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

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