arXiv subjects

Masashi Aiki

Publications and source records attributed to Masashi Aiki.

11 recordsLinked to original sources

arXiv2026

Lyapunov Unstable Motion Bifurcating from a Circular Vortex Filament

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 asymmetric perturbations, while identifying Lyapunov instability due to the linear growth of translation modes. Motivated by this result, we prove the existence of a family of closed solutions, which we call axial screw motions, that bifurcate from a circular filament. These solutions remain uniformly close to the orbit of the circle, but drift secularly away from the reference motion because their translation speed along the symmetry axis differs from that of the circular filament. In particular, they provide explicit non-trivial perturbations that satisfy orbital-stability estimates while failing Lyapunov stability, thereby realizing the gap between orbital stability and Lyapunov stability near the circular filament.

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arXiv2026

Stability of the Shape for Circular Vortex Filaments under Non-Symmetric Perturbations

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 author's recent work [Aiki, 2025] by removing symmetry assumptions on perturbations. While the circular filaments are known to be Lyapunov unstable due to linear growth of the translational mode, we prove that their shape remains globally stable modulo spatial translations and rotations about the symmetry axis. The crucial ingredient is a geometric stability lemma derived from the conservation of vector fluid impulse, which constrains the low-frequency modulations that are not covered by the relative energy. Finally, we relate the fluid impulse to an isoperimetric inequality, yielding a geometric constraint for closed filaments.

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arXiv2024

Long-time behavior of an Arc-shaped Vortex Filament and its Application to the Stability of a Circular Vortex Filament

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, which is an exact solution to an initial-boundary value problem for the Localized Induction Equation. An arc-shaped filament travels along an axis at a constant speed without changing its shape, and is oriented in such a way that the arc stays in a plane that is perpendicular to the axis. We prove that an arc-shaped filament is stable in the Lyapunov sense for general perturbations except in the axis-direction, for which the perturbation can grow linearly in time. We also show that this estimate is optimal. We then apply the obtained stability estimates to study the stability of a circular vortex filament under some symmetry assumptions on the initial perturbation. We do this by dividing the circular filament into arcs, apply the stability estimate to each arc-shaped filament, and combine the estimates to obtain estimates for the whole circle. The optimality of the stability estimates for an arc-shaped filament also shows that a circular filament is not stable in the Lyapunov sense, namely, certain perturbations can grow linearly in time.

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arXiv2022

The Hasimoto Transformation for a Finite Length Vortex Filament and its Application

We consider two nonlinear equations, the Localized Induction Equation and the cubic nonlinear Schrödinger Equation, and prove that the solvability of certain initial-boundary value problems for each equation is equivalent through the generalized Hasimoto transformation. As an application, we prove the orbital stability of plane wave solutions of the nonlinear Schrödinger equation based on stability estimates obtained for the Localized Induction Equation by the author in a paper in preparation. As far as the author knows, this is the first time that the analysis of the Localized Induction Equation, along with the Hasimoto transformation, provided new insight for the nonlinear Schrödinger equation.

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arXiv2020

Head-On Collision of a Pair of Coaxial Circular Vortex Filament

We consider the head-on collision of two coaxial vortex rings described as the motion of two circular vortex filaments under the localized induction approximation. We prove the existence of solutions to a system of nonlinear partial differential equations proposed by the author which exhibit head--on collision. We also give a necessary and sufficient condition for the initial configuration and parameters of the filaments for head-on collision to occur. Our results suggest that there exists a critical value $ γ_{\ast}>1 $ for the ratio $ γ$ of the absolute value of the circulations such that when $ γ\in [1,γ_{\ast}]$, two approaching rings will collide, and when $ γ\in (γ_{\ast},\infty) $, the ring with the larger circulation passes through the other and then separate indefinitely.

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arXiv2018

Motion of a Vortex Filament in an External Flow

We consider a nonlinear model equation describing the motion of a vortex filament immersed in an incompressible and inviscid fluid. In the present problem setting, we also take into account the effect of external flow. We prove the unique solvability, locally in time, of an initial value problem posed on the one dimensional torus. The problem describes the motion of a closed vortex filament.

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arXiv2018

On the Existence of Leapfrogging Pair of Circular Vortex Filaments

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 and prove that the condition is necessary and sufficient for the occurrence of leapfrogging.

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arXiv2017

Motion of a Vortex Filament on a Slanted Plane

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arXiv2012

Motion of a Vortex Filament with Axial Flow in the Half Space

We consider a nonlinear third order dispersive equation which models the motion of a vortex filament immersed in an incompressible and inviscid fluid occupying the three dimensional half space. We prove the unique solvability of initial-boundary value problems as an attempt to analyze the motion of a tornado.

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arXiv2012

Solvability of an Initial-Boundary Value Problem for a Second Order Parabolic System with a Third Order Dispersion Term

We consider a linear second order parabolic system with a third order dispersion term. This type of system arises when considering a nonlinear model equation describing the motion of a vortex filament with axial flow immersed in an incompressible and inviscid fluid. We prove the solvability of an initial-boundary value problem of the parabolic-dispersive system which allows application to the motion of a vortex filament. To do so, we propose a new regularization technique by adding a space-time derivative term.

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arXiv2010

Motion of a Vortex Filament in the Half Space

A model equation for the motion of a vortex filament immersed in three dimensional, incompressible and inviscid fluid is investigated as a humble attempt to model the motion of a tornado. We solve an initial-boundary value problem in the half space where we impose a boundary condition in which the vortex filament is allowed to move on the boundary.

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