Whitham modulation theory for the two-dimensional Benjamin-Ono equation

TL;DR

This paper develops a Whitham modulation theory for the 2D Benjamin-Ono equation, deriving a system of PDEs to analyze wave modulations and stability, including exact reductions and initial value problem formulations.

Contribution

It introduces a novel 2DBO-Whitham system that models wave modulations and transverse stability, extending Whitham theory to two-dimensional nonlocal equations.

Findings

Derived a system of five quasi-linear PDEs for 2DBO wave modulations.

Transformed the system into a singularity-free hydrodynamic-like form.

Discussed exact reductions and initial value problem formulations.

Abstract

Whitham modulation theory for the two dimensional Benjamin-Ono (2DBO) equation is presented. A system of five quasi-linear first-order partial differential equations is derived. The system describes modulations of the traveling wave solutions of the 2DBO equation. These equations are transformed to a singularity-free hyrdodynamic-like system referred to here as the 2DBO-Whitham system. Exact reductions of this system are discussed, the formulation of initial value problems is considered, and the system is used to study the transverse stability of traveling wave solutions of the 2DBO equation.