Ross, K. A.; Thompson, C. J. Iteration of some discretizations of the nonlinear Schrödinger equation. (English) Zbl 0664.65116 Physica A 135, 551-558 (1986). 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. Cited in 6 Documents 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 Keywords:iteration; nonlinear Schrödinger equation; symmetric difference equations; orbits; loops PDFBibTeX XMLCite \textit{K. A. Ross} and \textit{C. J. Thompson}, Physica A 135, 551--558 (1986; Zbl 0664.65116) Full Text: DOI References: [1] Fokas, A. S.; Ablowitz, M. J., Phys. Rev. Lett., 47, 1096 (1981) [2] Quispel, G. R.W.; Nijhoff, F. W.; Capel, H. W.; Van der Linden, J., Physica, 125A, 344 (1984) [3] Thompson, C. J.; Ross, K. A.; Thompson, B. J.P.; Lakshmanan, M., Physica, 133A, 330 (1985) [4] Potts, R. B., J. Austral. Math. Soc. (Series B), 23, 64 (1981) [5] Potts, R. B., J. Austral. Math. Soc. (Series B), 23, 349 (1982) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.