Nonautonomous Solitons And Soliton Collisions In A Nonlinear Media

It is now commonly acknowledged that nonlinear science is another scientific revolution after the theory of relativity and the Quantum mechanics, one has now gradually recognized that linear system is only approximate to the complex real physical world, but the nonlinear one can be better close to the nature of the world. People commonly use some nonlinear partial differential equation to describe the nonlinear system, equation solution are called soliton. People commonly use some nonlinear partial differential equation to describe the nonlinear system. Usually nonlinear evolution equations has a variety of types of solution, such as triangle function, Jacobi elliptic function solution, the solitary wave solution, etc. From mathematics on see, soliton is some nonlinear partial differential equations of the maintain the waveform and speed constantly. From a physical standpoint, it is the material nonlinearity effect and the dispersion effect to reach a balance is a kind of special products. Soliton theory has been widely used in BEC, nonlinear optics, electronics, photonics, biology, semiconductor, heat conduction of plasma, etc.For uniform nonlinear system, one can describe it nonlinear equations is generally constant coefficient, this type of equation have many types of solutions, and the waveform is not dependent any controllable perameer, which makes the spread of soliton evolution waveform cannot adjust. Practically, it is impossible to synthesis ideally homogeneous materials, and the outside environment is not always unalterable as well, there are always some factors affect the stable evolution of soliton. As for those non-uniform nonlinear systems, such as the non-autonomous soliton in BEC and in fiber communication, etc are both the typical macroscopic nonlinear phenomena, describing them nonlinear equations are generally variable coefficient, this type of equation of various factor generally is about coordinates or time function, as usual, the equation is generally not accumulate, we must find the equation integrable conditions, in integrable conditions, to seek equation of soliton solution. Usually soliton evolution waveform is related to the equation of coefficient, so, we can adjust the coefficient to control of soliton evolution waveform.Nonlinear schrodinger equation as the important physical model is widely used in many field of phusics. This thesis we will use similar transform method solve a series of Nonlinear schrodinger equations, and study the solution properties, the results obtained here and the methods used here may be helpful to provide some theoretical for studying the stable transmission of inhomogeneous fiber systems or optical soliton and other related fields.The first portion we study the generalized nonlinear nonautonmous schrodinger equation. By using the similarity reduction method, we construct a large bright one-soliton and bright two-soliton solutions. First, we reduce the the generalized nonlinear nonautonmous schrodinger equation into the standard nonlinear schrodinger equation, and obtain integrability conditions. Then we solve the bright soliton solution in linear potential, time dependent parabolic potential, lattice potential and fly-bird potential,respectively. And discuss the evolution of soliton and the collision characteristics between two solitons. By means of adjusting the controled parameters to change amplitude, width, speed and waveform of the soliton, So as to achieve the purpose of manipulating the soliton successfully. Finally, we use Fourier method to analyse the stability of soliton.The second portion by using the similarity reduction method, study the generalized nonautonmous cubic-quintic nonlinear schrodinger equation, obtain integrability conditions, and bright solitons and duck solitons solutions. We discussed deeply the evolution characteristics of bright soliton and dark soliton in time dependent parabolic potential, time and spatial modulation periodic potential, offered some useful and effective scheme which can modulate the soliton. Finally, we also discussed nonlinear tunneling of the soliton in the periodic potential.