Reduced Vortex Dynamics Of Ginzburg-Landau-Schr?dinger Equation

Vortices play an important role in a variety of physical systems,from or-dinary fluids to condensed matter to the early universe,and these changes are reflected in mathematical models that describe their formation,structure,and dynamics,the nonlinear Ginzburg-Landau-Schrodinger equation describes well the rich dynamic properties of quantum vortices.In this paper,we mainly study the GLSE under the reduced dynamic law,i.e.#12 where the zj=(xj,yj)T ? R2 represents the location of the center of the j-th vortices,the mj=+1 or-1 represents the winding number of the j-th vortices,and N is the total number of vortices,k1 and k2 are two constants greater than zero,#12Firstly,the analytical solutions are obtained under some special initial conditions:(1)When N like vortices the same winding numbers are uniformly dis-tributed on the circle with the origin as the center and a>0 as the radius,at any time,all vortices are distributed on the circle with the origin as the center and(?)as the radius,In particular,when mj=1,the vortices rotate counterclockwise,and when mj=-1,the vortices rotate clockwise;(2)When N-1 like vortices with the same winding numbers are uniformly distributed at the center of the circle at the origin,a>0 as the circumference of the radius,and one like vortices is located at the origin,(i)we can see that the vortex at the center of the circle stays stationary at the initial moment;(2)for any time,the other N-1 like vortices are uniformly distributed on the circumference of the circle with the origin as the center and(?)as the radius.In particular,when mj=1,the vortices rotate counterclockwise,when mj=-1,the vortices rotate clockwise;(3)When N-1 like vortices with the same winding numbers are uniformly distributed at the center of the circle at the origin and a>0 as the circumference of the radius,and one with the different winding number is located at the center of the circle,(1)we can see that the vortex at the center of the circle stays stationary at the initial moment;(2)when N=3,the two vortices at the initial moment of the circle are distributed around the circumference of the circle with the origin as the center and(?)as the radius.In particular,when mj=1,the vortices rotate clockwise,when mj=-1,the vortices rotate counterclockwise,and the three vortices collide when t=tc=a2/2k1;(3)when N=4,all the vortices remain at their initial position;(4)when N?5,at any time,N-1 vortices with the same winding numbers are distributed on the circumference of the circle with the origin as the center and(?)as the radius.In particular,when mj=1,the vortices rotate counterclockwise,when mj=-1,the vortices rotate clockwise.Secondly,we give a complete description of the dynamic behavior of two vortices under any initial conditions:(1)At any time,the pairs of vortices are distributed on the circumference of a circle with the center of the two vortices and the radius of(?),In particular,when mj=1,the vortices rotate counterclockwise,when mj=-1,the vortices rotate clockwise;(2)The two dipoles attract each other,the distance between the two vortices decreases with time,and when t=a2/ak1.the two vortices collide at(?).Finally,it is proved that N vortex centers with the same winding numbers are conservation of the mass center,and several first integrals are obtained.In addition,at least two vortices move to infinity as t??.