Related papers: Reduction of the planar 4-vortex system at zero mo…
We examine in detail the relative equilibria of the 4-vortex problem when three vortices have equal strength, that is, $\Gamma_{1} = \Gamma_{2} = \Gamma_{3} = 1$, and $\Gamma_{4}$ is a real parameter. We give the exact number of relative…
We investigate some properties of the dynamical system of point vortices on the hyperboloid. This system has noncompact symmetry SL(2, R) and a coadjoint equivariant momentum map J. The relative equilibrium conditions are found and the…
We examine in detail the relative equilibria in the four-vortex problem where two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and \Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is…
This paper addresses the question of existence of (not necessarily self-similar) solutions to the 4-vortex problem that lead to total or partial collision. We begin by showing that energy considerations alone imply that, for the general…
The motion of two pairs of counter-rotating point vortices placed in a uniform flow past a circular cylinder is studied analytically and numerically. When the dynamics is restricted to the symmetric subspace---a case that can be realized…
A system of three point vortices in an unbounded plane has a special family of self-similarly contracting or expanding solutions: during the motion, vortex triangle remains similar to the original one, while its area decreases (grows) at a…
We give an exact quantitative solution for the motion of three vortices of any strength, which Poincar\'e showed to be integrable. The absolute motion of one vortex is generally biperiodic: in uniformly rotating axes, the motion is…
We obtained new periodic solutions in the problems of three and four point vortices moving on a plane. In the case of three vortices, the system is reduced to a Hamiltonian system with one degree of freedom, and it is integrable. In the…
We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two…
We give a method to determine relative periodic orbits in point vortex systems: it consists mainly into perform a symplectic reduction on a fixed point submanifold in order to obtain a two-dimensional reduced phase space. The method is…
We investigate the dynamics of $N$ point vortices in the plane, in the limit of large $N$. We consider {\em relative equilibria}, which are rigidly rotating lattice-like configurations of vortices. These configurations were observed in…
We study the dynamics of $N$ point vortices on a rotating sphere. The Hamiltonian system becomes infinite dimensional due to the non-uniform background vorticity coming from the Coriolis force. We prove that a relative equilibrium formed of…
We studied formation of vortex with four-fold symmetry in a minimal model of self-propelled particles, confined inside a squared box, using computer simulations and also theoretical analysis. In addition to the vortex pattern, we observed…
In this article we considered the integrable problems of three vortices on a plane and sphere for noncompact case. We investigated explicitly the problems of a collapse and a scattering of vortices and obtained the conditions of its…
We study collinear relative equilibria of the planar four-vortex problem where three of the four vortex strengths are identical. The $S_3$ invariance obtained from the equality of vorticities is used to reduce the defining equations and…
Helmholtz's equations provide the motion of a system of N vortices which describes a planar incompressible fluid with zero viscosity. A relative equilibrium is a particular solution of these equations for which the distances between the…
We analyze existence, stability, and symmetry of point vortex relative equilibria with one dominant vortex and N vortices with infinitesimal circulation. The dimension of the problem can be reduced by taking an infinitesimal circulation…
The possibility of the appearance of the C$_{4}$ symmetry in the rotational bands is studied within the particle-rotor model. Allowing for a slight triaxiality of the rotor it is shown that the lowest states of the system for a sufficiently…
Numerical approximations of finite systems of vortices in the plane, collapsing to a point in finite time are studied. A fast method leading to highly accurate approximations is presented. The method is illustrated with an example of an…
We prove a sufficient condition for nonlinear stability of relative equilibria in the planar $N$-vortex problem. This result builds on our previous work on the Hamiltonian formulation of its relative dynamics as a Lie--Poisson system. The…