Spatiotemporal Soliton Solutions To The Ginzburg-Landau Equation And The Nonlinear Schr(?)dinger Equation

The origination and development of theory about spatiotemporal soliton(STS)is a very significant event in the nonlinear optics.Along the development of the research in liquid realm or optics,many nonlinear equations with soliton solution have been created.Therefore,one of the most important ways to understand STS's theory and realistic application is to gain the solution to these equations by various methods.This paper firstly summarizes the theory of three classical types of soliton,temporal soliton,spatial soliton and STS and explicitly analyzes their theoretical models and results.Then we briefly introduce several calculating methods to discuss series of nonlinear equations with stable soliton solution,such as split-step Fourier method(SSFM),F-expansion method(FEM)and Newton-Jacobi method(NJM).With these methods,in this paper,the works we do are listed below:1.We use SSFM to simulate the complex cubic-quintic Ginzburg-Landau equation to study the propagation characteristics and the collision between two STSs.We find that the nonlinear gain and dispersion have great effect on appearances of dissipative light bullet(LB).Altering the two parameters can get various soliton solutions,such as “bell-shaped” pulse,square pulse and “rocket” soliton,etc.These results could be reference for the experiment of dissipative soliton in cubic-quintic system.2.We do the research about collision of two LBs and the collision results with different initial conditions,including position,velocity and phase.We study both the head-on and the non-head-on collisions to manifest the influence of the inial geometrical relation on the final results.We achieve many phenomena,including fusion,elastic collision,non-interaction and fission.These results could arouse a new logic for all-optical switches in optical communications.3.We propose a new model which is the nonlinear Schr?dinger equation with parabolic potential in transverse direction.Using F-expansion method and homogenous balance principle,we gain the stable spatiotemporal soliton solution to the equation.We discuss the influence of spatial-teporal chirp on the soliton solution.We also utilize the property of parabolic potential to control the broadening speed of STS's beam.4.We calculate the second-order intensity moment to gain the pulse width and beam width of STS and analyze the singularity of soltion solution.By choosing the proper parameters,we achieve four propagation characteristics,such as stable,oscillating,blowing up and decaying propagation.These results may be applied to the experiment about soliton propagation in grid-index fiber.