A Unified Approach to Vortex Motion Laws of Complex Scalar Field Equations

In this short note, we give a unified rigorous derivation of vortex motion laws of nonlinear wave (NLW) and nonlinear heat (NLH) equations based on the fluid dynamic approach the authors recently developed in solving the nonlinear Schrödinger (NLS) equation. Hence in all three complex scalar field equations, the motion laws follow from the Euler-type equations, and the knowledge of the finite mass Radon defect measure. Appeared in Math Research Letters, 5, pp 1-6(1998). ∗Courant Institute, New York University, 251 Mercer Street, NY, NY 10012, USA. †Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA. 1 A Summary of Basic Facts Let us consider as ↓ 0 the two-dimensional complex scalar field equations: 1 log −1 u ,tt = ∆u + −2u (1− |u |), (1.1) the nonlinear wave (NLW) equation; 1 log −1 u ,t = ∆u + −2u (1− |u |), (1.2) the nonlinear heat (NLH) equation, or the Ginzburg-Landau equation; and iu ,t = ∆u + −2u (1− |u |), (1.3) the nonlinear Schrödinger (NLS) equation, on a bounded domain Ω ⊂ R with smooth boundary. The boundary condition is: u |∂Ω = g(x), with g : ∂Ω → S a smooth map of degree d > 0. The initial condition contains d vortices of degree one so that the total initial energy has the asymptotic expression: