Dynamics Investigation Of Nonautonomous Solitons

The autonomous and nonautonomous solitons in fibers communication and Bose-Einstein condensations are both typical macroscopic nonlinear phenomena which can be realized by the balance of dispersion and non-linearity. Their dynamics of such systems are descripted by the nonlinear Schrodinger equation. In this thesis, we mainly investigate the dynamics of autonomous and nonautonomous soliton in such two systems. The main contents are as follows:(i) In terms of the Hirota method, we obtain the exact grey soliton solutions of the higher-order nonlinear Schrodinger equation which descripts the propagation of ultrashort pulse under the continuous-wave background. The results reveal that the velocity of the grey soliton is clearly affected by the higher-order effects, yet the grey soliton propagates without any change in their shape and intensity. The higher-order term and the phase velocity play the important role for the maximum valley of grey soliton, i.e., the intensity of grey soliton. The analysis of the asymptotic behavior of two grey soliton solution shows the collision is elastic.(ii) In terms of darboux transformation and Hirota method, we present a family of nonautonomous bright and dark soliton solutions of Bose-Einstein condensates with the time-dependent scattering length in an expulsive parabolic potential. These solutions show that the amplitude, width, and velocity of soliton can be manipulated by adjusting the atomic scattering length via Feshbach resonance. Especially, we investigate the modulation instability process in uniform Bose-Einstein condensates with attractive interaction and nonvanishing background, and clarify that the procedure of pattern formation is in fact the superposition of the perturbed dark and bright solitary waves. At last, we give the analytical expressions of nonautonomous dark one- and two-soliton solutions for repulsive interaction, and investigate their properties analytically. (iii) In the presence of a linear potential with an arbitrary time-dependence, Hirota method is developed carefully for applying into the quasi-one-dimensional Bose-Einstein condensation with repulsive interaction. We obtain the exact nonautonomous soliton solutions. These solutions show that the time-dependent potential can affect the velocity of nonautonomous soliton solution. In some special cases the velocity has the character of both increase and oscillation with time. A detail analysis for the asymptotic behaviour of solutions shows that the collision of two nonautonomous soliton solution is elastic.(iv) We investigate the combined soliton solutions of two-component Bose–Einstein condensates with external potential. The"phase diagram"is obtained for the formation regions of different combined solitons. Our results show that the intraspecies (interspecies) interaction strengths and the external trapped potential clearly affect the formation of the combined soliton solutions. Especially, we find that the bright-bright (dark–dark) soliton can exist in the case of both repulsive (attractive) intraspecies interaction strengths in the presence of external potential. This novel phenomenon is completely different from the formation of soliton solution of onecomponent Bose–Einstein condensates without external potential.