Устойчивость правильных вихревых многоугольников в конденсате Бозе-Эйнштейна

Рассматривается задача об устойчивости вращающихся правильных вихревых $N$-угольников (томсоновских конфигурации) в конденсате Бозе-Эйнштейна в гармонической ловушке. Получена зависимость скорости вращения $\omega$ томсоновской конфигурации вокруг центра ловушки в зависимости от количества вихрей $N$ и радиуса конфигурации $R$. Выполнен анализ устойчивости движения таких конфигураций в линейном приближении. Для $N \leqslant 6$ построены области орбитальной устойчивости конфигураций в пространстве параметров. Показано, что вихревые $N$-угольники для $N > 6$ при любых параметрах системы неустойчивы

Stability of Relative Equilibria in the Planar N-Vortex Problem

We study the linear and nonlinear stability of relative equilibria in the planar N-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vorticities is also nonlinearly stable. Two symmetric families, the rhombus and the isosceles trapezoid, are analyzed in detail, with stable solutions found in each case.Comment: 23 pages, 3 figure

Transition of global dynamics of a polygonal vortex ring on a sphere with pole vortices

This paper deal with the motion of a polygonal ring of identical vortex points that are equally spaced at a line of latitude on a sphere with vortex points fixed at the both poles, which we call "N-ring". We give not only all the eigenvalues but also all the eigenvectors corresponding to them for the linearized steationary N-ring. Then, we also reduce the equations to those for a pair of two vortex points, when N is even. As a consequence of the mathematical and numerical studies of the reduced system, we obtain a transition of global periodic motions of the perturbed N-ring and the stability of these periodic motions

Deformation of geometry and bifurcation of vortex rings

We construct a smooth family of Hamiltonian systems, together with a family of group symmetries and momentum maps, for the dynamics of point vortices on surfaces parametrized by the curvature of the surface. Equivariant bifurcations in this family are characterized, whence the stability of the Thomson heptagon is deduced without recourse to the Birkhoff normal form, which has hitherto been a necessary tool.Comment: 26 page

Incompressible viscous fluid flows in a thin spherical shell

Linearized stability of incompressible viscous fluid flows in a thin spherical shell is studied by using the two-dimensional Navier--Stokes equations on a sphere. The stationary flow on the sphere has two singularities (a sink and a source) at the North and South poles of the sphere. We prove analytically for the linearized Navier--Stokes equations that the stationary flow is asymptotically stable. When the spherical layer is truncated between two symmetrical rings, we study eigenvalues of the linearized equations numerically by using power series solutions and show that the stationary flow remains asymptotically stable for all Reynolds numbers.Comment: 28 pages, 10 figure

Determination of stable branches of relative equilibria of the NN-vortex problem on the sphere

We consider the $N$-vortex problem on the sphere assuming that all vorticities have equal strength. We investigate relative equilibria (RE) consisting of $n$ latitudinal rings which are uniformly rotating about the vertical axis with angular velocity $\omega$. Each such ring contains $m$ vortices placed at the vertices of a concentric regular polygon and we allow the presence of additional vortices at the poles. We develop a framework to prove existence and orbital stability of branches of RE of this type parametrised by $\omega$. Such framework is implemented to rigorously determine and prove stability of segments of branches using computer-assisted proofs. This approach circumvents the analytical complexities that arise when the number of rings $n\geq 2$ and allows us to give several new rigorous results. We exemplify our method providing new contributions consisting in the determination of enclosures and proofs of stability of several equilibria and RE for $5\leq N\leq 12$.Comment: 60 page

Nonlinear stability of a latitudinal ring of point-vortices on a nonrotating sphere

Abstract. We study the nonlinear stability of relative equilibria of configurations of identical point-vortices on the surface of a sphere. In particular, we study how the stability changes as a function of the colatitude θ and of the number of vortices N. By using the integrals of motion, we view the system in a suitable corotating frame where the polygonal vortex configuration is at rest. Then after a sufficient criterion due to Dirichlet, the stability ranges are the θ-intervals for which the Hessian of the Hamiltonian—evaluated at the equilibrium configuration—is positive or negative definite. We find that the stability intervals coincide with those for linear stability determined by Polvani and Dritschel [J. Fluid Mech., 255 (1993), pp. 35–64]. For N = 3 we recover the result previously established by Pekarsky and Marsden [J. Math. Phys., 39 (1998), pp. 5894–5907]