Dynamics Behavior Of Deformed Soliton In An External Field

In recent decades, Bose-Einstein condensation(BEC) has been attracting physical scientist and an important researching object along. Because the plentiful par-ticles with similar dynamics behavior assemble on the same energy level so, not only does it supply a system of studying quantum mechanics essential question for us. but also it can be widely applied in different fields, such as atomic laser, quantum information, nonlinear physics, statistical physics, nuclear physics and as-trophysics. In our paper, under the frame of the mean field theory, we combine the exact compatibility conditions, considering the nonautonomous Gross-Pitaevskii (GP) equation as the main model, and study the exact solution of the GP equa-tion with sorts of external potential, i.e. study the exact solution of nonlinear Schroinger equation(NLSE) with sorts of external potential. During the studying, we obtain a class of exact deformed soliton solution of nonautonomous GP equa-tion. By means of the deformed soliton solution, we study the dynamics behavior and the ratchet effect of deformed soliton in the external potential field.The paper is composed of five different chapters. In the first chapter, we briefly introduce the basis course of BEC from theory to realization, the basis principle about BEC. the mean field theory describing the dynamics behavior of BEC, and the course of knowing and studying the soliton. In additional, we also introduce the autonomous and nonautonomous GP equation.In the second chapter, we consider the nonautonomous GP equation with external parabolic potential as the model. In terms of the exact solvable condition, we can confirm the parabolic potential and the corresponding nonlinearity at the same time. Inserting the exact deformed soliton solution into the nonautonomous GP equation, we obtain the concrete form of exact deformed soliton solution. Basing the exact deformed soliton solution, in the parabolic potential, for the different deformation factor, namely, the different condensation atom number and different initial condition, we study the evolvement formation of the nonlinearity and corresponding parabolic potential. Then, for the symmetry and asymmetry case, we study the directional transportation of deformed soliton and ratchet effect. In terms of the research result, we find, under the bi-frequency driving, for the asymmetric sawtooth case, the average velocity of deformed soliton in the external parabolic potential represent the ratchet effect, or anti-ratchet effect. Namely, the ratchet effect of deformed soliton can be induced or suppressed even reversed by adjusting the factor. But, for the symmetric case, the dynamics behavior of deformed soliton is simple, monotonously increase with the increasing of deformed factor, and is not represent the ratchet effect, because the changing of average velocity is very small, so, the dynamics behavior of soliton represent approximate localization without net drift.In the third chapter, we mainly study the dynamic behavior of exact deformed soliton in the different external parabolic potentials. For example, the nonlinear-ity linearly and exponentially changes with the time evolution near the feshbach resonance, and the nonlinearity direct proportion and inverse proportion with sine function, for the different nonlinearity and the corresponding parabolic potential, we study and obtain a series of soliton especial solutions. Then, for different de-formed factor and initial conditions, we study the dynamic behavior of deformed soliton in different parabolic potentials. We find, in different parabolic potential, the dynamic behavior and the change of deformation factor are nearly correlative. Namely, we can manipulate the position, velocity, and shape of deformed soliton by means of changing the condensation atom number in BEC and initial condition of system.In the forth chapter, we consider a general nonautonomous GP equation with the parabolic potential and linear potential as the model. In terms of exact com-patibility conditions, we find, the compatibility conditions is irrelative with the linear potential. So, we suggest a scheme to control the movement of deform soli-ton by means of the linear potential. Using the dipole laser seam to form the linear potential, we can adjust the dynamics behavior of deformed soliton by means of modulating the amplitude, frequency, and phase of laser seam. In this chapter, we set a bi-frequency linear driving, by which we manipulate the dynamic behavior of deformed soliton in the combination potential of parabolic potential and linear po-tential. We find, under the fixed the parabolic potential, the dynamics of deformed soliton is completely determined by the linear potential. For the symmetry case, it is differ from the null case, or trending to localization in the other literature, the average velocity of deformed soliton is not zero, moreover, it varies with the time quasi-periodically. But then, during the finite time, we can see the average velocity of deformed soliton is more and more slow. For the asymmetry case, the average velocity of deformed soliton represents perfect directional transportation result. In the coordinate system, we can distinctly see, with the increasing of time, the average velocity of directional transportation of deformed soliton is positive or negative. In additional, as we study the relation between the average velocity of deformed soliton and frequency of linear potential, we find that the average velocity of deformed soliton represent the ratchet effect, inverse-ratchet effect, or localization with the frequency variation. This point means that we can manipu-late the ratchet effect and dynamics localization behavior of deformed soliton by choosing different frequency of linear potential.Finally, in the fifth chapter, we briefly conclude the paper, and topics the possibility of advanced studying the dynamic behavior and ratchet effect of de-formed soliton in external field. For example, the exact solution exist only when the nonautonomous GP equation satisfy the solvable condition, if the external po-tential depart little from exact solvable condition, we can consider the departure as the perturbation, and treat the the exact solution as the zero level approximate solution. So, we can study the chaos behavior of deformed soliton in different external potential. This is our work in the future. In the paper, the main research work is concentrated in the second, third, and fourth chapters.