Point vortices on the sphere: a case with opposite vorticities

Existence and Stability of Four-Vortex Collinear Relative Equilibria with Three Equal Vorticities

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 organize the solutions into two distinct groups based on the ordering of the vortices along the line. The number and type of solutions are given, along with a discussion of the bifurcations that occur. The linear stability of all solutions is investigated rigorously and stable solutions are found to exist for cases where the vorticities have mixed signs. We emp…

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.

球面上$N$涡旋问题相对平衡稳定分支的确定

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

针对球面上等强度$N$涡旋问题，提出含极点涡旋的多纬圈相对平衡分支的存在性与轨道稳定性证明框架，结合计算机辅助证明规避解析难点，给出$5\leq N\leq12$范围内多个平衡态的稳定性新结果。

Equilibria for the $N$-vortex-problem in a general bounded domain

This article is concerned with the study of existence and properties of stationary solutions for the dynamics of $N$ point vortices in an idealised fluid constrained to a bounded two--dimen\-sional domain $Ω$, which is governed by a Hamiltonian system \[ \left\{\begin{aligned} Γ_i\frac{d x_i}{d t} &=\frac{\partial H_Ω}{\partial y_i}(z_1,\dots,z_N)\\ Γ_i\frac{d y_i}{d t} &=-\frac{\partial H_Ω}{\partial x_i}(z_1,\dots,z_N) \end{aligned} \hspace{2cm}\text{where}\ z_i=(x_i,y_i),\ i=1,\dots,N, \right. \] where $H_Ω(z):=\sum_{j=1}^NΓ_j^2h(z_j)+\sum_{i,j=1, i\not=j}^NΓ_iΓ_jG(z_i,z_j)$ is the so--call…

具有一个主导点涡和四个无穷小点涡的对称相对平衡态

Symmetric relative equilibria with one dominant and four infinitesimal point vortices

本文研究(1+4)-涡问题中一个主导涡与四个无穷小涡的对称相对平衡态，分类了矩形、梯形和风筝形等构型，并证明(1+4)-边形对角涡环量必须相等，证明依赖代数几何与Gröbner基计算。