On the stability of the solitary waves to the (generalized) kawahara equation

TL;DR

This paper studies the orbital stability of solitary waves in the generalized Kawahara equation, proving stability for certain nonlinearities and constructing two families of even solitary waves using spectral and variational methods.

Contribution

It introduces the first family of even solitons near explicit solutions and establishes their orbital stability in the energy space for specific nonlinearities.

Findings

Proved orbital stability of two branches of even solitary waves in gKW.

Constructed the first family of even solitons via implicit function theorem.

Identified a second family of traveling waves as rescaled perturbations of gKdV solitons.

Abstract

In this paper we investigate the orbital stability of solitary waves to the (generalized) Kawahara equation (gKW) which is a fifth order dispersive equation. For some values of the power of the nonlinearity, we prove the orbital stability in the energy space H 2 (R) of two branches of even solitary waves of gKW by combining the well-known spectral method introduced by Benjamin [3] with continuity arguments. We construct the first family of even solitons by applying the implicit function theorem in the neighborhood of the explicit solitons of gKW found by Dey et al. [8]. The second family consists of even travelling waves with low speeds. They are solutions of a constraint minimization problem on the line and rescaling of perturbations of the soliton of gKdV with speed 1.