Temporal growth and eventual periodicity for dispersive wave equations in a quarter plane

Jerry Bona,Jiahong Wu

doi:10.3934/dcds.2009.23.1141

Abstract

Studied here is thelarge-time behavior and eventual periodicity of solutions ofinitial-boundary-value problems for the BBM equation and the KdVequation, with and without a Burgers-type dissipation appended. Itis shown that the total energy of a solution of these problems growsat an algebraic rate which is in fact sharp for solutions of theassociated linear equations. We also establish that solutions of thelinear problems are eventually periodic if the boundary data areperiodic.

Highlights

Initial-boundary-value problems for the KdV equation or the BBM equation arise naturally in modeling small-amplitude long waves in a channel with a wavemaker mounted at one end, or in modeling coastal zone motions generated by long-crested waves propagating shoreward from deep water
We will address the question of energy growth, both locally and globally and the issue of what we term eventual periodicity which is exhibited by solutions of initialboundary-value problems (IBVP ) for the generalized BBM equation and the generalized KdV equation ut + ux + upux − uxxt = 0 for x, t ≥ 0, 2000
We study in detail the eventual periodicity properties for the more complex problems presented by the linear IBVP for the BBM-Burgers equation, the KdV-Burgers equation, and for their non-dissipative counterparts

Summary

Introduction

Initial-boundary-value problems for the KdV equation or the BBM equation arise naturally in modeling small-amplitude long waves in a channel with a wavemaker mounted at one end, or in modeling coastal zone motions generated by long-crested waves propagating shoreward from deep water (see, for example [9], [3], [4]). The total energy of solutions of the initial-boundary-value problems ut + ux + upux + μu − uxxt = 0 (4) The analysis in the linear case that appears here, in particular the integral representations, will very likely find use in a theory of eventual periodicity for the nonlinear problems

Results

Conclusion