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

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.

25 pages, 5 figures, minor typos corrected

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Cited by in corpus (5)

• Parabolic regularization of the gradient catastrophes for the Burgers-Hopf equation and Jordan chain
• On the plane into plane mappings of hydrodynamic type. Parabolic case
• Critical behaviour for scalar nonlinear waves
• Elliptic Euler-Poisson-Darboux equation, critical points and integrable systems
• Jordan form, parabolicity and other features of change of type transition for hydrodynamic type systems