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Double Degeneracy in Multiphase Modulation and the Emergence of the Boussinesq Equation

Ratliff, Daniel (2017) Double Degeneracy in Multiphase Modulation and the Emergence of the Boussinesq Equation Studies in Applied Mathematics.

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Abstract

In recent years, a connection between conservation law singularity, or more generally zero characteristics arising within the linear Whitham equations, and the emergence of reduced nonlinear partial differential equations (PDEs) from systems generated by a Lagrangian density has been made in conservative systems. Remarkably, the conservation laws form part of the reduced nonlinear system. Within this paper, the case of double degeneracy is investigated in multiphase wavetrains, characterized by a double zero characteristic of the linearized Whitham system, with the resulting modulation of relative equilibrium (which are a generalization of the modulation of wavetrains) leading to a vector two-way Boussinesq equation. The derived PDE adheres to the previous results (such as [1]) in the sense that all but one of its coefficients is related to the conservation laws along the relative equilibrium solution, which are then projected to form a corresponding scalar system. The theory is applied to two examples to highlight how both the criticality can be assessed and the two-way Boussinesq equation's coefficients are obtained. The first is the coupled Nonlinear Schrodinger (NLS) system and is the first time the two-way Boussinesq equation has been shown to arise in such a context, and the second is a stratified shallow water model which validates the theory against existing results.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Ratliff, Danield.ratliff@surrey.ac.ukUNSPECIFIED
Date : 21 August 2017
Identification Number : 10.1111/sapm.12189
Copyright Disclaimer : © 2017 Wiley Periodicals, Inc., A Wiley Company.
Uncontrolled Keywords : Whitham modulation; Lagrangian dynamics; Nonlinear waves; Partial differential equations; Asymptotic analysis
Depositing User : Clive Harris
Date Deposited : 30 Aug 2017 07:50
Last Modified : 30 Aug 2017 07:50
URI: http://epubs.surrey.ac.uk/id/eprint/842077

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