A Study Of Soliton Propagation System

The soliton solution to most completely integrable (1+1) nonlinear partial differential equations, especially to which has important physical applications, were found by inverse scattering transform (1ST) method. The derivative nonlinear Schrodinger (DNLS) equation is, however, an open and important issue both in theoretical and applied aspects. The DNLS equation with nonvanishing boundary conditions is solved by introducting an affine parameter and we find that it supports a rich set of soliton solutions. Both the breather, bright solitons and dark soltions can be observed by choosing different parameter.A practical soliton-transmission system is a nearly integrable one which should be studied by the perturbation theory. Direct perturbation theory is a systematic approach, which was applied to solve the complicated perturbative problems such as NLS dark soltion and DNLS/MNLS bright soltion. However, the calculation of the zero order correction of the DNLS/MNLS equation is intricate. We develope a symbolic computation technique to calculate them, based on the residue theorem.It is revealed numerically and experimentally that some sort of perturbations can induce the radiation. This phenomena should be investigated by considering the higher order correction. The first order correction can be conclued to a complicated integral which is unlikely to be solved exactly or numerically. The perturbation theory of NLS equation is reviewd based on the direct perturbation method in this dissertation and we find that the first order correction can be reformulated in a simple form by fourier transformation, which can be calculated by a fast algorithm numerically. The result reveals that the first order correction cannot explain the perturbation-induce radiation completely.Finally, we review the numerical simulation algorithms for soliton issues. The traditional split-step fourier method (SSF) is an effective algorithm for the simulation of hyperbolic-cosecant solitons. However, SSF is no longer valid when applied to the femtosecond solitons and the dark soltions. In this paper, we introduce an advanced SSF method and propose a scheme for dark soliton simulation based on the Chebyshev spectral method.