High-order compact finite difference scheme with two conserving invariants for the coupled nonlinear Schrödinger–KdV equations

In this paper, a new high-order accurate compact finite difference scheme for the coupled nonlinear Schrödinger–KdV (CNLS–KdV) equations is proposed. Conservation of the discrete number of plasmon and the discrete number of particle are given in detail. Convergence with second-order in time and fourth-order in space of the present scheme are proved by using "cut-off" function technique and discrete energy method. Numerical experiments are given to support the theoretical analysis.