Dark solitons and Bäcklund transformation for the (2+1)-dimensional coupled nonlinear Schrödinger equation with the variable coefficients in a graded-index waveguide

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Under investigation in this paper is a set of the (2 + 1)-dimensional coupled nonlinear Schrödinger equations, which describes the propagation of an optical beam inside a (2 + 1)-dimensional graded-index waveguide with the polarization effects. Employing the Hirota method and symbolic computation, we obtain the dark one-, two- and N-soliton solutions under the variable-coefficient constraints. Bäcklund transformation and the corresponding soliton solutions are derived. We graphically study the dark solitons with the influence of the profile function and nonlinearity coefficient. Parallel and period solitons are observed. Amplitudes of the dark solitons increase with the profile function increasing; When the profile function is a periodic function, amplitudes of the dark solitons change periodically. Decrease of the nonlinearity coefficient leads to the larger solitons' velocities, but does not affect the solitons' amplitudes.