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ON THE ELLIPTIC-HYPERBOLIC TRANSITION IN WHITHAM MODULATION THEORY

Bridges, Thomas and Ratliff, Daniel (2017) ON THE ELLIPTIC-HYPERBOLIC TRANSITION IN WHITHAM MODULATION THEORY SIAM Journal on Applied Mathematics.

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Abstract

The dispersionless Whitham modulation equations in one space dimension and time are generically hyperbolic or elliptic, and breakdown at the transition, which is a curve in the frequency-wavenumber plane. In this paper, the modulation theory is reformulated with a slow phase and different scalings resulting in a phase modulation equation near the singular curves which is a geometric form of the two-way Boussinesq equation. This equation is universal in the same sense as Whitham theory. Moreover, it is dispersive, and it has a wide range of interesting multiperiodic, quasiperiodic and multi-pulse localized solutions. This theory shows that the elliptic-hyperbolic transition is a rich source of complex behaviour in nonlinear wave fields. There are several examples of these transition curves in the literature to which the theory applies. For illustration the theory is applied to the complex nonlinear Klein-Gordon equation which has two singular curves in the manifold of periodic travelling waves.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Bridges, ThomasT.Bridges@surrey.ac.ukUNSPECIFIED
Ratliff, Danield.ratliff@surrey.ac.ukUNSPECIFIED
Date : 29 September 2017
Copyright Disclaimer : © 2017, Society for Industrial and Applied Mathematics
Depositing User : Melanie Hughes
Date Deposited : 29 Jun 2017 09:45
Last Modified : 29 Jun 2017 09:45
URI: http://epubs.surrey.ac.uk/id/eprint/841504

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