Nonlinear theory for coalescing characteristics in multiphase Whitham modulation theory
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 twoway 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 twophase travellingwave solutions of coupled nonlinear Schrödinger equations, highlighting how collisions can be identified and the relevant dispersive dynamics constructed.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.01022
 Bibcode:
 2020arXiv200501022B
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Dynamical Systems;
 35A15;
 76B15;
 37K05;
 35A30;
 37K45
 EPrint:
 40 pages, 2 figures