Whitham modulation theory for the Kadomtsev-Petviashvili equation

Abstract

The genus-1 KP-Whitham system is derived for both variants of the Kadomtsev-Petviashvili (KP) equation (namely, the KPI and KPII equations). The basic properties of the KP-Whitham system, including symmetries, exact reductions, and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg-deVries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable while all genus-1 solutions of KPII {are linearly stable within the context of Whitham theory.

Significantly revised version

References in corpus (5)

• Dispersive shock waves and modulation theory
• Inverse Scattering Problem for Vector Fields and the Cauchy Problem for the Heavenly Equation
• On the solutions of the dKP equation: nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breaking
• Semiclassical limit of the focusing NLS: Whitham equations and the Riemann-Hilbert Problem approach
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Cited by in corpus (12)

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• Whitham modulation theory for the defocusing nonlinear Schrodinger equation in two and three spatial dimensions
• Modulation theory for soliton resonance and Mach reflection
• Oblique interactions between solitons and mean flows in the Kadomtsev-Petviashvili equation
• Integrability, exact reductions and special solutions of the KP-Whitham equations
• Whitham modulation theory for the Zakharov-Kuznetsov equation and transverse instability of its periodic traveling wave solutions
• Obliquely interacting solitary waves and wave wakes in free-surface flows
• Two-dimensional reductions of the Whitham modulation system for the Kadomtsev-Petviashvili equation
• Initial-Boundary Value Problem for Stimulated Raman Scattering Model: Solvability of Whitham Type System of Equations Arising in Long-Time Asymptotic Analysis