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.

arXiv comment25 pages, 5 figures, minor typos corrected

References in corpus (24)

• Integrable Hierarchies and Dispersionless Limit
• Conformal maps and dispersionless integrable hierarchies
• Integrable hydrodynamic chains
• SDiff(2) Toda equation -- hierarchy, $\tau$ function, and symmetries
• Laplacian Growth and Whitham Equations of Soliton Theory
• On the stability of a singular vortex dynamics
• Local Geometric Invariants of Integrable Evolution Equations
• Universality for the focusing nonlinear Schroedinger equation at the gradient catastrophe point: Rational breathers and poles of the tritronquee solution to Painleve I
• The Semiclassical Modified Nonlinear Schroedinger Equation I: Modulation Theory and Spectral Analysis
• Nonlinear steepest descent asymptotics for semiclassical limit of integrable systems: Continuation in the parameter space
• Finite-gap Solutions of the Vortex Filament Equation: Isoperiodic Deformations
• Hodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler-Poisson-Darboux equation
• Hasimoto Transformation and Vortex Soliton Motion Driven by Fluid Helicity
• Differential Geometry of the Vortex Filament Equation
• The double scaling limit method in the Toda hierarchy
• Generalized Hasimoto Transform of One-Dimensional Dispersive Flows into Compact Riemann Surfaces
• On the singular sector of the Hermitian random matrix model in the large N limit
• The semiclassical limit of focusing NLS for a family of non-analytic initial data
• Semiclassical limit of the scattering transform for the focusing Nonlinear Schr\" odinger Equation
• Singular sectors of the 1-layer Benney and dToda systems and their interrelations
• The Zero-Dispersion Limit for the Odd Flows in the Focusing Zakharov-Shabat Hierarchy
• Gradient catastrophe and flutter in vortex filament dynamics
• The hodograph method applicability in the problem of long-scale nonlinear dynamics of a thin vortex filament near a flat boundary
• Generalized Local Induction Equation, Elliptic Asymptotics, and Simulating Superfluid Turbulence

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