Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System

W. Abou Salem

Mathematical Modelling of Natural Phenomena (2012)

  • Volume: 7, Issue: 2, page 1-12
  • ISSN: 0973-5348

Abstract

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

How to cite

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Abou Salem, W.. "Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System." Mathematical Modelling of Natural Phenomena 7.2 (2012): 1-12. <http://eudml.org/doc/222294>.

@article{AbouSalem2012,
abstract = {The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vries equations over a slowly varying random bottom is rigorously studied. One motivation for studying such a system is better understanding the unidirectional motion of interacting surface and internal waves for a fluid system that is formed of two immiscible layers. It was shown recently by Craig-Guyenne-Sulem [1] that in the regime where the internal wave has a large amplitude and a long wavelength, the dynamics of the surface of the fluid is described by the Schrödinger equation, while that of the internal wave is described by the Korteweg-de Vries equation. The purpose of this letter is to show that in the presence of a slowly varying random bottom, the coupled waves evolve adiabatically over a long time scale. The analysis covers the cases when the surface wave is a stable bound state or a long-lived metastable state.},
author = {Abou Salem, W.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {Korteweg-de Vries equation; Schrödinger equation; coupled waves; effective dynamics; adiabatic theorem},
language = {eng},
month = {2},
number = {2},
pages = {1-12},
publisher = {EDP Sciences},
title = {Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System},
url = {http://eudml.org/doc/222294},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Abou Salem, W.
TI - Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System
JO - Mathematical Modelling of Natural Phenomena
DA - 2012/2//
PB - EDP Sciences
VL - 7
IS - 2
SP - 1
EP - 12
AB - The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vries equations over a slowly varying random bottom is rigorously studied. One motivation for studying such a system is better understanding the unidirectional motion of interacting surface and internal waves for a fluid system that is formed of two immiscible layers. It was shown recently by Craig-Guyenne-Sulem [1] that in the regime where the internal wave has a large amplitude and a long wavelength, the dynamics of the surface of the fluid is described by the Schrödinger equation, while that of the internal wave is described by the Korteweg-de Vries equation. The purpose of this letter is to show that in the presence of a slowly varying random bottom, the coupled waves evolve adiabatically over a long time scale. The analysis covers the cases when the surface wave is a stable bound state or a long-lived metastable state.
LA - eng
KW - Korteweg-de Vries equation; Schrödinger equation; coupled waves; effective dynamics; adiabatic theorem
UR - http://eudml.org/doc/222294
ER -

References

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