Numerical study of the  Kadomtsev  Petviashvili equation and dispersive shock waves

Tamara Grava; Christian Klein; Giuseppe Pitton

doi:10.1098/rspa.2017.0458

Numerical study of the Kadomtsev Petviashvili equation and dispersive shock waves

Tamara Grava
, Christian Klein
, Giuseppe Pitton

Research output: Contribution to journal › Article (Academic Journal) › peer-review

28 Citations (Scopus)

333 Downloads (Pure)

Abstract

A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev–Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.

Original language	English
Article number	20170458
Number of pages	20
Journal	Proceedings of the Royal Society A: Mathematical and Physical Sciences
Volume	474
Issue number	2210
Early online date	14 Feb 2018
DOIs	https://doi.org/10.1098/rspa.2017.0458
Publication status	Published - 28 Feb 2018

Keywords

Dispersive Shock waves
Kadomtsev Petviashvili equatuon

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10.1098/rspa.2017.0458

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Cite this

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title = "Numerical study of the Kadomtsev Petviashvili equation and dispersive shock waves",

abstract = "A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev–Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schr{\"o}dinger equation in the semiclassical limit. ",

keywords = "Dispersive Shock waves, Kadomtsev Petviashvili equatuon",

author = "Tamara Grava and Christian Klein and Giuseppe Pitton",

year = "2018",

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doi = "10.1098/rspa.2017.0458",

language = "English",

volume = "474",

journal = "Proceedings of the Royal Society A: Mathematical and Physical Sciences",

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TY - JOUR

T1 - Numerical study of the Kadomtsev Petviashvili equation and dispersive shock waves

AU - Grava, Tamara

AU - Klein, Christian

AU - Pitton, Giuseppe

PY - 2018/2/28

Y1 - 2018/2/28

N2 - A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev–Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.

AB - A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev–Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.

KW - Dispersive Shock waves

KW - Kadomtsev Petviashvili equatuon

UR - https://arxiv.org/abs/1706.04104

UR - http://hdl.handle.net/20.500.11767/67822

U2 - 10.1098/rspa.2017.0458

DO - 10.1098/rspa.2017.0458

M3 - Article (Academic Journal)

SN - 0962-8444

VL - 474

JO - Proceedings of the Royal Society A: Mathematical and Physical Sciences

JF - Proceedings of the Royal Society A: Mathematical and Physical Sciences

IS - 2210

M1 - 20170458

ER -