On the Rosenau equation: Lie symmetries, periodic solutions and solitary wave dynamics

TL;DR

This paper analyzes the Rosenau equation's symmetries, finds explicit solutions, and studies solitary wave dynamics using analytical methods and numerical simulations, including the Fourier pseudo-spectral method.

Contribution

It provides a comprehensive analysis of the Rosenau equation, including Lie symmetry classification, explicit elliptic solutions, and numerical investigation of solitary wave interactions.

Findings

Lie symmetry algebra identified for the quadratic case

Explicit periodic solutions in elliptic functions obtained

Numerical simulations of solitary wave evolution and collisions conducted

Abstract

In this paper, we first consider the Rosenau equation with the quadratic nonlinearity and identify its Lie symmetry algebra. We obtain reductions of the equation to ODEs, and find periodic analytical solutions in terms of elliptic functions. Then, considering a general power-type nonlinearity, we prove the non-existence of solitary waves for some parameters using Pohozaev type identities. The Fourier pseudo-spectral method is proposed for the Rosenau equation with this single power type nonlinearity. In order to investigate the solitary wave dynamics, we generate the solitary wave profile as an initial condition by using the Petviashvili's method. Then the evolution of the single solitary wave and overtaking collision of solitary waves are investigated by various numerical experiments.