Dispersive shock waves in Three Dimensional Benjamin-Ono equation

TL;DR

This paper investigates dispersive shock waves in the three-dimensional Benjamin-Ono equation by reducing it to a spherical form, deriving Whitham equations, and comparing numerical solutions to understand DSW behavior along paraboloid fronts.

Contribution

The study introduces a reduction of the 3D Benjamin-Ono equation to a spherical form and derives Whitham modulation equations to analyze DSW evolution.

Findings

Good agreement between Whitham system solutions and direct numerical simulations.

Reduction to sBO equation effectively captures DSW behavior in 3D Benjamin-Ono equation.

Provides qualitative insights into DSW dynamics along paraboloid fronts.

Abstract

Dispersive shock waves (DSWs) in the three dimensional Benjamin- Ono (3DBO) equation is studied with step-like initial condition along a paraboloid front. By using a similarity reduction, problem of studying DSWs in three space one time (3+1) dimensions reduces to finding DSW solution of a (1+1) dimensional equation. By using a special ansatz, the 3DBO equation exactly reduces to the spherical Benjamin-Ono (sBO) equation. Whitham modulation equations are derived which describes DSW evolution in the sBO equation by using a perturbation method and these equations are written in terms of appropriate Riemmann type variables to obtain the sBO- Whitham system. DSW solution which obtained from the numerical solutions of the Whitham system and the direct numerical solution of the sBO equation are compared. In this comparison, a good agreement is found between these solutions. Also, some physical qualitative results about DSWs in sBO equation are presented. It is concluded that DSW solutions in the reduced sBO equation provide some information about DSW behaviour along the paraboloid fronts in the 3DBO equation.