Topological classification of integrable geodesic flows on orientable two-dimensional Riemannian manifolds with additional integral depending on momenta linearly or quadratically

Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 27, No. 3, pp. 42–56, July–September, 1993.     

Pairs of commuting Hamiltonians quadratic in the momenta

In the case of two degrees of freedom, we consider pairs of Hamiltonians quadratic in the momenta and commuting with respect to the standard Poisson bracket. We find new multiparameter families of such pairs and present a universal scheme for constructing a complete solution of the Hamilton-Jacobi equation in terms of integrals over an algebraic curve. For the most complicated examples, this curve is a nonhyperelliptic covering of an elliptic curve. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 147–160, November, 2006.     

On the Pesin set of expansive geodesic flows in manifolds with no conjugate points

In this paper, we show that the Pesin set of an expansive geodesic flow in compact manifold with no conjugate points and bounded asymptote coincides a.e with an open and dense set of the unit tangent bundle. We also show that the set of hyperbolic periodic orbits is dense in the unit tangent bundle.     

一族新的可积系及其双约束流的Hamilton正则表示

基于一个新的等谱问题，按屠格式导出了一族新的可积系，具有双Hamilton结构，通过建立双对称约束，得到了该方程族的两组约束流，并将其化为正则的Hamilton系统。     

The Euler problem in solid body dynamics and the Jacobi problem about geodesics on an ellipsoid are not topologically conjugate

Two integrable problems are considered: the geodesic flow of an ellipsoid (the Jacobi problem) and the rotation of a solid about its center of mass (the Euler problem). It is proved that transforming the dynamical system of the Euler problem into the dynamical system of the Jacobi problem by a continuous change of coordinates is impossible. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 252–258, February, 1997. Translated by M. A. Shishkova     

Orbital classification of geodesic flows on two-dimensional ellipsoids. The Jacobi problem is orbitally equivalent to the integrable Euler case in rigid body dynamics

Moscow State University, Department of Mathematics and Mechanics. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 29, No. 3, pp. 1–15, July–September, 1995.     

A topological classification of linear differential equations on Banach spaces

In the present work, we give necessary and sufficient conditions for linear differential non-autonomous equations on a Banach space, that satisfy exponential dichotomy, to be topologically equivalent. We shall also prove that such equations are structurally stable.     

New Families of Conservative Systems on S2 Possessing an Integral of Fourth Degree in Momenta

There is a well-known example of an integrable conservative system on S2, the case of Kovalevskaya in the dynamics of a rigid body, possessing an integral of fourth degree in momenta. In this paper we propose new families of examples of conservative systems on S2 possessing an integral of fourth degree in momenta.     

Permutations of tori in integrable Hamiltonian systems and spectral series of pseudodifferential operators

In the present paper, we study integrable Hamiltonian systems with two degrees of freedom, whose regular level sets consist of several Liouville tori, and the bifurcation diagram has an isolated point. We study assumptions under which going around the singular point causes a permutation of the tori. We also consider a quantum analog of this situation and give model examples.     

Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations

We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is equivalent to describing all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. We prove that every such Hamiltonian operator (or the corresponding submanifold) specifies a pencil of compatible Poisson brackets, generates bihamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that there is a natural special class of such Hamiltonian operators (submanifolds) exactly described by the associativity equations of two-dimensional topological quantum field theory (the Witten-Dijkgraaf-Verlinde-Verlinde and Dubrovin equations). We show that each N-dimensional Frobenius manifold can locally be represented by a special flat N-dimensional submanifold with flat normal bundle in a 2N-dimensional pseudo-Euclidean space. This submanifold is uniquely determined up to motions.     

Evolution of Measures in the Phase Space of Nonlinear Hamiltonian Systems

We establish the existence of weak limits of solutions (in the class L _p, p%thinsp;1) of the Liouville equation for nondegenerate quasihomogeneous Hamilton equations. We find the limit probability distributions in the configuration space. We give conditions for a uniform distribution of Gibbs ensembles for geodesic flows on compact manifolds.     

关于代换系统拓扑序列熵的注记

Goodman证明了对两符号等长代换系统, 如果代换规则中0和1对应的词只有一个位置不同,那么对应的代换系统为null的, 即此系统沿着任意正整数序列的序列熵均为0. 在本文中, 我们针对系统的结构特征, 通过考察因子系统, 给出了此经典结果的另外一种证明. 同时, 对此类代换系统沿着给定序列的复杂性, 我们得到了比Goodman更为精确的估计.     

Topological classification of supertransitive flows on closed nonorientable surfaces. Part I. Necessary conditions for topological equivalence

In this paper a topological invariant is introduced for the class of supertransitive flows on closed nonorientable surfacesM of negative Euler characteristic. We describe properties of this invariant and prove that it provides necessary conditions for the topological equivalence of flows belonging to the above-mentioned class of supertransitive flows. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 461–470, September, 2000.     

A topological characterization of the ω-limit sets for analytic flows on the plane, the sphere and the projective plane

In this paper the ω-limit sets for analytic and polynomial differential equations on the plane are characterized up to homeomorphisms. The analogous problem is solved in full detail for analytic flows on the sphere and the projective plane. We also explain how to carry on the same program for analytic flows defined on open subsets of these surfaces.     

Soliton solutions of an integrable boundary problem on the half-line for the discrete toda chain

We write formulas for soliton solutions of the discrete Toda chain and pose the integrable boundary value problem for this chain. We find conditions for the parameters (discrete spectrum points, transmission coefficients, and the corresponding factors) whereby solutions of the integrable boundary value problem are selected from all soliton solutions. As a result, we construct two hierarchies of soliton solutions of the specified problem with even and odd soliton numbers and find an explicit form of the conditions for the parameters. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 387–397, September, 2006.     

Recent advances in the monodromy theory of integrable Hamiltonian systems

The notion of monodromy was introduced by J.J. Duistermaat as the first obstruction to the existence of global action coordinates in integrable Hamiltonian systems. This invariant was extensively studied since then and was shown to be non-trivial in various concrete examples of finite-dimensional integrable systems. The goal of the present paper is to give a brief overview of monodromy and discuss some of its generalizations. In particular, we will discuss the monodromy around a focus–focus singularity and the notions of quantum, fractional and scattering monodromy. The exposition will be complemented with a number of examples and open problems.     

A minimum-stencil difference scheme for computing two-dimensional axisymmetric gas flows: Examples of pulsating flows with instabilities

A minimum-stencil difference scheme for computing two-dimensional axisymmetric gas flows is described. The scheme is explicit, conservative, and second-order accurate in space and time. The numerical results obtained for pulsating flows and contact discontinuity instabilities are discussed. The mechanisms of flow pulsation and instability generation are described.