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

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.

COMMUNICATED BY:

• Adrian Constantin

Keywords:

• Approximation
• nonlocal wave equation
• improved Boussinesq equation
• long-wave limit

AMS CLASSIFICATIONS:

• 35Q53
• 35Q74
• 74J30
• 35C20

Disclosure statement

No potential conflict of interest was reported by the authors.