Difference Schemes for Solving the Generalized Nonlinear Schrödinger Equation

This paper studies finite difference schemes for solving the generalized nonlinear Schrödinger (GNLS) equationiut−uxx+q(|u|2)u=f(x,t)u. A new linearlized Crank–Nicolson-type scheme is presented by applying an extrapolation technique to the real coefficient of the nonlinear term in the GNLS equation. Several schemes, including Crank–Nicolson-type schemes, Hopscotch-type schemes, split step Fourier scheme, and pseudospectral scheme, are adopted for solving three model problems of GNLS equation which arise from many physical problems. withq(s)=s2,q(s)=ln(1+s), andq(s)=−4s/(1+s), respectively. The numerical results demonstrate that the linearized Crank–Nicolson scheme is efficient and robust.