Flux singularities in multiphase wavetrains and the Kadomtsev‐Petviashvili equation with applications to stratified hydrodynamics

This paper illustrates how the singularity of the wave action flux causes the Kadomtsev‐Petviashvili (KP) equation to arise naturally from the modulation of a two‐phased wavetrain, causing the dispersion to emerge from the classical Whitham modulation theory. Interestingly, the coefficients of the r...

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Bibliographic Details
Published in:Studies in applied mathematics (Cambridge) 2019-02, Vol.142 (2), p.109-138
Main Author: Ratliff, Daniel J.
Format: Article
Language:English
Subjects:
asymptotic analysis
Fluid dynamics
Fluid flow
Flux
Hydrodynamics
Lagrangian dynamics
Modulation
nonlinear waves
Singularity (mathematics)
Stratification
Uniform flow
Water waves
water waves and fluid dynamics
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Summary:This paper illustrates how the singularity of the wave action flux causes the Kadomtsev‐Petviashvili (KP) equation to arise naturally from the modulation of a two‐phased wavetrain, causing the dispersion to emerge from the classical Whitham modulation theory. Interestingly, the coefficients of the resulting KP are shown to be related to the associated conservation of wave action for the original wavetrain, and therefore may be obtained prior to the modulation. This provides a universal form for the KP as a dispersive reduction from any Lagrangian with the appropriate wave action flux singularity. The theory is applied to the full water wave problem with two layers of stratification, illustrating how the KP equation arises from the modulation of a uniform flow state and how its coefficients may be extracted from the system.
ISSN:0022-2526
1467-9590
DOI:10.1111/sapm.12242