Numerical simulation of the soliton solutions for a complex modified Korteweg–de Vries equation by a finite difference method

In this paper, a Crank–Nicolson-type finite difference method is proposed for computing the soliton solutions of a complex modified Korteweg–de Vries (MKdV) equation (which is equivalent to the Sasa–Satsuma equation) with the vanishing boundary condition. It is proved that such a numerical scheme has the second-order accuracy both in space and time, and conserves the mass in the discrete level. Meanwhile, the resulting scheme is shown to be unconditionally stable via the von Nuemann analysis. In addition, an iterative method and the Thomas algorithm are used together to enhance the computational efficiency. In numerical experiments, this method is used to simulate the single-soliton propagation and two-soliton collisions in the complex MKdV equation. The numerical accuracy, mass conservation and linear stability are tested to assess the scheme’s performance.