The Integrability Of A Class Of 3-component Gross-Pitaevskii System

This dissertation considers the integrability of the following(1+1)-dimensional 3-component Gross-Pitaevskii system(?) This system arises from the study of 3-component Bose-Einstein condensates in the field of ultra-cold atomic physics,where u1(x,t),u2(x,t),u3(x,t)are complex functions in x and t,which denote the three wave functions of the atoms in each component,respectively.f(t)is the parameter representing effective mass,g(t)is the parameter denoting the strength of interaction between different components,V(x,t)is the trapped potential,σij(i,j=1,2,3)takes+1 or-1.In this thesis,through the Painlev’e test,the integrability condition of the system is obtained.Under this condition,a nonlinear transformation is found,which can convert the above Gross-Pitaevskii system into the standard 3-component nonlinear Schr?dinger system of Manakov type,then from the solutions of Manakov system,rich kinds of solutions of the Gross-Pitaevskii system can be obtained.