Vortex dynamics on a cylinder

Point vortices on a cylinder (periodic strip) are studied geometrically. The Hamiltonian formalism is developed, a non-existence theorem for relative equilibria is proved, equilibria are classified when all vorticities have the same sign, and several results on relative periodic orbits are established, including as corollaries classical results on vortex streets and leapfrogging.Comment: LaTeX2e, 13 pages, 5 figure

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

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.Comment: 10 page

The spectral properties and singularities of monodromy-free Schroedinger operators

The main object of study is the theory of Schrödinger operators with meromorphic potentials, having trivial monodromy in the complex domain. In the first part we study the spectral properties of a class of such operators related to the classical Whittaker-Hill equation (-d^2/dx^2+Acos2x+Bcos4x)ψ=λψ. The equation, for special choices of A and B, is known to have the remarkable property that half of the gaps eventually become closed (semifinite-gap operator). Using the Darboux transformation we construct new trigonometric examples of semifinite-gap operators with real, smooth potentials. A similar technique applied to the Lamé operator gives smooth, real, finite-gap potentials in terms of classical Jacobi elliptic functions. In the second part we study the singular locus of monodromy-free potentials in the complex domain. A particular case is given by the zeros of Wronskians of Hermite polynomials, which are studied in detail. We introduce a class of partitions (doubled partitions) for which we observe a direct qualitative relationship between the pattern of zeros and the shape of the corresponding Young diagram. For the Wronskians W(H_n,H_{n+k}) we give an asymptotic formula for the curve on which zeros lie as n → ∞. We also give some empirical formulas for asymptotic behaviour of zeros of Wronskians of 3 and 4 Hermite polynomials. In the last chapter we apply the theory of monodromy-free operators to produce new vortex equilibria in the periodic case and in the presence of background flow

Vortex crystals

Vortex crystals is one name in use for the subject of vortex patterns that move without change of shape or size. Most of what is known pertains to the case of arrays of parallel line vortices moving so as to produce an essentially two-dimensional flow. The possible patterns of points indicating the intersections of these vortices with a plane perpendicular to them have been studied for almost 150 years. Analog experiments have been devised, and experiments with vortices in a variety of fluids have been performed. Some of the states observed are understood analytically. Others have been found computationally to high precision. Our degree of understanding of these patterns varies considerably. Surprising connections to the zeros of 'special functions' arising in classical mathematical physics have been revealed. Vortex motion on two-dimensional manifolds, such as the sphere, the cylinder (periodic strip) and torus (periodic parallelogram) has also been studied, because of the potential applications, and some results are available regarding the problem of vortex crystals in such geometries. Although a large amount of material is available for review, some results are reported here for the first time. The subject seems pregnant with possibilities for further development.published or submitted for publicationis peer reviewe

Dynamique de N pôles à intensités variables

Thèse numérisée par la Direction des bibliothèques de l'Université de Montréal

Point vortex dynamics on K\"ahler twistor spaces

In this paper, we provide tools to study the dynamics of point vortex dynamics on $\mathbb{CP}^n$ and the flag manifold \mathbb{F}_{1,2}(\C^3). These are the only K\"ahler twistor spaces arising from 4-manifolds. We give an explicit expression for Green's function on $\mathbb{CP}^n$ {which enables us to} determine the Hamiltonian $H$ and the equations of motions for the point vortex problem on $\mathbb{CP}^n$. Moreover, we determine the momentum map \mu:\mathbb{F}_{1,2}(\C^3)\to \mathfrak{su}^*(3) on the flag manifold.Comment: 38 page

Periodic Vortex Streets and Complex Monodromy

The explicit constructions of periodic and doubly periodic vortex relative equilibria using the theory of monodromy-free Schrödinger operators are described. Several concrete examples with the qualitative analysis of the corresponding travelling vortex streets are given