Integrable Vortex Dynamics and Complex Burgers' Equation

Abstract

Integrable dynamical models of the point magnetic vortex interactions in the plane are studied. Reformulating the Euler equations for vorticity in the Helmholtz form, the Hamiltonian and Lax representations are found. Reduction of these equations for the point vortices to the Kirchho equations, and non-integrability of the system of N 4 hydrodynamical vortices are discussed.As an integrable model of planar motion with given vorticity for the stationary and its solutions are given. For non-stationary planar vortex diffusion and exactly solvable Initial Value Problem for the one dimensional Burgers equation are solved.By the complexied Cole-Hopf transformation, the complex Burgers equation with integrable N vortex dynamics is introduced and linearization of this equation in terms of the complex Schr odinger equation is found.This allows us to construct N vortex congurations in terms of the complex Hermite polynomials, the vortex chain lattices and study their mutual dynamics. Mapping of our vortex problem to N-particle problem, the complexied Calogero-Moser system, showing its integrability and Hamiltonian structure is given. As an applicaton of the general results, we consider the problem of magnetic vortices in a magnetic fluid model. The holomorphic reduction of topological magnetic system to the linear complex Schrodinger equation, allows us to apply all results on integrable vortex dynamics in the complex Burgers equation to the magnetic vortex evolution, including magnetic vortex lattices and the bound states of vortices.