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P. G. Grinevich

Publications and source records attributed to P. G. Grinevich.

42 records · Page 3Linked to original sources
arXiv1997

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.

dg-ga↗
arXiv1997

Conformal invariant functionals of immersions of tori into R^3

We show, that higher analogs of the Willmore functional, defined on the space of immersions M^2\rightarrow R^3, where M^2 is a two-dimensional torus, R^3 is the 3-dimensional Euclidean space are invariant under conformal transformations of R^3. This hypothesis was formulated recently by I.A.Taimanov (dg-ga/9610013). Higher analogs of the Willmore functional are defined in terms of the Modified Novikov-Veselov hierarchy. This soliton hierarchy is associated with the zero-energy scattering problem for the two-dimensional Dirac operator.

dg-ga↗
arXiv1995

Nonisospectral symmetries of the KdV equation and the corresponding symmetries of the Whitham equations

In our paper we construct a new infinite family of symmetries of the Whitham equations (averaged Korteveg-de-Vries equation). In contrast with the ordinary hydrodynamic-type flows these symmetries are nonhomogeneous (i.e. they act nontrivially at the constant solutions), are nonlocal, explicitly depend upon space and time coordinates and form a noncommutative algebra, isomorphic to the algebra of the polynomial vector fields in the complex plane (Virasoro algebra with the zero central charge).

solv-int↗
arXiv1995

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 Schrödinger equation.

solv-int↗
arXiv1995

Nonselfintersecting magnetic orbits on the plane. Proof of Principle of the Overthrowing of the Cycles.

Beginning from 1981 one of the present authors (S.Novikov) published a series of papers, (some of them in collaboration with I.Schmelzer and I.Taimanov) dedicated to the development of the analog of Morse theory for the closed 1-forms -- multivalued functions and functionals -- on the finite - and infinite-dimensional manifolds ({\bf Morse-Novikov Theory}). The notion of ``Multivalued action'' was understood and ``Topological quantization of the coupling constant'' for them was formulated by Novikov in 1981 as a Corollary from the requirement, that the Feinmann Amplitude should be one-valued on the space of fields-maps. Very beautiful analog of this theory appeared also in the late 80-ies in the Symplectic Geometry and Topology, when the so-called Floer Homology Theory was discovered. A very first topological idea of this theory, formulated in early 80-ies, was the so-called ``Principle of the Overthrowing of the Cycles''. It led to the results which were not proved rigorously until now. Our goal is to prove some of them. We study the motion of a classical charged particle on the Euclidean plane in a magnetic field orthogonal to this field. The trajectories of this motion can be characterized as extremals of the ``Maupertui--Fermat'' functional. We show that for any smooth everyvhere positive double periodic magnetic field for any fixed energy there exist at least two different periodic convex extremals, such that the value of the Maupertui-Fermat functional is positive for them. If all such extremals are nondegenerate in the sense of Morse in the space of nonparameterized curves then for any energy there exist at least 4 periodic convex extremals with the Morse indices (1,2,2,3).

solv-int↗