The Painlevé Test And Analytic Solutions Of The System Of Spin-1 BEC With Zeeman Effect

We consider the following 1 dimensional Gross-Pitaevskii system of the spin-1 Bose-Einstein condensates where c0,c1,p,q are four arbitrary constants,Φ±1(X,t),Φ0(x,t)are complex func-tions in x and t,and Φ*j(x,t)are the complex conjugated functions of Φj(x,t)(j=0,±1).By applying the methods of Painleve test,we prove t,hat when c1 =0 p=q,or c0 = c1,p= q,or c0=-2c1,p = q,the system is Painleve integrable.For the case c0=-2c1,p=q,we also present two kinds of analytic solutions.