Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg-de Vries Equation

TL;DR

This paper investigates the orbital stability of periodic traveling wave solutions in the generalized Korteweg-de Vries equation, providing conditions linking spectral and orbital stability through the analysis of the classical action's Hessian.

Contribution

It establishes new sufficient conditions for orbital stability of periodic solutions based on the Hessian of the classical action, connecting spectral and orbital stability in this context.

Findings

Derived stability conditions using the Hessian of the classical action.

Proved equivalence of spectral and orbital stability for KdV solutions.

Extended stability analysis to solutions near homoclinic and equilibrium states.

Abstract

In this paper, we study the orbital stability for a four-parameter family of periodic stationary traveling wave solutions to the generalized Korteweg-de Vries equation. In particular, we derive sufficient conditions for such a solution to be orbitally stable in terms of the Hessian of the classical action of the corresponding traveling wave ordinary differential equation restricted to the manifold of periodic traveling wave solution. We show this condition is equivalent to the solution being spectrally stable with respect to perturbations of the same period in the case of the Korteweg-de Vries equation, and in neighborhoods of the homoclinic and equilibrium solutions in the case of a power-law nonlinearity.