Bilinear Bäcklund transformation and soliton solutions to a (3+1)-dimensional coupled nonlinear Schrödinger equation with variable coefficients in optical fibers

Optical fiber plays a crucial role in modern information and communication technology. The birefringent fiber can allow multiple independent data streams to be transmitted simultaneously in the same fiber, which significantly improves the bandwidth utilization rate of the communication system and has an important impact on the realization of future high-speed, efficient and low-power communication systems. In this paper, with the help of Hirota bilinear method, the (3+1)-dimensional coupled nonlinear Schrödinger equation with variable coefficients is studied carefully, which shows the evolution of two polarization envelopes in birefringent fibers and plays a significant role in the development of optical communications. At the beginning, the bilinear form of the above equation along with bilinear Bäcklund transformation are derived by the Hirota bilinear method. Subsequently, a diverse range of bright–dark alternating soliton solutions, such as U-type, S-type, kink-type and so on, can be constructed by deciding the appropriate special values of the undetermined parameters. It is worthy to note that the above results have never appeared in previous references. The impacts of some parameters on soliton solutions have been discussed and analyzed in detail by comparing the graphs corresponding to multitudinous soliton solutions obtained above. Furthermore, the situations before and after the collision of two solitons in special cases have been studied by using asymptotic analysis.