5 research outputs found
Quasi-classical approximation in vortex filament dynamics. Integrable systems, gradient catastrophe and flutter
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.Comment: 25 pages, 5 figures, minor typos correcte
Systems of quasilinear equations and their applications to gas dynamics
This book is essentially a new edition, revised and augmented by results of the last decade, of the work of the same title published in 1968 by "Nauka." It is devoted to mathematical questions of gas dynamics. Topics covered include Foundations of the Theory of Systems of Quasilinear Equations of Hyperbolic Type in Two Independent Variables; Classical and Generalized Solutions of One-Dimensional Gas Dynamics; Difference Methods for Solving the Equations of Gas Dynamics; and Generalized Solutions of Systems of Quasilinear Equations of Hyperbolic Type