  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.
