Soliton interactions in perturbed nonlinear Schrödinger equations

James A. Besley*, Peter D. Miller, Nail N. Akhmediev

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    19 Citations (Scopus)

    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 Schrödinger 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 soliton amplitudes 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.

    Original languageEnglish
    Pages (from-to)7121-7133
    Number of pages13
    JournalPhysical Review E
    Volume61
    Issue number6
    DOIs
    Publication statusPublished - 2000

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