Abstract
Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg–de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically.
Highlights
There is a multitude of physical examples where multiphase wavetrains arise naturally
The second example shows how a single Korteweg–de Vries (KdV) equation can emerge at criticality for coupled nonlinear Schrödinger (NLS) equations, showing how the mathematical theory can evoke new results for the physical system, without the need to understand the physical mechanism behind the criticality condition for NLS
The multiphase modulation theory of the paper is used to show that there is a large range of parameters where the coupled NLS equation can be reduced to a single KdV equation
Summary
There is a multitude of physical examples where multiphase wavetrains arise naturally. The strategy, in this paper, for generating dispersion in modulation equations deduced from the multiphase conservation of wave action is motivated by the theory for the single-phase case studied in [5,6,7], but has subtle new features. Projection of (1.12) onto the complement of the kernel of DkB provides an equation determining α, which would be used at the order This ansatz (1.10) looks like a straightforward generalization of the ansatz in the single-phase case, there are subtle non-trivial differences in the multiphase theory. The theory starts with a conservative PDE, generated by a Lagrangian in canonical form (1.1)–(1.2), with a four-parameter family of two-phase wavetrains (1.4) When this fourparameter family has a simple degeneracy (1.8), the modulation ansatz (1.10) satisfies the.
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