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Adiabatic Evolution of Coupled Waves for aSchrödinger-Korteweg-de Vries System

Published online by Cambridge University Press:  29 February 2012

W. Abou Salem
W. Abou Salem*
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, SK, S7N 5E6, Canada
*
⋆ Corresponding author. E-mail: abousalem@math.usask.ca
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Abstract

The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vriesequations over a slowly varying random bottom is rigorously studied. One motivation forstudying such a system is better understanding the unidirectional motion of interactingsurface and internal waves for a fluid system that is formed of two immiscible layers. Itwas shown recently by Craig-Guyenne-Sulem [1] thatin the regime where the internal wave has a large amplitude and a long wavelength, thedynamics of the surface of the fluid is described by the Schrödinger equation, while thatof the internal wave is described by the Korteweg-de Vries equation. The purpose of thisletter is to show that in the presence of a slowly varying random bottom, the coupledwaves evolve adiabatically over a long time scale. The analysis covers the cases when thesurface wave is a stable bound state or a long-lived metastable state.

Keywords

Korteweg-de Vries equationSchrödinger equationcoupled waveseffective dynamicsadiabatic theorem
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 2: Solitary waves , 2012 , pp. 1 - 12
Copyright
© EDP Sciences, 2012

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