Existence, uniqueness and analyticity of space-periodic solutions to the regularised long-wave equation

TL;DR

This paper investigates the existence, uniqueness, and analyticity of space-periodic solutions to the regularised long-wave equation, highlighting the role of damping and providing results on travelling waves and their convergence.

Contribution

It establishes the existence, uniqueness, and smoothness of solutions, and demonstrates the importance of damping for well-posedness of travelling-wave solutions.

Findings

Existence and uniqueness of evolutionary solutions for smooth initial data.

Global in time spatial analyticity of solutions with analytical initial conditions.

Travelling-wave solutions exist and are unique for small norms.

Abstract

We consider space-periodic evolutionary and travelling-wave solutions to the regularised long-wave equation (RLWE) with damping and forcing. We establish existence, uniqueness and smoothness of the evolutionary solutions for smooth initial conditions, and global in time spatial analyticity of such solutions for analytical initial conditions. The width of the analyticity strip decays at most polynomially. We prove existence of travelling-wave solutions and uniqueness of travelling waves of a sufficiently small norm. The importance of damping is demonstrated by showing that the problem of finding travelling-wave solutions to the undamped RLWE is not well-posed. Finally, we demonstrate the asymptotic convergence of the power series expansion of travelling waves for a weak forcing.