Solitons for the (3+1)-dimensional variable-coefficient coupled nonlinear Schrödinger equations in an optical fiber

In this paper, the (3 + 1)-dimensional variable-coefficient coupled nonlinear Schrödinger equations are investigated, which describe the evolution of two polarization envelopes in an optical fiber with birefringence. Under the integrable constraint on the variable coefficients, with the aid of the Hirota method and auxiliary function, bilinear forms and soliton solutions are derived. In addition, propagation and interaction of the solitons are discussed graphically. Linear- and cubic-type solitons are obtained when the diffraction coefficient α(t) is a constant or a square function of the local time t, and we find that α(t) can affect the soliton velocity, but the soliton amplitude remains unchanged. Two parabolic-type solitons are obtained when α(t) is a linear function, and we notice that the interaction between the two solitons do not affect the amplitudes and velocities of each soliton, except for a phase shift, indicating that the interaction between the two solitons is elastic.