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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  1. W. Craig, P. Guyenne, C. Sulem. Coupling between internal and surface waves, Natural Hazards, Special Issue on “Internal waves in the oceans and estuaries : modeling and observations”, (2010), doi :, 26pp.  URI10.1007/s11069-010-9535-4
  2. W. Craig, P. Guyenne, C. Sulem. A Hamiltonian approach to nonlinear modulation analysis. Wave Motion47 (2010), 552–563.  
  3. E. van Groesen, S. R. Pudjaprasetya. Uni-directional waves over slowly varying bottom. I. Derivation of a KdV-type of equation. Wave Motion18 (1993), 345–370.  
  4. S. B. Yoon, Philip L.-F. Liu. A note on Hamiltonian for long water waves in varying depth. Wave Motion20 (1994), 359–370.  
  5. S.I Dejak, I.M. Segal. Long time dynamics of KdV solitary waves over a variable bottom. Comm. Pure Appl. Math.59 (2006), 869–905.  
  6. S.I. Dejak, B.L.G Jonsson. Long time dynamics of variable coefficient mKdV solitary waves. J. Math. Phys. 47 (2006), 072703, 16pp.  
  7. J. Holmer. Dynamics of KdV solitons in the presence of a slowly varying potential. IMRN (2011), doi :, 31pp.  URI10.1093/imrn/rnq284
  8. J. Holmer, G. Perelman, M. Zworski. Effective dynamics of double solitons for perturbed mKdV. Commun. Math. Phys.305 (2011), 363–425.  
  9. C. Munoz. On the soliton dynamics under a slowly varying medium for generalized KdV equations. arxiv.org arXiv :0912.4725 [math.AP] (2009). To appear in Analysis and PDE.  
  10. W. Abou Salem, J. Fröhlich. Adiabatic theorems for quantum resonances. Commun. Math. Phys.273 (2007), 651–675.  
  11. T. Kato. On the adiabatic theorem of quantum mechanics. Phys. Soc. Jap.5 (1958), 435–439.  
  12. J.E. Avron, A. Elgart. Adiabatic theorem without a gap condition. Commun. Math. Phys.203 (1999), 445–463.  
  13. S. Teufel. A note on the adiabatic theorem without a gap condition. Lett. Math. Phys.58 (2002), 261–266.  
  14. A. Joye. General adiabatic evolution with a gap condition. Commun. Math. Phys.275 (2007), 139-162.  
  15. V. S. Buslaev, C. Sulem. Linear adiabatic dynamics generated by operators with continuous spectrum. Asymptotic Anal.58 (2008), 17–45.  
  16. A. Elgart, G. A. Hagedorn. An adiabatic theorem for resonances. Comm. Pure Appl. Math.64 (2011), 1029–1058.  
  17. J.L. Bona, P.E. Souganidis, W.A. Strauss. Stability and instability of solitary waves of Korteweg de Vries type. Proc. Roy. Soc. London Ser. A411 (1987), 395–412.  
  18. L. Guillopé, M. Zworski. Upper bounds on the number of resonances on noncompact Riemann e surfaces. J. Func. Anal.129 (1995), 364–389.  
  19. T. Kato. Perturbation Theory for Linear Operators. Springer-Verlag New York, 1991.  
  20. W. Hunziker. Resonances, metastable states and exponential decay laws in perturbation theory. Commun. Math. Phys.132 (1990), 177.  
  21. C.E. Kenig, G. Ponce, L. Vega. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math.46 (1993), 527–620.  
  22. J. Holmer, M. Zworski. Soliton interaction with slowly varying potentials. IMRN (2008), doi : , 36 pp.  URI10.1093/imrn/rnn026
  23. Y. Martel, F. Merle. Asymptotic stability of solitons for subcritical gKdV equations revisited. Nonlinearity18 (2005), 55–80.  

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