Quantized Vortex Dynamics of the Nonlinear Schr\"odinger Equation with Wave Operator on the Torus

TL;DR

This paper rigorously derives the vortex dynamics law for a nonlinear Schrödinger equation with wave operator on a torus, revealing a mixed vortex motion law and confirming convergence through numerical simulations.

Contribution

It introduces a new reduced dynamical law for vortex motion in the nonlinear Schrödinger equation with wave operator, combining laws from related equations.

Findings

Derived the reduced vortex dynamics law as vortex core size approaches zero.

Proved the vortex law is a mixture of laws from nonlinear wave and Schrödinger equations.

Numerical simulations confirm convergence to the vortex motion law.

Abstract

We derive rigorously the reduced dynamical law for quantized vortex dynamics of the nonlinear Schr\"odinger equation with wave operator on the torus when the core size of vortex $\varepsilon \to 0$. It is proved that the reduced dynamical law of the nonlinear Schr\"odinger equation with wave operator is a mixed state of the vortex motion laws for the nonlinear wave equation and the nonlinear Schr\"odinger equation. We will also investigate the convergence of the reduced dynamical law of the nonlinear Schr\"odinger equation with wave operator to the vortex motion law of the nonlinear Schr\"odinger equation via numerical simulation.