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A numerical study of the nonlinear Schrödinger equation. (English) Zbl 0555.65060
The paper studies the numerical solution of the nonlinear cubic Schrödinger equation $$i(\partial u/\partial t)+(\partial^ 2u/\partial x^ 2)+u| u|^ 2=0$$. By writing $$u=v+iw$$ this can be transformed to the vector equation $$\partial u/\partial t+A(\partial^ 2u/\partial x^ 2)+f(\underline u)=0$$ where A is a $$2\times 2$$ matrix. The method of solution proposed is a finite element Galerkin method where an approximation U is written in the form $$U(x,t)=\sum^{N}_{j=1}\alpha_ j(t)\psi_ j(x)$$ for some specified functions $$\psi_ j$$ and the $$\alpha_ j$$ are found from the iterative scheme $$(M+rS)\alpha^{m+1}=[M-rS]\alpha^ m-\tau MF((\alpha^*+\alpha^ m))$$. The step lengths for x and t are h and $$\tau$$ and $$r=\tau /h^ 2$$. M is a mass matrix, S a stiffness matrix and F is a nonlinear vector function. This scheme uses an intermediate time $$t^*=(m+\beta)\tau$$, $$\beta >0$$ to aid the treatment of F. It is found that the best choice of $$\beta$$ is $$1+0(\tau)$$ and that the scheme is stable to small local perturbations if $$\beta \leq 1+0(\tau)$$. Numerical results are given for three example problems and results are also given for the finite difference scheme when M is replaced by I. This corresponds to the usual Crank-Nicolson scheme. The analysis uses the discrete least squares norm but it is pointed out that some nodal values may grow as t increases.
Reviewer: B.Burrows

##### MSC:
 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35Q99 Partial differential equations of mathematical physics and other areas of application
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