Numerical simulations of the complex modified Korteweg–de Vries equation

In this paper implementations of three numerical schemes for the numerical simulation of the complex modified Korteweg-de Vries (CMKdV) equation are reported. The first is an integrable scheme derived by methods related to the Inverse Scattering Transform (IST). The second is derived from the first and is called the local IST scheme. The third is a standard finite difference scheme for the CMKdV equation. Travelling-wave solution as well as a double homoclinic orbit are used as initial conditions. Numerical experiments have shown that the standard scheme is subject to instability and the numerical solution becomes unbounded in finite time. In contrast the integrable IST scheme does not suffer from any instabilities. The main difference among the three schemes is in the discretization of the nonlinear term in the CMKdV equation. This demonstrates the importance of proper discretization of nonlinear terms when a numerical method is designed for solving a nonlinear differential equation.