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

Mark J. Ablowitz; Ali Demirci; Yi Ping Ma

doi:10.1016/j.physd.2016.01.013

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

Mark J. Ablowitz, Ali Demirci^*, Yi Ping Ma

^*Corresponding author for this work

Department of Mathematics Engineering

University of Colorado Boulder

Research output: Contribution to journal › Article › peer-review

29 Citations (Scopus)

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.

Original language	English
Pages (from-to)	84-98
Number of pages	15
Journal	Physica D: Nonlinear Phenomena
Volume	333
DOIs	https://doi.org/10.1016/j.physd.2016.01.013
Publication status	Published - 15 Oct 2016

Bibliographical note

Publisher Copyright:
© 2016 Elsevier B.V.

Funding

This research was partially supported by the US Air Force Office of Scientific Research , under grant FA9550-16-1-0041 and by the National Science Foundation (NSF) under grant DMS-1310200 and by the Scientific and Technological Research Council of Turkey (TUBITAK) under grant 1059B-19140044 . We thank I. Rumanov for numerous valuables comments and D.E. Baldwin for MATLAB codes of version of the ETDRK4 method that we use in the study. This research was partially supported by the US Air Force Office of Scientific Research, under grant FA9550-16-1-0041 and by the National Science Foundation (NSF) under grant DMS-1310200 and by the Scientific and Technological Research Council of Turkey (TUBITAK) under grant 1059B-19140044. We thank I. Rumanov for numerous valuables comments and D.E. Baldwin for MATLAB codes of version of the ETDRK4 method that we use in the study.

Funders	Funder number
TUBITAK	1059B-19140044
National Science Foundation	DMS-1310200
Directorate for Mathematical and Physical Sciences	1310200
Air Force Office of Scientific Research	FA9550-16-1-0041
Türkiye Bilimsel ve Teknolojik Araştirma Kurumu
National Science Foundation

Keywords

Dispersive shock waves
Kadomtsev–Petviashvili equation
Two dimensional Benjamin–Ono equation

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10.1016/j.physd.2016.01.013

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title = "Dispersive shock waves in the Kadomtsev–Petviashvili and two dimensional Benjamin–Ono equations",

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.",

keywords = "Dispersive shock waves, Kadomtsev–Petviashvili equation, Two dimensional Benjamin–Ono equation",

author = "Ablowitz, \{Mark J.\} and Ali Demirci and Ma, \{Yi Ping\}",

note = "Publisher Copyright: {\textcopyright} 2016 Elsevier B.V.",

year = "2016",

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doi = "10.1016/j.physd.2016.01.013",

language = "English",

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AU - Ma, Yi Ping

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N2 - 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.

AB - 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.

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KW - Two dimensional Benjamin–Ono equation

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

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Bibliographical note

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Keywords

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