Modulation equations approach for solving vortex and radiation in   nonlinear Schrodinger equation
  We apply the modulation theory to study the vortex and radiation solution in the two-dimensional nonlinear Schr\"{o}dinger equation. The full modulation equations which describe the dynamics of the vortex and radiation separately are derived. A general algorithm is proposed to efficiently and accurately find vortices with different values of energy and spin index. The modulation equations are solved by accurate numerical method. Numerical tests and simulations of scattering are given. 
Avy Soffer, Xiaofei Zhao
math-ph cond-mat.quant-gas math.MP