Nonlinear Theory for Coalescing Characteristics in Multiphase Whitham Modulation Theory

Bridges, Thomas J. and Ratliff, Daniel (2021) Nonlinear Theory for Coalescing Characteristics in Multiphase Whitham Modulation Theory. Journal of Nonlinear Science, 31 (1). p. 7. ISSN 0938-8974

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Abstract

The multiphase Whitham modulation equations with N phases have 2N characteristics which may be of hyperbolic or elliptic type. In this paper, a nonlinear theory is developed for coalescence, where two characteristics change from hyperbolic to elliptic via collision. Firstly, a linear theory develops the structure of colliding characteristics involving the topological sign of characteristics and multiple Jordan chains, and secondly, a nonlinear modulation theory is developed for transitions. The nonlinear theory shows that coalescing characteristics morph the Whitham equations into an asymptotically valid geometric form of the two-way Boussinesq equation, that is, coalescing characteristics generate dispersion, nonlinearity and complex wave fields. For illustration, the theory is applied to coalescing characteristics associated with the modulation of two-phase travelling wave solutions of coupled nonlinear Schrödinger equations, highlighting how collisions can be identified and the relevant dispersive dynamics constructed.

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