Direct dynamical energy cascade in the modified KdV equation

Denys Dutykh; Elena Tobisch

doi:10.1016/j.physd.2015.01.002

Direct dynamical energy cascade in the modified KdV equation

Denys Dutykh, Elena Tobisch

Mathematics

Johannes Kepler University Linz

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

6 Scopus citations

Abstract

In this study we examine the energy transfer mechanism during the nonlinear stage of the Modulational Instability (MI) in the modified Korteweg-de Vries (mKdV) equation. The particularity of this study consists in considering the problem essentially in the Fourier space. A dynamical energy cascade model of this process originally proposed for the focusing NLS-type equations is transposed to the mKdV setting using the existing connections between the KdV-type and NLS-type equations. The main predictions of the D-cascade model are outlined and validated by direct numerical simulations of the mKdV equation using the pseudo-spectral methods. The nonlinear stages of the MI evolution are also investigated for the mKdV equation.

Original language	British English
Pages (from-to)	76-87
Number of pages	12
Journal	Physica D: Nonlinear Phenomena
Volume	297
DOIs	https://doi.org/10.1016/j.physd.2015.01.002
State	Published - 15 Mar 2015

Keywords

Energy cascade
Korteweg-de Vries equation
Modified KdV equation
Modulational instability
NLS equation

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10.1016/j.physd.2015.01.002

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title = "Direct dynamical energy cascade in the modified KdV equation",

abstract = "In this study we examine the energy transfer mechanism during the nonlinear stage of the Modulational Instability (MI) in the modified Korteweg-de Vries (mKdV) equation. The particularity of this study consists in considering the problem essentially in the Fourier space. A dynamical energy cascade model of this process originally proposed for the focusing NLS-type equations is transposed to the mKdV setting using the existing connections between the KdV-type and NLS-type equations. The main predictions of the D-cascade model are outlined and validated by direct numerical simulations of the mKdV equation using the pseudo-spectral methods. The nonlinear stages of the MI evolution are also investigated for the mKdV equation.",

keywords = "Energy cascade, Korteweg-de Vries equation, Modified KdV equation, Modulational instability, NLS equation",

author = "Denys Dutykh and Elena Tobisch",

note = "Funding Information: The authors would like to thank Professors Mat Johnson (University of Kansas) and Fritz Gesztesy (University of Missouri) for interesting discussions and most useful comments. This research has been supported by the Austrian Science Foundation (FWF) under projects P22943-N18 and P24671. Publisher Copyright: {\textcopyright} 2015 Elsevier B.V. All rights reserved.",

year = "2015",

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day = "15",

doi = "10.1016/j.physd.2015.01.002",

language = "British English",

volume = "297",

pages = "76--87",

journal = "Physica D: Nonlinear Phenomena",

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TY - JOUR

T1 - Direct dynamical energy cascade in the modified KdV equation

AU - Dutykh, Denys

AU - Tobisch, Elena

N1 - Funding Information: The authors would like to thank Professors Mat Johnson (University of Kansas) and Fritz Gesztesy (University of Missouri) for interesting discussions and most useful comments. This research has been supported by the Austrian Science Foundation (FWF) under projects P22943-N18 and P24671. Publisher Copyright: © 2015 Elsevier B.V. All rights reserved.

PY - 2015/3/15

Y1 - 2015/3/15

N2 - In this study we examine the energy transfer mechanism during the nonlinear stage of the Modulational Instability (MI) in the modified Korteweg-de Vries (mKdV) equation. The particularity of this study consists in considering the problem essentially in the Fourier space. A dynamical energy cascade model of this process originally proposed for the focusing NLS-type equations is transposed to the mKdV setting using the existing connections between the KdV-type and NLS-type equations. The main predictions of the D-cascade model are outlined and validated by direct numerical simulations of the mKdV equation using the pseudo-spectral methods. The nonlinear stages of the MI evolution are also investigated for the mKdV equation.

AB - In this study we examine the energy transfer mechanism during the nonlinear stage of the Modulational Instability (MI) in the modified Korteweg-de Vries (mKdV) equation. The particularity of this study consists in considering the problem essentially in the Fourier space. A dynamical energy cascade model of this process originally proposed for the focusing NLS-type equations is transposed to the mKdV setting using the existing connections between the KdV-type and NLS-type equations. The main predictions of the D-cascade model are outlined and validated by direct numerical simulations of the mKdV equation using the pseudo-spectral methods. The nonlinear stages of the MI evolution are also investigated for the mKdV equation.

KW - Energy cascade

KW - Korteweg-de Vries equation

KW - Modified KdV equation

KW - Modulational instability

KW - NLS equation

UR - http://www.scopus.com/inward/record.url?scp=84922572728&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2015.01.002

DO - 10.1016/j.physd.2015.01.002

M3 - Article

AN - SCOPUS:84922572728

SN - 0167-2789

VL - 297

SP - 76

EP - 87

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

ER -

Direct dynamical energy cascade in the modified KdV equation

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Keywords

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