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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 \[ q(s) = s^2, \qquad q(s) = \ln(1+s), \qquad q(s) = -4s/(1+s) \] 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:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
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