Under investigation is the (3+1)-dimensional coupled nonlinear Schrödinger equations, which describe the propagation of the soliton in the inhomogeneous parity-time (PT)-symmetric coupler with gain or loss. Employing the Hirota method and symbolic computation, we obtain the one- and two-soliton solutions under a variable-coefficient constraint. Bäcklund transformation and the corresponding one-soliton solutions are derived. Via graphic analysis, we observe the linear-, parabolic-, and periodic-shaped solitons with different values of the self-phase modulation and cross-phase modulation. Increase of the diffraction and dispersion leads to the increase of both the soliton amplitudes and the velocities. However, ϱ(z) and γ do not affect the soliton amplitude and velocity, with ϱ(z) being the coupling between the modes propagating in the two fibers and γ describing the PT-balanced gain or loss.
Investigation in this paper is a discrete Ablowitz–Ladik equation, which has certain applications in the electrical and optical systems. Based on the Hirota method, bilinear forms, dark one- and two-soliton solutions for such an equation are obtained. Soliton propagation and collision are graphically presented and analyzed: Dark one soliton is shown to maintain its original amplitude and width during the propagation, and discrete peaks of the propagating soliton are displayed. Overtaking collision between the two solitons with the different amplitudes is observed, and solitons have the same traveling direction with their amplitudes, velocities, and widths unchanged. A head-on collision between the two solitons is illustrated, and shapes of the two solitons keep invariant except for some phase shifts. Asymptotic analysis shows that the collision between the two solitons is elastic.