Quasi-classical approximation in vortex filament dynamics. Integrable systems, gradient catastrophe and flutter

TL;DR

This paper explores the quasiclassical approximation in vortex filament dynamics, revealing how gradient catastrophes lead to flutter-like oscillations and are regularized by Painleve I equations.

Contribution

It introduces a quasiclassical framework for vortex filament equations, linking gradient catastrophes to elliptic umbilic singularities and Painleve I regularization.

Findings

Gradient catastrophe causes flutter-like oscillations in vortex filaments.

Double scaling regularization is governed by Painleve I equation.

Elliptic system of PDEs describes slow curvature and torsion variations.

Abstract

Quasiclassical approximation in the intrinsic description of the vortex filament dynamics is discussed. Within this approximation the governing equations are given by elliptic system of quasi-linear PDEs of the first order. Dispersionless Da Rios system and dispersionless Hirota equation are among them. They describe motion of vortex filament with slow varying curvature and torsion without or with axial flow. Gradient catastrophe for governing equations is studied. It is shown that geometrically this catastrophe manifests as a fast oscillation of a filament curve around the rectifying plane which resembles the flutter of airfoils. Analytically it is the elliptic umbilic singularity in the terminology of the catastrophe theory. It is demonstrated that its double scaling regularization is governed by the Painleve' I equation.