Numerical approaches to solving the Schrödinger non-linear equations system for wave propagation in an optical fiber

The paper discusses approaches to the numerical integration of the second-kind Manakov equation system. Emphasis is placed on the transition from writing equations in dimensional quantities to equations in dimensionless units. A combined explicit-implicit finite-difference integration scheme based on the implicit Crank- Nicolson finite-difference scheme is proposed and substantiated, which allows integrating a non-linear system of equations with a choice of non-linear term at the previous integration step. An algorithm for leveling the disadvantage associated with the definition of the nonlinear term from the previous integration step is proposed. The approach of automatic selection of the integration step, which reduces the total number of integration steps while maintaining the required accuracy of the approximate solution, is substantiated. Examples of the calculation results for some values of the disturbance propagation are given. The limitations imposed by the scheme on the length of the integrable fiber section are described, and approaches are proposed that eliminate these limitations without the need to increase the dimensions of the finite-difference scheme arrays. Requirements for initial boundary conditions were discussed.