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arXiv · 2005.01022

Nonlinear theory for coalescing characteristics in multiphase Whitham modulation theory

Thomas J. Bridges·Daniel J. Ratliff
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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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Thomas J. Bridges, Daniel J. Ratliff. 2020-11-29. Nonlinear theory for coalescing characteristics in multiphase Whitham modulation theory. https://doi.org/10.1007/s00332-020-09669-y

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