Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional Benjamin–Ono equations

Ablowitz, Mark, Demirci, Ali and Ma, Yi-Ping (2016) Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional Benjamin–Ono equations. Physica D: Nonlinear Phenomena, 333. pp. 84-98. ISSN 0167-2789

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Official URL: http://dx.doi.org/10.1016/j.physd.2016.01.013

Abstract

Dispersive shock waves (DSWs) in the Kadomtsev–Petviashvili (KP) equation and two dimensional Benjamin–Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such 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 the 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 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 very good 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 good agreement. It is concluded that the (2+1) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced (1+1) dimensional equations.

Item Type: Article
Uncontrolled Keywords: Dispersive shock waves; Kadomtsev–Petviashvili equation; Two dimensional Benjamin–Ono equation
Subjects: F300 Physics
G100 Mathematics
Department: Faculties > Engineering and Environment > Mathematics and Information Sciences
Depositing User: Yi-Ping Ma
Date Deposited: 10 Nov 2016 08:48
Last Modified: 17 Nov 2016 08:33
URI: http://nrl.northumbria.ac.uk/id/eprint/28403

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