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.

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

“…The primary purpose of this work is to prove a comparison result of solutions to (2) in the weak dispersive regime and also is to show that the behavior of solutions is determined by the dispersive character of the kernel in the long-wave limit rather than the shape of the kernel function. In a recent work [4], a similar comparison result was given for the nonlocal bidirectional wave equations. Based mainly on an energy estimate with no loss of derivative, our present comparison result extends basically the notion of "kernel based comparison" introduced in [4] to the nonlocal unidirectional wave equation (2).…”

On the convergence of the nonlocal nonlinear model to the classical elasticity equation

“…We prove that as the kernel β approaches the Dirac delta measure, not only (1.1) approaches formally the classical elasticity equation, (1.2), but also the corresponding solutions converge strongly to the solution of (1.2). The notion of this comparison result is in the sense of that in [7].…”

“…The main step is to obtain uniform estimates of solutions with respect to the parameter characterizing dispersion and to estimate the difference between the solutions using the energy estimate. Nevertheless, the fact that our energy estimate does not involve loss of derivative allows us to get a stronger convergence result than that in [7].…”

“…A collection of articles [5,[7][8][9][10][11] focuses on various aspects of mathematical analysis of the initial-value problems associated with (1.1). In those studies it is assumed that the Fourier transform β of the kernel function β satisfies the ellipticity and boundedness 2 condition…”

“…In the present work we prove that the limit of (1.1) is (1.2) in the sense that solutions of (1.1) approximate to the solution of (1.2) in the limit of vanishing nonlocality. This is inspired by the comparison result established in [7] where two solutions corresponding to two different kernels were compared in the long-wave limit. Here we concentrate on the particular case where one of the kernels is the Dirac delta function corresponding to the zero-dispersion case.…”