On the steadiness of symmetric solutions to higher order perturbations of KdV

TL;DR

This paper investigates symmetric solutions to higher order perturbations of the KdV equation, proving conditions for traveling solutions and classifying symmetric solutions, with implications for water wave modeling.

Contribution

It classifies symmetric traveling solutions for higher order perturbed KdV equations and shows nonexistence of nontrivial symmetric solitary solutions under dissipation or shoaling.

Findings

Symmetric solutions must be traveling solutions in the Rosenau-Kawahara-RLW equation.

No nontrivial symmetric solitary solutions exist with dissipation or shoaling.

Results support the sharpness of the symmetry principle for solitary waves.

Abstract

We consider the traveling structure of symmetric solutions to the Rosenau-Kawahara-RLW equation and the perturbed R-KdV-RLW equation. Both equations are higher order perturbations of the classical KdV equation. For the Rosenau-Kawahara-RLW equation, we prove that classical and weak solutions with a priori symmetry must be traveling solutions. For the more complicated perturbed R-KdV-RLW equation, we classify all symmetric traveling solutions, and prove that there exists no nontrivial symmetric traveling solution of solitary type once dissipation or shoaling perturbations exist. This gives a new perspective for evaluating the suitableness of a model for water waves. In addition, this result illustrates the sharpness of the symmetry principle in [Int. Math. Res. Not. IMRN, 2009; Ehrnstrom, Holden \& Raynaud] for solitary waves.