Novel homoclinic solutions and the Bäcklund transformation to a (1 + 1)-dimensional coupled nonlinear Schrödinger system in birefringent optical fibers

Nonlinear coupled partial differential equations are widely used in various fields. Notably, in the field of optical transmission, the nonlinear coupled Schrödinger systems play an important role in ultra-long propagation. Therefore, studying the solutions of these systems conduces us to predict and control the optical signals in complex systems. This paper presents three major original contributions: first, we derive a specific homoclinic orbit form by introducing a special transformation into a general coupled nonlinear Schrödinger system. Second, by employing the improved Hirota bilinear method, we obtain various types of soliton solutions. Finally, we perform a Bäcklund transformation on this homoclinic orbit coupled nonlinear Schrödinger system to derive solutions under different coordinate frameworks. The main works on the obtained homoclinic orbit coupled nonlinear Schrödinger system are summarized as follows: First, we obtain the D-operator forms of this new system by using Hirota direct method. Then, we use an improved Hirota method to obtain the bright and dark 1-, 2-, 3-soliton solutions, multiple breathers for this system. In addition, we provide illustrations of different solutions and consider the impact of parameters on the solutions. Finally, the Bäcklund transformation, the properties of 2-breather before and after collision will be determined.