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Iteration of some discretizations of the nonlinear Schrödinger equation. (English) Zbl 0664.65116

We consider several discretizations of the nonlinear Schrödinger equation which lead naturally to the study of some symmetric difference equations of the form \(\phi_{n+1}+\phi_{n-1}=f(\phi_ n)\). We find a variety of interesting and exotic behaviour from simple closed orbits to intricate patterns of orbits and loops in the \((\phi_{n+1},\phi_ n)\) phase-plane. Some analytical results for a special case are also presented.

MSC:

65Z05 Applications to the sciences
65N06 Finite difference methods for boundary value problems involving PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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References:

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