Period preserving nonisospectral flows and the moduli space of periodic solutions of soliton equations

Flows on the moduli space of the algebraic Riemann surfaces, preserving the periods of the corresponding solutions of the soliton equations are studied. We show that these flows are gradient with respect to some indefinite symmetric flat metric arising in the Hamiltonian theory of the Whitham equations. The functions generating these flows are conserved quantities for all the equations simultaneously. We show that for 1+1 systems these flows can be imbedded in a larger system of ordinary nonlinear differential equations with a rational right-hand side. Finally these flows are used to give a complete description of the moduli space of algebraic Riemann surfaces corresponding to periodic solutions of the nonlinear Schr\"odinger equation.Comment: 35 pages, LaTex. Macros file elsart.sty is used (it was submitted by the authors to [email protected] library macroses),e-mail: [email protected], e-mail:[email protected]

Closed curves in R^3: a characterization in terms of curvature and torsion, the Hasimoto map and periodic solutions of the Filament Equation

If a curve in R^3 is closed, then the curvature and the torsion are periodic functions satisfying some additional constraints. We show that these constraints can be naturally formulated in terms of the spectral problem for a 2x2 matrix differential operator. This operator arose in the theory of the self-focusing Nonlinear Schrodinger Equation. A simple spectral characterization of Bloch varieties generating periodic solutions of the Filament Equation is obtained. We show that the method of isoperiodic deformations suggested earlier by the authors for constructing periodic solutions of soliton equations can be naturally applied to the Filament Equation.Comment: LaTeX, 27 pages, macros "amssym.def" use

Gradient catastrophe and flutter in vortex filament dynamics

Gradient catastrophe and flutter instability in the motion of vortex filament within the localized induction approximation are analyzed. It is shown that the origin if this phenomenon is in the gradient catastrophe for the dispersionless Da Rios system which describes motion of filament with slow varying curvature and torsion. Geometrically this catastrophe manifests as a rapid oscillation of a filament curve in a point that 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 Painlev\'e-I equation.Comment: 11 pages, 3 figures, typos corrected, references adde