Speaker:  Weizhu Bao
          Department of Mathematics
          National University of Singapore

Title:  The dynamics and interaction of quantized vortices in   
Ginzburg-Landau-Schroedinger equation

Abstract: 

   In this talk, we study the dynamical laws of quantized   
vortex interactions in the Ginzburg-Landau-Schroedinger   
equation (GLSE) analytically and numerically. 
We begin with a review of the reduced dynamic laws 
governing the motion of vortex centers in GLSE and solve 
the nonlinear ordinary differential equations (ODEs) of the 
reduced dynamic laws analytically with a few types of initial data. 
By adopting the polar coordinates so as to effectively match 
and resolve the nonzero  far-field conditions in phase space
and applying a time-splitting technique for decoupling the 
nonlinearity in the GLSE, we propose an efficient and accurate 
numerical method for solving GLSE in two dimensions  
with nonzero far field conditions. By directly simulating the GLSE with our   
new numerical method for GLSE, 
we can compare quantized vortex interaction patterns   
of GLSE with those from the reduced dynamic laws qualitatively and   
quantitatively. Some conclusive findings on issues   
such as the stability of quantized vortex, interaction of   
two vortices, dynamics of the quantized vortex lattice and the motion of   
vortex with an inhomogeneous external potential are   
obtained, and discussions on numerical and theoretical results are   
made to provide further understanding of vortex interactions in   
GLSE. Finally, the vortex motion under an inhomogeneous potential   
in GLSE is also studied.   

   This is a joint work with Qiang Du and Yanzhi Zhang.