Dispersive shock waves in the Kadomtsev-Petviashvili and Two Dimensional Benjamin-Ono equations

TL;DR

This paper demonstrates that dispersive shock waves in the (2+1) dimensional KP and 2DBO equations can be effectively studied by reducing them to (1+1) dimensional cylindrical KdV and Benjamin-Ono equations using a parabolic front ansatz, with results validated through numerical simulations.

Contribution

The study introduces an exact reduction method for (2+1)D DSWs to (1+1)D equations and derives Whitham modulation equations for these reduced models, providing a new approach to analyze complex wave phenomena.

Findings

Excellent agreement between reduced models and full equations in DSW behavior.

Numerical simulations confirm the effectiveness of the reduction approach.

Reduced (1+1)D equations accurately describe (2+1)D DSW evolution along parabolic fronts.

Abstract

Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and two dimensional Benjamin-Ono (2DBO) equation are considered using parabolic front initial data. Employing a front tracking type ansatz exactly reduces the study of DSWs in two space one time (2+1) dimensions to finding DSW solutions of (1+1) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to cylindrical Korteweg-de Vries (cKdV) and cylindrical Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived in general and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with excellent agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with remarkable agreement. It is concluded that the (2+1) DSW behavior along parabolic fronts can be effectively described by the DSW solutions of the reduced (1+1) dimensional equations.