Does solitary wave solution persist for the long wave equation with small perturbations?

TL;DR

This paper investigates whether solitary wave solutions of the regularized long wave equation persist under small perturbations, using geometric singular perturbation theory and numerical simulations to confirm the theoretical findings.

Contribution

It demonstrates the persistence of solitary waves under small perturbations and analyzes how different perturbations influence wave speed.

Findings

Solitary waves persist under small backward diffusion and Marangoni effects.

Perturbations affect the wave speed needed for persistence.

Numerical simulations confirm theoretical predictions.

Abstract

In this paper, persistence of solitary wave solutions of the regularized long wave equation with small perturbations are investigated by the geometric singular perturbation theory. Two different kinds of the perturbations are considered in this paper: one is the weak backward diffusion and dissipation, the other is the Marangoni effects. Indeed, the solitary wave persists under small perturbations. Furthermore, the different perturbations do affect the proper wave speed ensuring the persistence of the solitary waves. Finally, numerical simulations are utilized to confirm the theoretical results.