Bright solitons for the (2+1)-dimensional coupled nonlinear Schrödinger equations in a graded-index waveguide

In this paper, we investigate the (2+1)-dimensional coupled nonlinear Schrödinger equations with variable coefficients, which describe the propagation of an optical beam inside the two-dimensional graded-index nonlinear waveguide amplifier with the polarization effects. Under certain transformation and constraints, bilinear forms, bright one- and two-soliton solutions are obtained, while soliton propagation and collision with the function γ(t), representing the gain/loss coefficient, profile function and nonlinearity coefficient, are graphically presented and analyzed, where t is the propagation distance. One soliton is shown to maintain its amplitude and width during the propagation when γ(t) is a constant. Choosing γ(t) as a function, soliton's amplitude and width alter during the propagation and the periodic oscillating soliton is observed with γ(t) as a trigonometric function. Properties of the soliton collision is revealed based on the two solitons, and the elastic collision is presented when γ(t) is a constant. With γ(t) as a function, the two solitons' amplitudes and widths do not keep invariant during the collision.