| In this paper, a general propagation equation of ultrashort pulses in an arbitrary dispersive nonlinear medium is derived using a method based on a consistent and mathematically rigorous expansion of the linear dispersion relation including the medium.s nonlinear optical response. A specific case of Kerr media is studied. The obtained ultrashort pulse propagation equation which is called Generalized Nonlinear Schrodinger Equation has a very complicated form and looking for its solutions is usually a very difficult task. Theoretical methods to solve this equation are effective only for some special cases. For this reason, several numerical methods of finding approximate solutions are used. We focus mainly on the methods: Split-Step and Runge-Kutta algorithms. Some numerical experiments are implemented for soliton propagation and interacting high order solitons. These algorithms in connection with the variational method give several interesting results concerning solitons in optical nonlinear media. The results from nonlinear optics to atom optics and vice versa could be transferred by analogy between the propagation equation in the Kerr medium and the Gross-Pitajevski equation for Bose-Einstein condensates (BECs). |
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