dorsal/arxiv
View SchemaKinks in the Presence of Rapidly Varying Perturbations
| Authors | Yuri S. Kivshar, Niels Gr\{o}nbech-Jensen, Robert D. Parmentier |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9403004 |
| URL | https://arxiv.org/abs/patt-sol/9403004 |
Abstract
Dynamics of sine-Gordon kinks in the presence of rapidly varying periodic perturbations of different physical origins is described analytically and numerically. The analytical approach is based on asymptotic expansions, and it allows to derive, in a rigorous way, an effective nonlinear equation for the slowly varying field component in any order of the asymptotic procedure as expansions in the small parameter $\omega^{-1}$, $\omega$ being the frequency of the rapidly varying ac driving force. Three physically important examples of such a dynamics, {\em i.e.}, kinks driven by a direct or parametric ac force, and kinks on rotating and oscillating background, are analysed in detail. It is shown that in the main order of the asymptotic procedure the effective equation for the slowly varying field component is {\em a renormalized sine-Gordon equation} in the case of the direct driving force or rotating (but phase-locked to an external ac force) background, and it is {\em the double sine-Gordon equation} for the parametric driving force. The properties of the kinks described by the renormalized nonlinear equations are analysed, and it is demonstrated analytically and numerically which kinds of physical phenomena may be expected in dealing with the renormalized, rather than the unrenormalized, nonlinear dynamics. In particular, we predict several qualitatively new effects which include, {\em e.g.}, the perturbation-induced internal oscillations of the $2\pi-$kink in a parametrically driven sine-Gordon model, and generation of kink motion by a pure ac driving force on a rotating background.
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"date_created": "2026-03-02T18:00:29.160000Z",
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"abstract": "Dynamics of sine-Gordon kinks in the presence of rapidly varying periodic\nperturbations of different physical origins is described analytically and\nnumerically. The analytical approach is based on asymptotic expansions, and it\nallows to derive, in a rigorous way, an effective nonlinear equation for the\nslowly varying field component in any order of the asymptotic procedure as\nexpansions in the small parameter $\\omega^{-1}$, $\\omega$ being the frequency\nof the rapidly varying ac driving force. Three physically important examples of\nsuch a dynamics, {\\em i.e.}, kinks driven by a direct or parametric ac force,\nand kinks on rotating and oscillating background, are analysed in detail. It is\nshown that in the main order of the asymptotic procedure the effective equation\nfor the slowly varying field component is {\\em a renormalized sine-Gordon\nequation} in the case of the direct driving force or rotating (but phase-locked\nto an external ac force) background, and it is {\\em the double sine-Gordon\nequation} for the parametric driving force. The properties of the kinks\ndescribed by the renormalized nonlinear equations are analysed, and it is\ndemonstrated analytically and numerically which kinds of physical phenomena may\nbe expected in dealing with the renormalized, rather than the unrenormalized,\nnonlinear dynamics. In particular, we predict several qualitatively new effects\nwhich include, {\\em e.g.}, the perturbation-induced internal oscillations of\nthe $2\\pi-$kink in a parametrically driven sine-Gordon model, and generation of\nkink motion by a pure ac driving force on a rotating background.",
"arxiv_id": "patt-sol/9403004",
"authors": [
"Yuri S. Kivshar",
"Niels Gr\\{o}nbech-Jensen",
"Robert D. Parmentier"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"title": "Kinks in the Presence of Rapidly Varying Perturbations",
"url": "https://arxiv.org/abs/patt-sol/9403004"
},
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