Comparison of nonlocal nonlinear wave equations in the long-wave limit
Erbay, Hüsnü Ata and Erbay, Saadet and Erkip, Albert (2019) Comparison of nonlocal nonlinear wave equations in the long-wave limit. Applicable Analysis . ISSN 0003-6811 (Print) 1563-504X (Online) Published Online First http://dx.doi.org/10.1080/00036811.2019.1577393
Official URL: http://dx.doi.org/10.1080/00036811.2019.1577393
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.
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