# Comparison of nonlocal nonlinear wave equations in the long-wave limit

## TL;DR

This paper compares solutions of different nonlocal wave equations with similar dispersive properties in the long-wave limit, demonstrating that they can be approximated by Boussinesq-type equations over long times.

## Contribution

It provides a rigorous comparison of nonlocal wave equations with different kernels, establishing their solutions' closeness in the long-wave regime.

## Key findings

- Solutions remain close over long times in Sobolev norms
- Nonlocal equations can be approximated by Boussinesq-type equations
- Comparison holds for kernels with similar dispersive characteristics

## Abstract

We consider a general class of convolution-type nonlocal wave equations modeling bidirectional nonlinear wave propagation. The model involves two small positive parameters measuring the relative strengths of the nonlinear and dispersive effects. We take two different kernel functions that have similar dispersive characteristics in the long-wave limit and compare the corresponding solutions of the Cauchy problems with the same initial data. We prove rigorously that the difference between the two solutions remains small over a long time interval in a suitable Sobolev norm. In particular our results show that, in the long-wave limit, solutions of such nonlocal equations can be well approximated by those of improved Boussinesq-type equations.