Dynamics of solitons and quantized vortices in nonlinear dispersive media

Abstract

A wide range in the systems of nonlinear dispersive medium is governed by the nonlinear SchrÖdinger equation (NSE). The classes of soliton solutions in NSE are determined by the property of the nonlinear potential in NSE. In this paper, the competing cases, where both of attractive and repulsive nonlinearities act simultaneously as a potential, are investigated to examine the instability and interactions of bright solitons, dark solitons, kinks, and quantized vortices.In 1-dimensional space, the kinks play a role of domain wall dividing the space into self-focusing and self-defocusing regions. Numerical analysis shows that the bright (dark) soliton can be exchanged into a dark (bright) one upon collisions with kinks, directly showing the reciprocity between the two different classes of solitons. A quantized vortex in 2-dimensional space is represented by a coherent circular dip with a charge n, in a homogeneous background field with an intensity β. The background field takes a critical role on the vortex stability. For β ＞ $β_c$, a vortex is stable with a finite core size, exhibiting a certain critical behavior as β → $β_c$. For β ＜ $β_c$, vortices become unstable, turning into an ever-expanding circular kink. In fact, this kink turns out to be a circular set of the 1-dimensional kinks. These results give a useful standpoint on the soliton dynamics in such a physical system as saturable optical media or superfluids with multi-body atomic interactions.