A comparison of solutions of two convolution-type unidirectional wave equations

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 non-local 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 of the hyperbolic conservation law.