Interaction Between Two Nonlinear Schrödinger Solitons

Abstract

The interaction between nonlinear Schrijdinger solitons is derived by the least action principle approach as a potential function of the soliton's separation and their initial relative phase, which shows clearly how the solitons interact with each other. Two solitons with the same initial phase always attract each other, while those of opposite phase repel. The method developed in this paper can be extended t o deal with interaction between solitons of other nonlinear equations.

Collective coordinate approximation to the scattering of solitons in the (1+1) dimensional NLS model

“…Where ψ 1 and ψ 2 are solutions of (2.2) when they are far apart. Following from Zou and Yan, [9], we assume that the two solitons are of equal height, constant width, and move symmetrically around their centre of mass. So we take ψ 1 = ϕ 1 e −iθ 1 and ψ 2 = ϕ 2 e iθ 2 where…”

“…Hence, the equation has also been studied numerically [5], [6], [7], [8] and an attempt has been made to introduce a collective coordinate approximation to a two soliton field configuration [9]. Several other papers have also looked at NLS solitons perturbed by external fields or in interaction with them [10] but though very interesting, these papers have not approximated the dynamics of the system of solitons by a full Lagrangian based collective coordinate model [11], which has recently been shown [12], [13] (in relativistic models) to be a very good approximation for the investigation of soliton dynamics.…”

“…This paper is organised as follows: in section 2 we give a brief introduction to the NLS model, its basic symmetries and its 1-soliton solution. In section 3, for completeness, we say a few words about the collective coordinate approximation in general, and in section 4 we present our 2-soliton approximation ansatz (based on [9]) and use it to determine the equations of motion for our collective coordinates. We have solved these equations numerically using the 4th order Runge Kutta method, and in section 5, we present some of our results.…”

“…Having performed some numerical simulations of the scattering of two solitons in a class of modified NLS models [14] we have started thinking of a collective coordinate approximation to this process and we have found the paper by Zou and Yan [9]. As this paper does not present many explicit results we have modified its approach a little and have looked at the interaction of two solitons in some detail.…”

Collective coordinate approximation to the scattering of solitons in modified NLS and sine-Gordon models

“…Solving these ODEs describes the time evolution of the coordinates, which in turn tells us how the field evolves in time. In some cases, and sometimes with further simplifying assumptions, the equations of motion for the collective coordinates can be solved analytically; such is the case in [13]. In our work the equations of motion need to be solved numerically and for this we use a 4th order Runge-Kutta method.…”