Research About The Modified And Derivative Nonlinear Shr(?)dinger Equation

As is well known, the nonlinear Schroinger (NLS) equation well describes transmission of picosecond pulses in optical fibers. However, experiments and theories on the propagation of ultrashort pulses in the long monomode fibres have shown that dynamics of femtosecond pulses is not well governed by the NLS equation. As a result, the modified nonlinear Schrodinger (MNLS) equation of which the nonlinear terms are composed of an usual cubic term and a derivative cubic term was proposed to describe the short pulse propagation in long monomode optical fibers in consideration of the inherent property of asymmetric output pulse spectrum. For its application in nonlinear optics, the MNLS equation is still a topic of intense research. The MNLS equation of vanishing boundary conditions can be solved by the inverse scattering transform (1ST) method. But in the case of non-vanishing boundary, the solution of it (simply MNLS~+ equation) has never been given until now. For this reason, the background noise corresponding to non-vanishing boundary is dealt with as perturbation sometimes, and a great deal of numerical analysis has been done for suiting needs of practise.On the other hand, the derivative nonlinear Schrodinger (DNLS) equation was first proposed to describe nonlinear Alfven waves in plasma. In the case of vanishing boundary, the DNLS equation was solved by the inverse scattering transform(IST), or other approaches. Similarly to the MNLS equation, the DNLS equation with non-vanishing boundary (simply DNLS~+ equation) has never been solved exactly though some works have tried to do it. As the solution of DNLS equation represents the complex transverse magnetic field, the vanishing boundary conditions can only deal with waves exactly parallel to the ambient field, while the case of non-vanishing boundary condition are more general in physics. In addition, being one of a few well-known unsolved completely integrable nonlinear evolution equations, it is worth trying our best to find the final solution for the DNLS~+ equation. In addition, the DNLS equation involves only the derivative cubic term, but it can be related to the MNLS equation by a simple gauge-like transformation, which means the solutions of the MNLS equation can be obtained as long as that of the DNLS equation is derived.In the second chapter of this thesis, we construct the direct perturbation theory of DNLS equation with vanishing boundary conditions. Most of nonlinear equationscannot be solved exactly, and some kinds of perturbation theory are needed, which also extend the application of nonlinear equations. The direct perturbation theory of nonlinear equations was firstly proposed by McLaughlin and Scott in 1978 to deal with sine-Gordon equation. At that time, in order to emphasize the difference between the direct perturbation theory and those perturbation theories based on the 1ST method, they tried to avoid using results derived from the 1ST. In our opinion, there is no reason to exclude the results of 1ST method, with which the solutions of the linearized equation can be easily constructed as squared Jost solution of the multi-soliton case. In the direct perturbation theory, the linearized equation is derived from the adiabatic solution, and then its basic functions with orthogonality and completeness relation is obtained. At last, the secularity conditions are given. The key point of this procedure is to show the completeness of basic functions, but this is not easy. Noticing that the linearized equation is actually a linear equation, we apply the usual Green's function method to this problem as what Mann did, and then prove the completeness of the squared Jost solutions, which is the basis for the perturbation theory of DNLS equation.At the end of this thesis, we formulate the Hamiltonian theory of DNLS+ equation completely, and obtain the action-angle variable. This work is the research about the integrability of nonlinear equations, and it is beneficial to foregoing works. Especially, the form of DNLS+ in this thesis differs from the general form of it by a Galileo transformation in order to construct the Hamiltonian theory perfectly, which is very interesting as such a difference eases the work of its solving and perturbation theory.