Evolution And Control Of Solitons In The Inhomogeneous Systems

Nonlinear phenomena is very popular in life, it widely appeared in nonlinear optical, BEC, photonics, semiconductor, electronics, biology, heat conduction of plasma, LCD, etc. Some nonlinear partial differential equations are used to describe these nonlinear system and the corresponding solutions are called solitons. From the physical standpoint, soliton is a special product when the balance between nonlinearity and dispersion effect is reached. In mathematics, it is the steady and limited energy solution of some nonlinear partial differential equations. That is to say, it can maintain the waveform and speed constantly. When collided with each other, solitons can still keep their shapes and velocities as particles, so it is also calledsolitary wave.Nonlinear system mainly divided into uniform and nonuniform models. For uniform nonlinear system, the describing management coefficients are usually constant and the corresponding soliton solutions which have simple forms are easy to find. For constant coefficients of nonlinear equations, the profiles of solitons does not rely on any controllable parameter, just on coordinates or a function of time, which makes the evolution of the soliton can not be easily controlled. But in fact, people are more wanted to obtain some controllable and adjusted solitons. On the other hand, it is impossible to maintain the outside environment unalterable in the experiment, which leads the difficulty to obtain the stable soliton and makes us turn the attentions to the variable coefficients nonlinear systems. In nonuniform nonlinear systems, the describing management coefficients are usually varied with space or time, and even with the external potentials. In this case, the nonlinear system is not integrable. Therefore, we should first look for the integrability conditions of the equation when it has solutions. Under the the integrability conditions, we can use the inverse scattering method, Backlund transformation, bilinear method and some similarity transformation method to find the soliton solutions of the equation. In these solutions, some variable parameters are included and it provides the way to control the evolution of the solitons. Then we can compare our results with the corresponding experiments and obtain some useful properties. In these nonlinear equation, the most interesting model is the nonlinear Schrodinger equation. In this paper, we will use similar transform method to solve a class of variable coefficients nonlinear Schrodinger equations, and expect to obtain some exact solutions and study their properties. The thesis research contents include:(1) By using the modified CK direct method, we constructed a lot of exact solution of the one-dimensional nonlinear Schrodinger equation. Firstly, we use the modified CK direct method to reduce the nonlinear Schrodinger equation to the ordinary differential equation, and obtain a relation between the and the parameters and the governing equation. Then by choosing different functions of the parameters, we obtain some meaningful external potentials which are related to the scientific research. Finally we obtain the explicit solution of nonlinear Schrodinger equation under different external potentials.(2) We construct a large family of analytical solitary wave solutions to the generalized nonautonomous cubic-quintic nonlinear Schrodinger equation with time-and space-dependent distributed coefficients and external potentials. First by using the similarity transformation method, we reduce the generalized non-autonomous high order nonlinear Schrodinger equation into the standard one, and obtain the integrability conditions that the analytical solution can be existed. Then, a meaningful result is obtained that a more general expression of the external potential, which simulates some interesting periodic potentials, such as the SQ potential, OL potential, FB potential and potential barrier (well). Then, abundant of exact solitary wave solutions have been found under these different types of external potentials, including decaying solitary waves, snakelike solitary waves, and solitary waves in an OL potential. Finally, properties of some solutions are also studied intensively, and the control of the widths, amplitudes, speeds and center positions of some solitary waves.(3) We study the generalized nonlinear nonautonomous Schrodinger equation in the Fourier-synthesized lattice potential and obtain some soliton solutions including soliton collisions between two solitons. First we use the similar transformation method to reduce the generalized nonlinear nonautonomous Schrodinger equation into the standard one, and obtain the constraint conditions on the coefficients depicting dispersion, nonlinearity, and gain(or loss). Then various shapes of bright solitons and interesting interactions between two solitons are observed. Phenomena of a few solitons and physical applications of interest to the field are discussed.