Dark Soliton Transmission Of The Nonlinear Schr(?)dinger Equation

The nonlinear Schrodinger (NLS) equation is an integrable model with most widely applications in nonlinear physics, describing nonlinear wave problems in optical fibers, optical waveguides, magnetic systems, and Bose-Einstein condensates systems, etc. under certain conditions. The NLS equation has two forms depending on the signs of the second order derivative term and the nonlinear term. If they possess the same sign, the equation (called the NLS equation) admits bright soliton solutions under vanishing boundary conditions. Otherwise, the equation (called the NLS~+ equation) admits dark soliton solutions under non-vanishing boundary conditions. Soliton solutions can be found by the inverse scattering transform(IST) method. For practical issues in physics, small perturbations violating the integrability inevitably exist. Behavior of solitons around completely integrable models can be worked out by the perturbation theory. The IST's and perturbation theories for both of the NLS and the NLS~+ equations have been basically solved. However, the non-vanishing boundary conditions make troubles for the NLS~+ equation by resulting double-valued problem in IST and divergences in relevant soliton perturbation theories. The double-valued problem can be simplified by introducing an appropriate affine parameter, namely, mapping the two Riemann sheets on the plane of the spectral parameter to the affine parameter space. The origin and the infinite point on the plane of the affine parameter correspond to the infinite points of the two Reimann sheets. As boundary of the Jost solutions, Jost solutions at these points should have substantial contributions in IST. However, there exist some errors in the present literature in treating asymptotic behavior of Jost solutions at the origin, yielding incorrectness in continuous spectrum despite the fact that they cause no errors in discrete spectrum and soliton solutions. Similar errors also spoil the completeness of the squared Jost solutions in the direct soliton perturbation theory. These errors should be corrected. This thesis mainly include the following parts:1. The IST for the NLS~+ equation is re-deduced, correcting errors in treating asymptotic behavior at the origin in the literature. It is found that the integral for continuous spectrum in the inverse scattering equation should be corrected as Cauchy principal.2. The direct perturbation theory for dark solitons of the NLS~+ equation is re-deduced, giving a proof for the completeness of the squared Jost solutions simply by correcting the treatment in the asymptotic behavior at the origin. It is found that the integral for the continuous spectrum in expression for the first order correction should also be corrected as Cauchy principal.3. Completeness of the squared Jost solutions of the DNLS equation under vanishing boundary conditions is proved by the similar way.