Interaction of sine-Gordon kinks with defects: The two-bounce resonance

Roy H. Goodman; Richard Haberman

doi:10.1016/j.physd.2004.04.002

Interaction of sine-Gordon kinks with defects: The two-bounce resonance

Roy H. Goodman, Richard Haberman

Mathematical Sciences

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59 Scopus citations

Abstract

A model of soliton-defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown to be exponentially small for weak defects. Matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem, which are then used to derive analytical formulas for the locations of the well known two- and three-bounce resonance windows, as well as several other phenomena seen in numerical simulations.

Original language	English (US)
Pages (from-to)	303-323
Number of pages	21
Journal	Physica D: Nonlinear Phenomena
Volume	195
Issue number	3-4
DOIs	https://doi.org/10.1016/j.physd.2004.04.002
State	Published - Aug 15 2004

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Mathematical Physics
Condensed Matter Physics
Applied Mathematics

Keywords

Sine-Gordon kinks
Singular perturbation theory
Two-bounce resonance

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10.1016/j.physd.2004.04.002

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title = "Interaction of sine-Gordon kinks with defects: The two-bounce resonance",

abstract = "A model of soliton-defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown to be exponentially small for weak defects. Matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem, which are then used to derive analytical formulas for the locations of the well known two- and three-bounce resonance windows, as well as several other phenomena seen in numerical simulations.",

keywords = "Sine-Gordon kinks, Singular perturbation theory, Two-bounce resonance",

author = "Goodman, {Roy H.} and Richard Haberman",

note = "Funding Information: We would like to thank Phil Holmes, Michael Weinstein, Greg Kriegsmann, and Chris Raymond for helpful discussions. RG was supported by NSF DMS-0204881 and by an SBR grant from NJIT. ",

year = "2004",

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doi = "10.1016/j.physd.2004.04.002",

language = "English (US)",

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T1 - Interaction of sine-Gordon kinks with defects

T2 - The two-bounce resonance

AU - Goodman, Roy H.

AU - Haberman, Richard

N1 - Funding Information: We would like to thank Phil Holmes, Michael Weinstein, Greg Kriegsmann, and Chris Raymond for helpful discussions. RG was supported by NSF DMS-0204881 and by an SBR grant from NJIT.

PY - 2004/8/15

Y1 - 2004/8/15

N2 - A model of soliton-defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown to be exponentially small for weak defects. Matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem, which are then used to derive analytical formulas for the locations of the well known two- and three-bounce resonance windows, as well as several other phenomena seen in numerical simulations.

AB - A model of soliton-defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown to be exponentially small for weak defects. Matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem, which are then used to derive analytical formulas for the locations of the well known two- and three-bounce resonance windows, as well as several other phenomena seen in numerical simulations.

KW - Sine-Gordon kinks

KW - Singular perturbation theory

KW - Two-bounce resonance

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DO - 10.1016/j.physd.2004.04.002

M3 - Article

AN - SCOPUS:3342892305

SN - 0167-2789

VL - 195

SP - 303

EP - 323

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 3-4

ER -

Interaction of sine-Gordon kinks with defects: The two-bounce resonance

Abstract

All Science Journal Classification (ASJC) codes

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