On the nonexistence of asymptotically free solutions for a coupled nonlinear Schrِdinger system

We study the asymptotic behavior of the solution for the initial value problem for a coupled nonlinear Schrödinger system that describes some physical phenomena such as the propagation in birefringent optical fibers, Kerr-like photo refractive media in optics and Bose–Einstein condensates. First, we prove the nonexistence of asymptotically free solutions for the coupled nonlinear Schrödinger system following an idea of R. Glassey. Then, we show the asymptotic behavior of some numerical examples using a Crank–Nicolson scheme. This numerical scheme preserves the densities and the energy of the solution, and it is proved the convergence of its sequence of solutions.