Periodic traveling--wave solutions of nonlinear dispersive evolution equations

Hongqiu Chen; Jerry L. Bona

doi:10.3934/dcds.2013.33.4841

Article Contents

2013, Volume 33, Issue 11&12: 4841-4873. Doi: 10.3934/dcds.2013.33.4841

This issue Previous Article Singularity formation and blowup of complex-valued solutions of the modified KdV equation Next Article On the asymptotic behavior of variational inequalities set in cylinders

Periodic traveling--wave solutions of nonlinear dispersive evolution equations

Hongqiu Chen^1, and
Jerry L. Bona^2,

1.
Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee, 38152
2.
Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607

Received: January 31, 2012

Revised: October 31, 2012

Published: October 2013

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Abstract

For a general class of nonlinear, dispersive wave equations, existence of periodic, traveling-wave solutions is studied. These traveling waveforms are the analog of the classical cnoidal-wave solutions of the Korteweg-de Vries equation. They are determined to be stable to perturbation of the same period. Their large wavelength limit is shown to be solitary waves.

Keywords:

Period traveling waves,
generalized cnoidal waves,
solitary waves,
generalized Korteweg--de Vries equations,
non-local dispersion relations,
stability theory.

Mathematics Subject Classification: Primary: 35C07, 35C08, 35Q35, 35Q51, 35Q53, 35S10, 76B15, 76B25; Secondary: 35C10, 35Q86, 86A05.

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