Related papers: Existence of corotating asymmetric vortex pairs fo…
We study pairs of co-axial vortex rings starting from the action for a classical bosonic string in a three-form background. We complete earlier work on the phase diagram of classical orbits by explicitly considering the case where the…
In this paper, we study the uniformly rotating vortex patch solutions for the 2D incompressible Euler equations. Specifically, we prove that if the patch solution is close to the Rankine vortex in a certain weak topology, it is either the…
We show that horizontally symmetric water waves are traveling waves. The result is valid for the Euler equations, and is based on a general principle that applies to a large class of nonlinear partial differential equations, including some…
The main goal of this paper is to explore the leapfrogging phenomenon in the inviscid planar flows. We show for 2d Euler equations that under suitable constraints, four concentrated vortex patches leapfrog for all time. When observed from a…
We provide a variational construction of special solutions to the generalized surface quasi-geostrophic equations. These solutions take the form of N vortex patches with N-fold symmetry , which are steady in a uniformly rotating frame.…
A {\em vortex pair} solution of the incompressible $2d$ Euler equation in vorticity form $$ \omega_t + \nabla^\perp \Psi\cdot \nabla \omega = 0 , \quad \Psi = (-\Delta)^{-1} \omega, \quad \hbox{in } \mathbb{R}^2 \times (0,\infty)$$ is a…
The aim of the present work is to examine the role of discreteness in the interaction of both co-winding and counter-winding vortices in the context of the nonlinear Schr{\"o}dinger equation. Contrary to the well-known rotation of same…
We develop a coarse-grained description of the point-vortex model, finding that a large number of planar vortices and antivortices behave as an inviscid non-Eulerian fluid at large scales. The emergent binary vortex fluid is subject to…
We construct a series of patch type solutions for incompressible Euler equation on $\mathbb S^2$, which constitutes the regularization for steady or traveling point vortex systems. We first prove the existence of $k$-fold symmetric patch…
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…
In this paper, we obtain families of two-fold doubly-connected uniformly rotating vortex patches of the 2-D incompressible Euler equations emanating from some specific annuli. The main difficulty comes from strong degeneracy of the problem,…
We rigorously construct the first steady traveling wave solutions of the 2D incompressible Euler equation that take the form of a contiguous vortex-patch dipole, which can be viewed as the vortex-patch counterpart of the well-known…
We reveal the existence of asymmetric vortex solitons in ideally symmetric periodic lattices, and show how such nonlinear localized structures describing elementary circular flows can be analyzed systematically using the energy-balance…
We construct traveling wave solutions to the 3d Euler equations by axisymmetric Beltrami fields with a non-constant proportionality factor. They form a vortex ring with nested invariant tori consisting of level sets of the proportionality…
We consider massive vortices in binary condensates, where the immiscibility condition entails the trapping of the minority component in the vortex cores of the majority component. We study such vortices by means of a 2D point-like model,…
The motion of a two-dimensional buoyant vortex patch, i.e. a vortex patch with a uniform density different from the uniform density of the surrounding fluid, is analyzed in terms of evolution equations for the motion of its centroid,…
We propose and analyze a system of nonlinear partial differential equations describing the motion of a pair of vortex filaments. Furthermore, for a pair of coaxial circular vortex filaments, we derive a condition for leapfrogging to occur…
Vortices symmetric with respect to simultaneous parity and time reversing transformations are considered on the square lattice in the framework of the discrete nonlinear Schr\"{o}dinger equation. The existence and stability of vortex…
The problem of vortex pair motion in two-dimensional plane radial flow is solved. Under certain conditions for flow parameters, the vortex pair can reverse its motion within a bounded region. The vortex-pair translational velocity decreases…
We prove the existence of time-periodic leapfrogging vortex rings for the three-dimensional incompressible Euler equations, thereby providing a rigorous realization of a phenomenon first conjectured by Helmholtz (1858). In the leapfrogging…