Abstract

We address traveling single-humped localized solutions in the spatial discretizations of the nonlinear Schrodinger (NLS) equation. By using the Implicit Function Theorem for solution of the differential advance-delay equation in exponentially weighted spaces, we develop a mathematical technique for analysis of persistence of traveling solutions. The technique is based on a number of assumptions on the linearization spectrum, which are checked numerically in the general case. We apply the technique to a wide class of discrete NLS equations with general cubic nonlinearity which includes the Salerno model, the translationally invariant and the Hamiltonian NLS lattices as special cases. We show that the traveling solutions terminate in the Salerno model and they persist generally in the other two NLS lattices as a one-parameter family of the relevant differential advance-delay equation. These results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails. Analysis of persistence also predicts the spectral stability of the one-parameter family of traveling solutions in the time evolution of the discrete NLS equation
Original languageEnglish
Publication statusPublished - 5 Apr 2007

Bibliographical note

Sponsorship: D.P. was supported by the EPSRC and the Humboldt Research Foundation. T.R.O is supported by a studentship from
the EPSRC

EPSRC
Humboldt Research Foundation

Keywords

  • lattices
  • travelling waves
  • solitons
  • Melnikov integral

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