Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg-de Vries equation

Ratliff, Daniel and Bridges, Thomas (2016) Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg-de Vries equation Proceedings of the Royal Society A, 472 (2196), 0160456.

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Abstract

Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is nondegenerate) modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar KdV equation. The coefficients in the emergent KdV equation have a geometric interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically.

Item Type:	Article
Subjects :	Mathematics
Divisions :	Faculty of Engineering and Physical Sciences > Mathematics
Authors :	Name Email ORCID Ratliff, Daniel d.ratliff@surrey.ac.uk Bridges, Thomas T.Bridges@surrey.ac.uk
Date :	7 December 2016
DOI :	10.1098/rspa.2016.0456
Copyright Disclaimer :	Copyright 2016 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited.
Uncontrolled Keywords :	nonlinear waves, wave action, modulation
Related URLs :	Publisher
Depositing User :	Symplectic Elements
Date Deposited :	25 Nov 2016 10:23
Last Modified :	16 Jan 2019 17:10
URI:	http://epubs.surrey.ac.uk/id/eprint/812957

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