Behaviour of Cubic Nonlinear Schrödinger Equation by Using the Symplectic Method

The dynamic properties of cubic nonlinear Schrödinger equation are investigated numerically by using the symplectic method. We show that the trajectories in phase space will exhibit different behaviour (elliptic orbit or homoclinic orbit) with the increase of nonlinear perturbation. We illustrate this phenomenon by mean of linearized stability analysis. The theoretical analysis is consistent with the numerical results.