This paper delves into the soliton solutions of the \((2+1)\) -dimensional variable-coefficient coupled nonlinear Schrödinger equations within a graded-index waveguide. The propagation of light beams in two-dimensional graded-index nonlinear waveguide amplifiers, incorporating polarization effects, is described by these equations. Utilizing the Hirota bilinear method, novel exact solutions for one-soliton, two-soliton and three-soliton scenarios are derived and graphically represented to facilitate a detailed investigation of the influence exerted by variable coefficients on solitons. By assigning different values to the parameters, breather-like solutions as well as dark-soliton-breather-like solutions are obtained. It is noteworthy that this paper, for the first time, has obtained bright-dark soliton solutions, breather-like solutions and dark-soliton-breather-like solutions using the Hirota bilinear method, all exhibiting alternating bright and dark characteristics, which have not been seen in previous literature.
