Bright-dark solitons for a set of the general coupled nonlinear Schrödinger equations in a birefringent fiber

Abstract

Under investigation in this paper is the coupled nonlinear Schrödinger equations with the four-wave mixing term, which describe the optical solitons in a birefringent fiber. Via the Kadomtsev-Petviashvili hierarchy reduction, we obtain the N-bright-dark soliton solutions in terms of the Gram determinant. Propagation and interaction of the solitons corresponding to the electric fields in the two orthogonal polarizations are discussed and presented graphically. We find that the one bright-dark soliton possesses the periodic oscillation and exhibits the breather-like profile, which is different from that in the previous literature. Besides, for the one soliton, we observe that the larger velocity leads to the fiercer oscillation. Elastic interactions including the head-on and overtaking interactions between the two bright-dark solitons are demonstrated. Particularly, we find the oblique inelastic interaction between the two bright-dark solitons, which possess the V-shape profile in the zero background component and the Y-shape profile in the nonzero background component. Besides, we present two cases of the bound-state solitons. For the one case, the two solitons interact with each other all the time along a direction and for the other case, the resonance phenomenon is raised.

Numerical analysis for coupled systems of two-dimensional time-space fractional Schrödinger equations with trapping potentials

“…The problem will be considered in both classical and general forms. General form of the model is able to include terms representing the four wave mixing effect as [7] Here, a and c are real constants scaling self-phase modulation and cross-phase modulation respectively, whereas complex constant b denotes four wave mixing effect [8] and V(x,y) is a real valued trapping potential. The four-wave mixing effect can be defined as an inter modulation phenomenon in nonlinear optics, whereby the interactions between the two or three wavelengths produce one or two new wavelengths [9].…”