Related papers: Stability of closed solutions to the VFE hierarchy…
In this paper, we consider the evolution of the Vortex Filament equation (VFE): \begin{equation*} \mathbf X_t = \mathbf Xs \wedge \mathbf Xss, \end{equation*} taking $M$-sided regular polygons with nonzero torsion as initial data. Using…
In this paper, we study the evolution of the vortex filament equation (VFE), $$\mathbf X_t = \mathbf X_s \wedge \mathbf X_{ss},$$ with $\mathbf X(s, 0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical…
We consider a nonlinear model equation, known as the Localized Induction Equation, describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. We show stability estimates for an arc-shaped vortex filament,…
We review some recent results concerning the evolution of a vortex filament and its relation to the cubic non-linear Schr\"odinger equation. Selfsimilar solutions and questions related to their stability are studied.
For the class of quasi-periodic solutions of the vortex filament equation, we study connections between the algebro-geometric data used for their explicit construction and the geometry of the evolving curves. We give a complete description…
The vortex filament equations (VFE) in 1+1 and 2+1 dimensions are considered. Some of these equations are integrable. Also the VFE with potentials and with self-consistent potentials are presented. Finally several examples of integrable…
This paper investigates the dynamics of closed vortex filaments in $\R^3$ governed by the Localized Induction Equation. Recently, Aiki and Higaki (2026) established the nonlinear orbital stability of circular vortex filaments under…
We establish the nonlinear orbital stability of circular vortex filaments governed by the Localized Induction Equation (LIE) under non-symmetric perturbations, within the framework of [Tani-Nishiyama, 1997]. This result extends the first…
We address the existence and stability of vortex-soliton (VS) solutions of the fractional nonlinear Schr\"odinger equation (NLSE) with competing cubic-quintic nonlinearities and the L\'evy index (fractionality) taking values 1…
This article describes a Hitchin-Kobayashi style correspondence for the Vafa-Witten equations on smooth projective surfaces. This is an equivalence between a suitable notion of stability for a pair $(\mathcal{E}, \varphi)$, where…
We consider as base flow the stationary vortex filament solution obtained by Castillo- Castellanos et al. (2021) in the far-wake of a rotor with tip-splitting blades. The cases of a single blade and of two blades with a hub vortex are…
In this paper we continue our investigation about selfsimilar solutions of the vortex filament equation, also known as the binormal flow (BF) or the localized induction equation (LIE). Our main result is the stability of the selfsimilar…
The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow…
We investigate stability properties of a type of periodic solutions of the $N$-vortex problem on general domains $\Omega\subset \mathbb{R}^2$. The solutions in question bifurcate from rigidly rotating configurations of the whole-plane…
This paper studies the stability of velocity-pressure mixed approximations of the Stokes problem when different finite element (FE) spaces for each component of the velocity field are considered. We consider some new combinations of…
The system of a closed vortex filament is an integrable Hamiltonian one, namely, a Hamiltonian system with an infinite sequense of constants of motion in involution. An algebraic framework is given for the aim of describing differential…
We consider solutions of the Navier-Stokes equations in $3d$ with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth…
We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement…
In this paper we use bifurcation methods to construct a new family of solutions of the binormal flow, also known as the vortex filament equation, which do not change their form. Our examples are complementary to those obtained by S. Kida in…
It is known that given a stable holomorphic pair $(E ,\phi)$, where $E$ is a holomorphic vector bundle on a compact K\"ahler manifold $X$ and $\phi$ is a holomorphic section of $E$, the vector bundle $E$ admits a Hermitian metric solving…