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I. A. Taimanov

Publications and source records attributed to I. A. Taimanov.

At least 37 records · Page 2Linked to original sources
arXiv2016

On an integrable magnetic geodesic flow on the two-torus

We completely integrate the magnetic geodesic flow on a flat two-torus with the magnetic field $F = \cos (x) dx \wedge dy$ and describe all contractible periodic magnetic geodesics. It is shown that there are no such geodesics for energy $E \geq 1/2$, for $E< 1/2$ simple periodic magnetic geodesics form two $S^1$-families for which the (fixed energy) action functional is positive and therefore there are no periodic magnetic geodesics for which the action functional is negative.

math.DS↗
arXiv2014

On a high-dimensional generalization of Seifert fibrations

The notion of generalized Seifert fibration is introduced, it is shown that the projections of certain Eschenburg $7$-manifolds onto ${\mathbb C} P^2$ define such fibrations, and for them the characteristic classes corresponding to the generators of $H^2(B(U(2)/{\mathbb Z}_{2n});{\mathbb Z})$ are defined.

math.GT↗
arXiv2013

Faddeev eigenfunctions for two-dimensional Schrodinger operators via the Moutard transformation

We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying potentials of operators with non-trivial kernel and for deformed potentials which correspond to blowing up solutions of the Novikov-Veselov equation.

math-ph↗
arXiv2011

Singular spectral curves in finite gap integration

We expose some applications of the finite gap integration method which involve singular spectral curves. They include orthogonal curvilinear coordinate systems corresponding to spectral curves with geometrical genus zero, finite gap Frobenius manifolds, and soliton deformations of spectral curves corresponding to equations with self-consistent sources. This is an extended version of the talk given at the conference "Geometry, Dynamics, Integrable Systems --- GDIS 2010" (Serbia, September 7--13, 2010).

math-ph↗
arXiv2010

Role of the mean curvature in the geometry of magnetic confinement configurations

Examples are presented of how the geometric notion of the mean curvature is used for general magnetic field configurations and magnetic surfaces. It is shown that the mean magnetic curvature is related to the variation of the absolute value of the magnetic field along its lines. Magnetic surfaces of constant mean curvature are optimum for plasma confinement in multimirror open confinement systems and rippled tori.

physics.plasm-ph↗
arXiv2010

The type numbers of closed geodesics

A short survey on the type numbers of closed geodesics, on applications of the Morse theory to proving the existence of closed geodesics and on the recent progress in applying variational methods to the periodic problem for Finsler and magnetic geodesics

math.DG↗
arXiv2010

Periodic magnetic geodesics on almost every energy level via variational methods

For strong exact magnetic fields the action functional (i.e., the length plus the linear magnetic term) is not bounded from below on the space of closed contractible curves and the lower estimates for critical levels are derived by using the principle of throwing out cycles. It is proved that for almost every energy level the principle of throwing out cycles gives periodic magnetic geodesics on the critical levels defined by the "thrown out" cycles.

math.DG↗
arXiv2009

The Moutard transformation: an algebraic formalism via pseudodifferential operators and applications

We consider the Moutard transformation which is a two-dimensional version of the well-known Darboux transformation. We give an algebraic interpretation of the Moutard transformation as a conjugation in an appropriate ring and the corresponding version of the algebro-geometric formalism for two-dimensional Schroedinger operators. An application to some problems of the spectral theory of two-dimensional Schroedinger operators and to the $(2+1)$-dimensional Novikov--Veselov equation is sketched.

math-ph↗
arXiv2009

Surfaces in three-dimensional Lie groups in terms of spinors

This is a survey of results on surfaces in noncommutative three-dimensional Lie groups obtained by using the Weierstrass (spinor) representation of surfaces. It is based on the talk given at the conference "Geometry related to the theory of integrable systems" (RIMS, Kyoto, September 2007).

math.DG↗
arXiv2008

Two-dimensional rational solitons and their blow-up via the Moutard transformation

By using the Moutard transformation of two-dimensional Schroedinger operators we derive a procedure for constructing explicit examples of such operators with rational fast decaying potentials and degenerate $L_2$-kernels (this construction was sketched in arXiv:0706.3595) and show that if we take some of these potentials as the Cauchy data for the Novikov-Veselov equation (a two-dimensional version of the Korteweg-de Vries equation), then the corresponding solutions blow up in a finite time

math-ph↗
arXiv2008

Spectral conservation laws for periodic nonlinear equations of the Melnikov type

We consider the nonlinear equations obtained from soliton equations by adding self-consistent sources. We demonstrate by using as an example the Kadomtsev-Petviashvili equation that such equations on periodic functions are not isospectral. They deform the spectral curve but preserve the multipliers of the Floquet functions. The latter property implies that the conservation laws, for soliton equations, which may be described in terms of the Floquet multipliers give rise to conservation laws for the corresponding equations with self-consistent sources. Such a property was first observed by us for some geometrical flow which appears in the conformal geometry of tori in three- and four-dimensional Euclidean spaces (math/0611215).

math-ph↗
arXiv2006

Infinitesimal Darboux transformations of the spectral curves of tori in the four-space

We study the action of conformal transformations of the ambient space on the Dirac operator coming into the Weierstrass (or spinor) representation of a torus in the Euclidean four-space. It is showed that such an action generates a flow acting on the potential of the operator, that this flow is described by a nonlinear system of the Melnikov type and that it preserves the Floquet multipliers of the Dirac operator with double-periodic potential. However this flow is only almost isospectral since it does not preserve the spectral curve in general and its action may result in adding or removing multiple points corresponding to the same multipliers. We demonstrate that in some important geometrical examples after reparameterization of the temporary variable such flows are governed by integrable systems on whiskered tori.

math.DG↗
arXiv2006

On some algebraic examples of Frobenius manifolds

We construct some explicit quasihomogeneous algebraic solutions to the associativity (WDVV) equations by using analytical methods of the finite gap integration theory. These solutions are expanded in the uniform way to non-semisimple Frobenius manifolds.

math-ph↗