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Nonlinear modulation near the Lighthill instability threshold in 2+1 Whitham theory

Bridges, Thomas and Ratliff, Daniel (2018) Nonlinear modulation near the Lighthill instability threshold in 2+1 Whitham theory Philosophical Transactions A.

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Abstract

The dispersionless Whitham modulation equations in 2+1 (two space dimensions and time) are reviewed and the instabilities identified. The modulation theory is then reformulated, near the Lighthill instability threshold, with a slow phase, moving frame, and different scalings. The resulting nonlinear phase modulation equation near the Lighthill surfaces is a geometric form of the 2+1 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. For illustration the theory is applied to a complex nonlinear 2+1 Klein-Gordon equation which has two Lighthill surfaces 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 : 9 April 2018
Funders : EPSRC
Identification Number : 10.1098/rsta.2017.0194
Copyright Disclaimer : Copyright 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, stability, phase dynamics, Boussinesq equation
Depositing User : Melanie Hughes
Date Deposited : 09 Jan 2018 14:59
Last Modified : 09 Jan 2018 14:59
URI: http://epubs.surrey.ac.uk/id/eprint/845572

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