Whitham modulation theory for (2+1)-dimensional equations of Kadomtsev-Petviashvili type

Abstract

Whitham modulation theory for certain two-dimensional evolution equations of Kadomtsev-Petviashvili (KP) type is presented. Three specific examples are considered in detail: the KP equation, the two-dimensional Benjamin-Ono (2DBO) equation and a modified KP (m2KP) equation. A unified derivation is also provided. In the case of the m2KP equation, the corresponding Whitham modulation system exhibits features different from the other two. The approach presented here does not require integrability of the original evolution equation. Indeed, while the KP equation is known to be a completely integrable equation, the 2DBO equation and the m2KP equation are not known to be integrable. In each of the cases considered, the Whitham modulation system obtained consists of five first-order quasilinear partial differential equations. The Riemann problem (i.e. the analogue of the Gurevich-Pitaevskii problem) for the one-dimensional reduction of the m2KP equation is studied. For the m2KP equation, the system of modulation equations is used to analyze the linear stability of traveling wave solutions.

arXiv comment27 pages, 2 figures

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Cited by in corpus (7)

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• Whitham modulation theory for the defocusing nonlinear Schrodinger equation in two and three spatial dimensions
• Oblique interactions between solitons and mean flows in the Kadomtsev-Petviashvili equation
• Nonlinear Modulation of Periodic Waves in the Cylindrical Gardner 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
• Two-dimensional reductions of the Whitham modulation system for the Kadomtsev-Petviashvili equation