Phase dynamics of periodic waves leading to the Kadomtsev–Petviashvili equation in 3+1 dimensions

Ratliff, Daniel and Bridges, Thomas J. (2015) Phase dynamics of periodic waves leading to the Kadomtsev–Petviashvili equation in 3+1 dimensions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 471 (2178). p. 20150137. ISSN 1364-5021

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Official URL: https://doi.org/10.1098/rspa.2015.0137

Abstract

The Kadomstev–Petviashvili (KP) equation is a well-known modulation equation normally derived by starting with the trivial state and an appropriate dispersion relation. In this paper, it is shown that the KP equation is also the relevant modulation equation for bifurcation from periodic travelling waves when the wave action flux has a critical point. Moreover, the emergent KP equation arises in a universal form, with the coefficients determined by the components of the conservation of wave action. The theory is derived for a general class of partial differential equations generated by a Lagrangian using phase modulation. The theory extends to any space dimension and time, but the emphasis in the paper is on the case of 3+1. Motivated by light bullets and quantum vortex dynamics, the theory is illustrated by showing how defocusing NLS in 3+1 bifurcates to KP in 3+1 at criticality. The generalization to N>3 is also discussed.

Item Type: Article
Uncontrolled Keywords: nonlinear waves, Lagrangian systems, travelling waves, wave action, modulation
Subjects: G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics, Physics and Electrical Engineering
Depositing User: John Coen
Date Deposited: 17 Aug 2020 10:51
Last Modified: 17 Aug 2020 11:00
URI: http://nrl.northumbria.ac.uk/id/eprint/44104

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