N-fold Darboux Transformation And Conservation Law For The Nonautonomous GP System

Bose-Einstein condensate (BEC) is a special kind of macroscopic quantum phenomena. Under the mean field theory, while the scalar BEC is described by the Gross-Pitaevskii (GP) equation, the spinor BEC is described by the coupled Gross-Pitaevskii (GP) system. The nonautonomous GP system is used to study the quantum manipulation in the condensate, its soliton solutions are very helpful for understanding the dynamics of the spinor BEC. In this dissertation, we consider a kind of nonautonomous GP system, with the help of nonisospectral Lax pair, we get the integrability conditions and the N-fold Darboux transformation of the system has been constructed. The N-fold soliton-like solutions and infinite many conserva-tion laws are presented. In particular, the first three conservation laws have been written down explicitly.