Dynamics of poles with position-dependent strengths and its optical analogues

TL;DR

This paper explores a generalized vortex dynamics system with position-dependent and complex strengths, revealing optical analogues like Snell's law and reflection, and demonstrating rich nonlinear behavior with conservation laws.

Contribution

It introduces a novel vortex model with position-dependent strengths and complex values, uncovering optical analogues and conservation laws in a nonlinear dynamical system.

Findings

Motion of vortex pairs mimics ray behavior in optics

Exact solutions include Snell's law and reflection laws

Numerical experiments show striking vortex dynamics

Abstract

The dynamics of point vortices is generalized in two ways: first by making the strengths complex, which allows for sources and sinks in superposition with the usual vortices, second by making them functions of position. These generalizations lead to a rich dynamical system, which is nonlinear and yet has conservation laws coming from a Hamiltonian-like formalism. We then discover that in this system the motion of a pair mimics the behavior of rays in geometric optics. We describe several exact solutions with optical analogues, notably Snell's law and the law of reflection off a mirror, and perform numerical experiments illustrating some striking behavior.