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.
Highlights
The theory of modulation, Whitham modulation theory, takes the existing nonlinear waves, such as finite-amplitude periodic travelling waves, and provides a Communicated by Peter Miller
We find that the form of the two-way Boussinesq equation (1.9) carries over to the case of coalescing characteristics with nonzero speed, but there is a discrepancy between the fact that (1.9) is scalar-valued but the Whitham modulation equations (WMEs) in the multiphase case have 2N equations
There is a conservation law associated with each phase of the wavetrain, and multisymplectic Noether theory implies the existence of functions A j, B j satisfying
Summary
Journal of Nonlinear Science (2021) 31:7 framework for studying the dynamical implications of perturbing the basic properties of the nonlinear wave. The properties of the basic state (wavenumber, frequency, mean flow) are allowed to depend on space and time, and partial differential equations (PDEs) are derived for these parameters. The Whitham modulation equations (WMEs) can either be hyperbolic (real characteristics) or elliptic (complex characteristics) and the transition signals a change of stability of the underlying periodic waves (Whitham 1965, 1974; Bridges and Ratliff 2017, 2018). It is this change of type, and its generalization to multiphase wavetrains, and its nonlinear implications, that are the main themes of this paper. The complexity is due to the wide range of known localized, multi-pulse, quasiperiodic, and extreme value solutions of the two-way Boussinesq equation
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