Vortex dynamics for nonlinear Klein–Gordon equation

Abstract

We consider the vortex motion laws governed by the nonlinear Klein-Gordon equation. Three limits are determined when the light speed c → ∞ and → 0. It is shown that the vortex motion law governed by the nonlinear Klein-Gordon equation is an intermediate state that connects both the Kirchhoff's law for the Schrödinger equation and the vortex motion law for the wave equation.

Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity

“…where L 0 is defined by (26). Observe that φ 0 belongs to H with H defined in (23) because of our assumption on ψ n .…”

“…For the case on smooth bounded domain, C. Lin and K. Wu [19] studied the singular limits (including semiclassical ( → 0), nonrelativistic (c → 0), and nonrelativistic-semiclassical limits) of the cauchy problem for (1). Later on, Y. Yu [26] concerned the vortex dynamics for (1) on two dimensional smooth bounded domain, and established that the solitary vortex motion law was an intermediate state that connects both the Kirchhoff's law for the Schrödinger equation and the vortex motion law for the wave equation. In the present paper, we will construct traveling solitary wave solutions with various vortex structures(vortex pairs, vortex rings) for (1).…”