N-soliton solutions of four-wave mixing coupled Schrödinger equations based on Riemann–Hilbert method

In this paper, a class of variable-coefficient coupled nonlinear Schrödinger equations with four-wave mixing effect is studied. Firstly, the constraint conditions that the function should satisfy when the equation is integrable are given by Painlevé analysis, and then the $N$-soliton solution of the equation with variable coefficients was directly given by using the variable substitution technique and the Riemann–Hilbert method. On this basis, the evolution figure of the 1, 2-soliton solution was given by numerical simulation. The effects of functions and related parameters on the soliton solution dynamics are analyzed and summarized. Through our research, we can provide a certain theoretical basis for the control and application of solitons in practice.