On the Elliptic-Hyperbolic Transition in Whitham Modulation Theory

Bridges, Thomas J. and Ratliff, Daniel (2017) On the Elliptic-Hyperbolic Transition in Whitham Modulation Theory. SIAM Journal on Applied Mathematics, 77 (6). pp. 1989-2011. ISSN 0036-1399

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Official URL: https://doi.org/10.1137/17M1111437

Abstract

The dispersionless Whitham modulation equations in one space dimension and time are generically hyperbolic or elliptic and break down 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, quasi-periodic, and multipulse localized solutions. This theory shows that the elliptic-hyperbolic transition is a rich source of complex behavior 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 traveling waves.

Item Type: Article
Uncontrolled Keywords: nonlinear waves, modulation, Lagrangian, multisymplectic, traveling waves
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
Depositing User: John Coen
Date Deposited: 17 Aug 2020 09:25
Last Modified: 17 Aug 2020 10:33
URI: http://nrl.northumbria.ac.uk/id/eprint/44099

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