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Difference schemes for solving the generalized nonlinear Schrödinger equation. (English) Zbl 0923.65059

The authors study different finite difference schemes for solving the generalized nonlinear Schrödinger (GNLS) equation

iu t -u x u+q(|u| 2 )u=f(x,t)u·

A new linearized Crank-Nicolson-type scheme is presented by applying an extrapolation technique to the real coefficients of the nonlinear terms in the GNLS equations.

Three particular model situations with

q(s)=s 2 ,q(s)=ln(1+s),q(s)=-4s/(1+s)

are studied. The authors present results of numerical experiments, where the proposed scheme is compared with other Crank-Nicolson-type schemes, Hopscotch-type schemes, split step Fourier schemes, and with spectral schemes. The numerical experiments presented at the end of the paper demonstrate the efficiency and robustness of the proposed linearized Crank-Nicolson scheme for solving GNLS equations.

MSC:
65M06Finite difference methods (IVP of PDE)
35Q55NLS-like (nonlinear Schrödinger) equations