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

TL;DR

This paper develops a unified Whitham modulation theory for (2+1)-dimensional KP-type equations, including non-integrable cases, deriving systems of PDEs to analyze wave stability and dynamics.

Contribution

It provides a unified derivation of Whitham systems for KP, 2DBO, and m2KP equations, including non-integrable cases, with new insights into their features.

Findings

Derived five-equation Whitham systems for each equation

Analyzed the Riemann problem for m2KP

Studied linear stability of traveling waves

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.