Study Of The Propagation Properties Of Solitons And Quantum Control In Restricted Systems

Soliton is one of the most amazing phenomena in non-linear science. In recent yeares, the rapid development of the soliton physics has been extended to a great deal of extensions, such as nonlinear optics, Bose-Einstein condensation, biology, photonics, semiconductor electronics, plasma physics, thermal conductivity, liquid crystal and so on. As an important branch of solitons, it is because of nonlinear effects and the diffraction effects balance and the beam of the media environment in the absence of the boundary. Spatial solitons are a self-trapped or self-guided state with the stability of the transmission.The beam width and shape of optical soliton keep unchange along propagation distance. Collision between the solitons may produce the phenomenon of fractal, merge and spin. For different medium, it has been testified to exist in many of the nonlinear optical response mechanisms. It is due to their robustness and well-defined shapes that make them attractive as a basic bit in the future data transmission and processing schemes. Simultaneity, it is well known that optical soliton may offer a powerful way for creating reconfigurable all-optical circuits where light is controlled and guided by light itself.On the other impoint hand, matter wave solitons in Bose-Einstein condensates (BEC) have been anattractive subjects in recent decades. They not only offer the perfect macroscipic quantum systems to research many foundmental Problems in quantum mechanics but also have extensively application foregrounds such as quantum computation and in atom laser. In the famework of mean-field theory, the Bose-Einstein condensates is govonered by the Gross-Pitaeviskii equation. Based on the Gross-Pitaeviskii quation we shall study the dynamical behaviors of matter waves solitons in BEC. The work of this thesis mainly concentrates on the spatial solitons. The principal achievements obtained are as the following:1. Two-dimensional Kummer-Guass soliton clusters in strongly nonlocal nonlinear mediaIn the second chapter, the 2D strongly nonlocal nonlinear Schr(o|¨)dinger equation model is studied. Then based on a self-similar method, the model is solved by in polar coordinate system. We find that there exists a class of Kummer-Guass soliton clusters that propagate in a self-similar manner. Furthermore, a family of new spatial solitary waves has been found. It is interesting that the spatial soliton profile and its width remain invariabilitial with increasing propagation distance, and big phase shift is found.2. Three-dimensional Bessel and Hermite-Gaussian soliton clusters in strongly nonlocal nonlinear mediaIn recent years, a new research field has begun to arrest the people in nonlocal nonlinear soliton physics. In the field, it has been maded great achievements in the theoretical and experimental research. In the chapter, exact self-similar soliton solutions to the strongly nonlocal nonlinear Schr(o|¨)dinger equation have been gotten. The propagation of three-dimensional soliton cluster has been investigated numerically and analytically in strongly nonlocal nonlinear media. It is found that the fantastic soliton cluster solutions which are made of Bessel and Hermite-Gaussian functions. The stability of these solutions is confirmed by direct numerical simulation. It is shown that eerie higher-order spatial soliton clusters can occur in various forms .3. Exact spatio-temporal soliton solutions to three-dimensional extended nonlinear Schr(o|¨)dinger equation with varied coefficients.Three-dimensional nonlinear Schr(o|¨)dinger equation with varied coefficients is very important nonlinear model. It can describe many nonlinear phenomena in physics, such as matter wave in Bose-Einstein condensate, nonlinear optics in optical pulse transmission, fluid mechanics and plasma physics etc.We have used a homogeneous balance principle and extended F-expansion techniques to construct exact periodic wave solutions to the generalized three-dimensional nonlinear Schr(o|¨)dinger equation with distributed dispersion/ diffraction, nonlinearity, and gain or loss. In the limiting situation of parameters, these periodic Jacobi wave solutions can reduced to localized spatial soliton solutions. Under certain cases, such solutions occur, and it is needed that constraint is satisfied on the functions describing nonlinearity, dispersion/diffraction, and gain or loss. Furthermore it is found that the method of solution can be applied to other nonlinear partial differential equations in nolinear mathematical physics.4. Spatial solitons in utmost bleak atom gasl.Based on extended F-expansion techniques, exact periodic wave solutions of the generalized three-dimensional nonlinear Schr(o|¨)dinger equation with distributed dispersion, nonlinearity, gain or loss and in a harmonica potential are gotten. In the limiting cases of parameters, two kind of the bright and dark of localized spatial soliton solutions have been gotten. Such solutions exist only under certain cases and impose constraints on the functions describing dispersion, nonlinearity, gain or loss and harmonica potential. It is interesting that the matter-wave can be controlled by applying the external magnetic field and a background.