Periodic and Chaotic Breathers in the Nonlinear Schrödinger Equation

The breathers in the cubic nonlinear Schr\"odinger equation are investigated numerically by using the symplectic method. We show that the solitonlike wave, the periodic, quasiperiodic and chaotic breathers can be observed with the increase of cubic nonlinear perturbation. Finally, we discuss the breathers in the cubic-quintic nonlinear Schrödinger equation with the increase of quintic nonlinear perturbation.