Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis

N. M. Ercolani; D. W. McLaughlin; H. Roitner

doi:10.1007/BF02429875

Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis

N. M. Ercolani, D. W. McLaughlin, H. Roitner

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

Abstract

In this paper we rigorously show the existence and smoothness in ε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameter ε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions.

Original language	English (US)
Pages (from-to)	477-539
Number of pages	63
Journal	Journal of Nonlinear Science
Volume	3
Issue number	1
DOIs	https://doi.org/10.1007/BF02429875
State	Published - Dec 1993

Keywords

attractors
nearly integrable systems
numerical methods
spectral transform
stability
traveling waves

ASJC Scopus subject areas

Modeling and Simulation
General Engineering
Applied Mathematics

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10.1007/BF02429875

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title = "Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis",

abstract = "In this paper we rigorously show the existence and smoothness in ε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameter ε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions.",

keywords = "attractors, nearly integrable systems, numerical methods, spectral transform, stability, traveling waves",

author = "Ercolani, {N. M.} and McLaughlin, {D. W.} and H. Roitner",

year = "1993",

month = dec,

doi = "10.1007/BF02429875",

language = "English (US)",

volume = "3",

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journal = "Journal of Nonlinear Science",

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TY - JOUR

T1 - Attractors and transients for a perturbed periodic KdV equation

T2 - A nonlinear spectral analysis

AU - Ercolani, N. M.

AU - McLaughlin, D. W.

AU - Roitner, H.

PY - 1993/12

Y1 - 1993/12

N2 - In this paper we rigorously show the existence and smoothness in ε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameter ε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions.

AB - In this paper we rigorously show the existence and smoothness in ε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameter ε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions.

KW - attractors

KW - nearly integrable systems

KW - numerical methods

KW - spectral transform

KW - stability

KW - traveling waves

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DO - 10.1007/BF02429875

M3 - Article

AN - SCOPUS:51249165861

SN - 0938-8974

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SP - 477

EP - 539

JO - Journal of Nonlinear Science

JF - Journal of Nonlinear Science

IS - 1

ER -

Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis

Abstract

Keywords

ASJC Scopus subject areas

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