dorsal/arxiv
View SchemaSoliton Interactions in Perturbed Nonlinear Schroedinger Equations
| Authors | James A. Besley, Peter D. Miller, Nail N. Akhmediev |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9906009 |
| URL | https://arxiv.org/abs/patt-sol/9906009 |
| DOI | 10.1103/PhysRevE.61.7121 |
Abstract
We use multiscale perturbation theory in conjunction with the inverse scattering transform to study the interaction of a number of solitons of the cubic nonlinear Schroedinger equation under the influence of a small correction to the nonlinear potential. We assume that the solitons are all moving with the same velocity at the initial instant; this maximizes the effect each soliton has on the others as a consequence of the perturbation. Over the long time scales that we consider, the amplitudes of the solitons remain fixed, while their center of mass coordinates obey Newton's equations with a force law for which we present an integral formula. For the interaction of two solitons with a quintic perturbation term we present more details since symmetries -- one related to the form of the perturbation and one related to the small number of particles involved -- allow the problem to be reduced to a one-dimensional one with a single parameter, an effective mass. The main results include calculations of the binding energy and oscillation frequency of nearby solitons in the stable case when the perturbation is an attractive correction to the potential and of the asymptotic "ejection" velocity in the unstable case. Numerical experiments illustrate the accuracy of the perturbative calculations and indicate their range of validity.
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"abstract": "We use multiscale perturbation theory in conjunction with the inverse\nscattering transform to study the interaction of a number of solitons of the\ncubic nonlinear Schroedinger equation under the influence of a small correction\nto the nonlinear potential. We assume that the solitons are all moving with the\nsame velocity at the initial instant; this maximizes the effect each soliton\nhas on the others as a consequence of the perturbation. Over the long time\nscales that we consider, the amplitudes of the solitons remain fixed, while\ntheir center of mass coordinates obey Newton\u0027s equations with a force law for\nwhich we present an integral formula. For the interaction of two solitons with\na quintic perturbation term we present more details since symmetries -- one\nrelated to the form of the perturbation and one related to the small number of\nparticles involved -- allow the problem to be reduced to a one-dimensional one\nwith a single parameter, an effective mass. The main results include\ncalculations of the binding energy and oscillation frequency of nearby solitons\nin the stable case when the perturbation is an attractive correction to the\npotential and of the asymptotic \"ejection\" velocity in the unstable case.\nNumerical experiments illustrate the accuracy of the perturbative calculations\nand indicate their range of validity.",
"arxiv_id": "patt-sol/9906009",
"authors": [
"James A. Besley",
"Peter D. Miller",
"Nail N. Akhmediev"
],
"categories": [
"patt-sol",
"nlin.PS"
],
"doi": "10.1103/PhysRevE.61.7121",
"title": "Soliton Interactions in Perturbed Nonlinear Schroedinger Equations",
"url": "https://arxiv.org/abs/patt-sol/9906009"
},
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