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Pseudo-spectral solution of nonlinear Schrödinger equations. (English) Zbl 0691.65090
This paper compares four discretization methods for solving the generalized nonlinear Schrödinger equation $$iu_ t+u_{xx}+q_ c| u|^ 2u+q_ q| u|^ 4u+iq_ m| u|^ 2_ xu+iq_ u| u|^ 2u_ x=0$$ where $$q_ c$$, $$q_ q$$, $$q_ m$$ and $$q_ u$$ are real parameters. An initial value problem is considered so that $$u(x,0)=u_ 0(x)$$ is specified. The solution may be represented in a Fourier series where the coefficients depend on time and the methods differ on their formalism connecting the time variable with the space function discretization at n collocation points. Numerical examples are given.
Reviewer: B.Burrows

##### MSC:
 65Z05 Applications to the sciences 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 35Q99 Partial differential equations of mathematical physics and other areas of application 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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