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Modulational Instability and Rogue Waves in Shallow Water Models

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New Approaches to Nonlinear Waves

Part of the book series: Lecture Notes in Physics ((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.

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics, University College London, London, UK

    R. Grimshaw

  2. Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong

    K. W. Chow & H. N. Chan

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  1. R. Grimshaw
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  2. K. W. Chow
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  3. H. N. Chan
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Correspondence to R. Grimshaw .

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Editors and Affiliations

  1. Institute for Analysis, Johannes Kepler University, Linz, Austria

    Elena Tobisch

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Grimshaw, R., Chow, K.W., Chan, H.N. (2016). Modulational Instability and Rogue Waves in Shallow Water Models. In: Tobisch, E. (eds) New Approaches to Nonlinear Waves. Lecture Notes in Physics, vol 908. Springer, Cham. https://doi.org/10.1007/978-3-319-20690-5_5

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