic eigenvalues on the real axis. This index is a generalization of the one which governs the stability of the solitary wave. The second index provides a necessary and sufficient condition for the existence of a long-wavelength instability. This index is essentially the quantity calculated by Haragus and Kapitula in the small amplitude limit. Both of these quantities can be expressed in terms of the map between the constants of integration for the ordinary differential equation defining the traveling waves and the conserved quantities of the partial differential equation. These two indices together provide a good deal of information about the spectrum of the linearized operator. We sketch the connection of this calculation to a study of the linearized operator - in particular we perform a perturbation calculation in terms of the Floquet parameter. This suggests geometric interpretations attached to the vanishing of the modulational instability index previously mentioned.