Split-step methods for the solution of the nonlinear Schrödinger equation. (English) Zbl 0597.76012

Summary: A split-step method is used to discretize the time variable for the numerical solution of the nonlinear Schrödinger equation (NLS). The space variable is discretized by means of a finite difference and a Fourier method. These methods are analyzed with respect to various physical properties represented in the NLS. In particular it is shown how a conservation law, dispersion and instability are reflected in the numerical schemes. Analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived. These results are demonstrated by numerical examples.


76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
76M99 Basic methods in fluid mechanics
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