Propagation of Ultrashort Pulses in Nonlinear Media

In this paper, a general propagation equation of ultrashort pulses in an arbitrary dispersive nonlinear medium derived in [9] has been used for the case of Kerr media. This equation which is called Generalized Nonlinear Schroedinger Equation usually has very complicated form and looking for its solutions is usually a very difficult task. Theoretical methods reviewed in this paper to solve this equation are effective only for some special cases. As an example we describe the method of developed elliptic Jacobi function expansion and its expended form: F-expansion Method. Several numerical methods of finding approximate solutions are briefly discussed. We concentrate mainly on the methods: Split-Step, Runge-Kutta and Imaginary-time algorithms. Some numerical experiments are implemented for soliton propagation and interacting high order solitons. We consider also an interesting phenomenon, namely the collapse of solitons, where the variational formalism has been used