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

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)
2 Downloads (Pure)

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.
Original languageEnglish
Pages (from-to)48-77
Number of pages30
JournalStudies in Applied Mathematics
Volume140
Issue number1
Early online date21 Aug 2017
DOIs
Publication statusPublished - 1 Jan 2018
Externally publishedYes

Fingerprint Dive into the research topics of 'Double Degeneracy in Multiphase Modulation and the Emergence of the Boussinesq Equation: Double Degeneracy and the Boussinesq Equation'. Together they form a unique fingerprint.

Cite this