The finite element method for the coupled Schrödinger–KdV equations

In this Letter, we employ finite element method to study a periodic initial value problem for the coupled Schrödinger–KdV equations. For the case of one dimension, this problem is reduced to a system of ordinary differential equations by using a semi-discrete scheme. The conservation properties of this scheme, the existence and uniqueness of the discrete solutions, and error estimates are presented. In numerical experiments, the resulting system of ordinary differential equations are solved by Runge–Kutta method at each time level. The superior accuracy of this scheme is shown by comparing the numerical solutions with the exact solutions.