Numerical Simulation Of Soliton Solutions Of Complex Modified Korteweg-de Vries Equation Based On A Finite Difference Method

The Sasa-Satsuma(SS)equation is an integrable version of the higher-order nonlinear Schrodinger equation,which can be used to describe the propagation of femtosecond optical solitons in optics fibers and the internal solitary wave phenomenon in deep water.The complex modified Korteweg-de Vries(MKdV)equation is the equivalent form of SS equation.Firstly,we develop a conservative numerical difference scheme for the complex MKdV equation based on the Crank-Nicolson(CN)scheme.Also,we prove that the CN-type finite difference scheme has the second-order accuracy both in space and time,and employ the von Neumann method to show that such a scheme enjoys the unconditionally linear stability.Secondly,we use the CN-type finite difference method to simulate the propagation of the one-soliton solutions and the interactions of two-soliton solutions for the complex MKdV equation.It turns out that the interactions between the single-hump and double-hump solitons or between two multi-hump solitons meet the standard elastic properties in the sense that the interacting solitons keep their velocities and shapes unchanged after the interaction.Differently,when interacting with a multi-hump soliton,the double-hump soliton becomes a multi-hump one,but still keeps the mass conserved,which displays a shape-changing elastic interaction.Finally,we numerically check the accuracy and linear stability of the finite difference scheme,which shows the consistency with the result in theory.