Vortices in complex scalar fields

Abstract Consider a spatially extended field which evolves in time according to a PDE. The solutions contain particle-like defects whose motions are parts of the full dynamics. We inquire into the formulation of an asymptotic “particle + field” defect dynamics which derives from the full PDE. We carry out this program for two complex scalar field equations in two space dimensions, the nonlinear Schrodinger equation and the nonlinear heat equation. The topological defects are zeros of the complex scalar field with nonzero integer winding numbers, called vortices . Vortices evolving under the nonlinear Schrodinger equation behave like point vortices in ideal fluid. Pairs of vortices evolving under the nonlinear heat equation with like (opposite) winding numbers undergo a repulsive (attractive) interaction.