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Modulational Instability and Rogue Waves in Shallow Water Models

  • R. GrimshawEmail author
  • K. W. Chow
  • H. N. Chan
Part of the Lecture Notes in Physics book series (LNP, volume 908)

Abstract

It is now well known that the focussing nonlinear Schrödinger equation allows plane waves to be modulationally unstable, and at the same time supports breather solutions which are often invoked as models for rogue waves. This suggests a direct connection between modulation instability and the existence of rogue waves. In this chapter we review this connection for a suite of long wave models, such as the Korteweg-de Vries equation, the extended Korteweg-de Vries (Gardner) equation, often used to describe surface and internal waves in shallow water, a Boussinesq equation and, also a coupled set of Korteweg-de Vries equations.

Keywords

Modulation Instability Rogue Wave Darboux Transformation Linear Dispersion Relation Breather Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Partial financial support has been provided by the Research Grants Council through contract HKU 711713E.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK
  2. 2.Department of Mechanical EngineeringUniversity of Hong KongPokfulamHong Kong

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