Stability of solitary and cnoidal traveling wave solutions for a fifth order Korteweg-de Vries equation

TL;DR

This paper proves the nonlinear stability of solitary and cnoidal traveling wave solutions for a fifth order KdV equation, extending stability results to more complex dispersive wave solutions.

Contribution

It establishes the orbital stability of both solitons and cnoidal waves for a fifth order KdV equation, using Fourier transform positivity properties.

Findings

All families of solutions are orbitally stable.

Stability is proven for both solitary and periodic solutions.

The analysis relies on Fourier transform positivity properties.

Abstract

We establish the nonlinear stability of solitary waves (solitons) and periodic traveling wave solutions (cnoidal waves) for a Korteweg-de Vries (KdV) equation which includes a fifth order dispersive term. The traveling wave solutions which yield solitons for zero boundary conditions and wave-trains of cnoidal waves for nonzero boundary conditions are analyzed using stability theorems, which rely on the positivity properties of the Fourier transforms. We show that all families of solutions considered here are (orbitally) stable.