Numerical Study of Quantized Vortex Interactions in the Nonlinear Schrödinger Equation on Bounded Domains

Abstract. In this paper, we study numerically quantized vortex dynamics and their interactions in the two-dimensional (2D) nonlinear Schrödinger equation (NLSE) with a dimensionless parameter ε > 0 proportional to the size of the vortex core on bounded domains under either a Dirichlet or a homogeneous Neumann boundary condition (BC). We begin with a review of the reduced dynamical laws for time evolution of quantized vortex centers and show how to solve these nonlinear ordinary differential equations numerically. Then we outline some efficient and accurate numerical methods for discretizing the NLSE on either a rectangle or a disk under either Dirichlet or homogeneous Neumann boundary condition. Based on these efficient and accurate numerical methods for NLSE and the reduced dynamical laws, we simulate quantized vortex interactions of NLSE with different ε and different initial setups including single vortex, vortex pair, vortex dipole, and vortex cluster, compare them with those obtained from the corresponding reduced dynamical laws, and examine the validity of the reduced dynamical laws. Finally, we investigate radiation and generation of sound waves as well as their impact on vortex interactions in the NLSE dynamics.