Homoclinic orbits and chaos in discretized perturbed NLS systems. I: Homoclinic orbits
The authors study the \(N\)-particle dynamical system \[ iq_{n} = (1/h^{2}) [ q_{n+1} - 2q_{n} + q_{n-1} ] + |q_{n}|^{2}(q_{n+1} + q_{n-1}) \] \[ -2\omega^{2}q_{n} + i\epsilon [ -\alpha q_{n} + (\beta / h^{2}) (q_{n+1} - 2q_{n} + q_{n-1}) + \Gamma ], \quad q_{n+N} = q_{n}, q_{N-n} = q_{n}, \] where \(i = \sqrt{-1}\), which is a finite difference discretization of a nonlinear Schrödinger equation. The existence of homoclinic orbits in the \(2(M+1)\)-dimensional finite difference approximation is established for any finite \(M\), where \( M = N/2 \;(N \text{ even})\), or \(M = (N-1)/2 \;(N \text{ odd})\). A symbol dynamics studied in other papers is given in the references.
scientific article; zbMATH DE number 1016994