A Comparison of Solutions of Two Convolution-Type Unidirectional Wave Equations

TL;DR

This paper establishes a comparison framework for nonlinear dispersive unidirectional wave equations, demonstrating that solutions with similar dispersive kernels remain close over time, with applications to water and elastic wave models.

Contribution

It introduces a comparison result for solutions of a broad class of nonlocal wave equations, linking kernel similarity to solution proximity in Sobolev norms.

Findings

Solutions with similar kernels stay close in Sobolev norms

Energy estimates are derived for the nonlocal wave equations

Application to hyperbolic conservation law approximations

Abstract

In this work, we prove a comparison result for a general class of nonlinear dispersive unidirectional wave equations. The dispersive nature of one-dimensional waves occurs because of a convolution integral in space. For two specific choices of the kernel function, the Benjamin-Bona-Mahony equation and the Rosenau equation that are particularly suitable to model water waves and elastic waves, respectively, are two members of the class. We first prove an energy estimate for the Cauchy problem of the nonlocal unidirectional wave equation. Then, for the same initial data, we consider two distinct solutions corresponding to two different kernel functions. Our main result is that the difference between the solutions remains small in a suitable Sobolev norm if the two kernel functions have similar dispersive characteristics in the long-wave limit. As a sample case of this comparison result, we provide the approximations to the hyperbolic conservation law.