Soliton interactions in perturbed nonlinear Schrödinger equations
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 onedimensional 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.
 Publication:

Physical Review E
 Pub Date:
 June 2000
 DOI:
 10.1103/PhysRevE.61.7121
 arXiv:
 arXiv:pattsol/9906009
 Bibcode:
 2000PhRvE..61.7121B
 Keywords:

 42.65.Tg;
 42.81.Dp;
 02.30.Jr;
 Optical solitons;
 nonlinear guided waves;
 Propagation scattering and losses;
 solitons;
 Partial differential equations;
 Nonlinear Sciences  Pattern Formation and Solitons
 EPrint:
 28 pages, 7 figures, Submitted to Phys Rev E Revised: 21 pages, 6 figures, To appear in Phys Rev E (many displayed equations moved inline to shorten manuscript)