相关论文: Leapfrogging vortex rings in the Landau-Lifshitz e…
A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid $3$-dimensional fluid. Helmholtz (1858) observed that a pair of similar thin, coaxial…
The evolution of highly concentrated vorticity around rings in the three-dimensional axisymmetric Euler equations is studied in a regime for which the leapfrogging dynamics predicted by Helmholtz is expected to occur. We provide in this…
Vortex ring solutions are presented for the Landau-Lifshitz equation, which models the dynamics of a three-dimensional ferromagnet. The vortex rings propagate at constant speed along their symmetry axis and are characterized by the…
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…
The dynamics of vortex ring pairs in the homogeneous nonlinear Schr\"odinger equation is studied. The generation of numerically-exact solutions of traveling vortex rings is described and their translational velocity compared to revised…
We consider an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside $N$ small disjoint rings of thickness $\varepsilon$, each one of vorticity mass and…
In the present note, we solved numerically the viscous vorticity equation of the initial-boundary value problem describing the classic Helmholtz phenomena of vortex interaction. In the leapfrogging of vortex pairs, we demonstrate the fact…
Solitons which have the form of a vortex-antivortex pair have recently been found in the Landau-Lifshitz equation which is the standard model for the ferromagnet. We simulate numerically head-on collisions of two vortex-antivortex pairs and…
Leapfrogging motion of vortex rings sharing the same axis of symmetry was first predicted by Helmholtz in his famous work on the Euler equation for incompressible fluids. Its justification in that framework remains an open question to date.…
The leapfrogging of coaxial vortex rings is a famous effect which has been noticed since the times of Helmholtz. Recent advances in ultra-cold atomic gases show that the effect can now be studied in quantum fluids. The strong confinement…
Vortex rings are remarkably stable structures occurring in numerous systems: for example in turbulent gases, where they are at the origin of weather phenomena [1]; in fluids with implications for biology [2]; in electromagnetic discharges…
We study the Landau-Lifshitz-Gilbert equation for the dynamics of a magnetic vortex system. We present a PDE-based method for proving vortex dynamics that does not rely on strong well-preparedness of the initial data and allows for…
The field of spin hydrodynamics aims to describe magnetization dynamics from a fluid perspective. For ferromagnetic materials, there is an exact mapping between the Landau-Lifshitz equation and a set of dispersive hydrodynamic equations.…
Systems of coaxial vortex pairs in an inviscid flow give rise to complex dynamics, with motions ranging from ordered to chaotic. This complexity arises due to the problem's high nonlinearity and numerous degrees of freedom. We analyze the…
The dynamics of N point vortices in a fluid is described by the Helmholtz-Kirchhoff (HK) equations which lead to a completely integrable Hamiltonian system for N=2 or 3 but chaotic dynamics for N>3. Here we consider a generalization of the…
In this work we study the dynamical behavior of two interacting vortex pairs, each one of them consisting of two point vortices with opposite circulation in the 2d plane. The vortices are considered as effective particles and their…
The Landau-Lifshitz-Gilbert (LLG) equation is a fascinating nonlinear evolution equation both from mathematical and physical points of view. It is related to the dynamics of several important physical systems such as ferromagnets, vortex…
We study the Landau-Lifshitz equation of ferromagnetism on $\mathbb{R}^2$ with an easy-axis anisotropy. We give the necessary condition for the existence of the finite energy vortex solutions and show the behaviors of the solutions.
The motion of a vortex-(anti)vortex pair is studied numerically in the framework of a dynamical Ginzburg-Landau model, relevant to the description of a superconductor or of an idealized bosonic plasma. It is shown that up to a fine…
We establish vortex dynamics for the time-dependent Ginzburg-Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we…