Noncommutative integrable systems on <Emphasis Type="Italic">b</Emphasis>-symplectic manifolds

In this paper we study noncommutative integrable systems on b-Poisson manifolds. One important source of examples (and motivation) of such systems comes from considering noncommutative systems on manifolds with boundary having the right asymptotics on the boundary. In this paper we describe this and other examples and prove an action-angle theorem for noncommutative integrable systems on a b-symplectic manifold in a neighborhood of a Liouville torus inside the critical set of the Poisson structure associated to the b-symplectic structure.     

Algebraic integrability: The Adler-Van Moerbeke approach

In this paper, I present an overview of the active area of algebraic completely integrable systems in the sense of Adler and van Moerbeke. These are integrable systems whose trajectories are straight line motions on abelian varieties (complex algebraic tori). We make, via the Kowalewski-Painlevé analysis, a study of the level manifolds of the systems. These manifolds are described explicitly as being affine part of abelian varieties and the flow can be solved by quadrature, that is to say their solutions can be expressed in terms of abelian integrals. The Adler-Van Moerbeke method’s which will be used is devoted to illustrate how to decide about the algebraic completely integrable Hamiltonian systems and it is primarily analytical but heavily inspired by algebraic geometrical methods. I will discuss some interesting and well known examples of algebraic completely integrable systems: a five-dimensional system, the Hénon-Heiles system, the Kowalewski rigid body motion and the geodesic flow on the group SO(n) for a left invariant metric.     

Haantjes Manifolds with Symmetry

We introduce the definition of Haantjes manifolds with symmetry and explain why these manifolds appear in the theory of integrable systems of hydrodynamic type and in topological field theories.     

Self-Dual Hamiltonians as Deformations of Free Systems

We formulate the problem of finding self-dual Hamiltonians (associated with integrable systems) as deformations of free systems given on various symplectic manifolds and discuss several known explicit examples including the recently found double elliptic Hamiltonians. We consider self-duality as the basic principle, while duality in integrable systems (of the Toda/Calogero/Ruijsenaars type) comes as a secondary notion (degenerations of self-dual systems).     

Lax方程与耦合Poisson流形的动力r-矩阵

对于一个与Poisson流形耦合的动力r-矩阵,我们在相应的Lie双代数胚上构造出一类Lax方程和一族守恒量,希望利用该方法进一步研究可积Hamilton系统.     

Fomenko-Zieschang invariants of integrable systems with symplectic singularities

The paper is devoted to the study of integrable Hamiltonian systems which arise on symplectic manifolds with degenerate singularities. The results obtained extend the subject domain of the Fomenko-Zieschang theory of invariants.     

The modular hierarchy of the Toda lattice

The modular vector field plays an important role in the theory of Poisson manifolds and is intimately connected with the Poisson cohomology of the space. In this paper we investigate its significance in the theory of integrable systems. We illustrate in detail the case of the Toda lattice both in Flaschka and natural coordinates.     

Self-similar Hessian and conformally Kähler manifolds

Annals of Global Analysis and Geometry - We study a natural class of LCK manifolds that we call integrable LCK manifolds: those where the anti-Lee form $$\eta $$ corresponds to an integrable...     

Integrable systems and the topology of isospectral manifolds

We briefly review known results concerning the study of isospectral manifolds using integrable systems. We then describe new results concerning the topology of isospectral manifolds of zero-diagonal Jacobi matrices. This topology is studied using the Volterra system. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 140–146, April, 2008.     

The Gelfand-Cetlin system and quantization of the complex flag manifolds

The construction of angle action variables for collective completely integrable systems is described and the associated Bohr-Sommerfeld sets are determined. The quantization method of Sniatycki applied to such systems gives formulas for multiplicities. For the Gelfand-Cetlin system on complex flag manifolds we show that these formulas give the correct answers for the multiplicities of the associated representations.     

Curvature properties of anti-Kähler–Codazzi manifolds

In this paper we shall consider a new class of integrable almost anti-Hermitian manifolds, which will be called anti-Kähler–Codazzi manifolds, and we will investigate their curvature properties.     

Lie-algebraic structure of integrable nonlinear dynamical systems on extended functional manifolds

We consider the general Lie-algebraic scheme of construction of integrable nonlinear dynamical systems on extended functional manifolds. We obtain an explicit expression for consistent Poisson structures and write explicitly nonlinear equations generated by the spectrum of a periodic problem for an operator of Lax-type representation.     

The Mirror Systems of Integrable Equations

We provide an algorithm to convert integrable equations to regular systems near noncharacteristic, movable singularity manifolds of solutions. We illustrate how the algorithm is equivalent to the Painlevé test. We also use thealgorithm to prove the convergence of the Laurent series obtained from the Painlevé test.     

Invariants of four- and three-dimensional singularities of integrable systems

A relationship between invariants of four-dimensional singularities of integrable Hamiltonian systems (with two degrees of freedom) and invariants of two-dimensional foliations on three-dimensional manifolds being the “boundaries” of these four-dimensional singularities is discovered. Nonequivalent singularities which, nevertheless, have equal three-dimensional invariants are found.     

Symplectic method for the construction of ergodic measures on invariant submanifolds of nonautonomous hamiltonian systems: Lagrangian manifolds, their structure, and mather homologies

We develop a new approach to the study of properties of ergodic measures for nonautonomous periodic Hamiltonian flows on symplectic manifolds, which are used in many problems of mechanics and mathematical physics. Using Mather’s results on homologies of invariant probability measures that minimize some Lagrangian functionals and the symplectic theory developed by Floer and others for the investigation of symplectic actions and transversal intersections of Lagrangian manifolds, we propose an analog of a Mather-type β-function for the study of ergodic measures associated with nonautonomous Hamiltonian systems on weakly exact symplectic manifolds. Within the framework of the Gromov-Salamon-Zehnder elliptic methods in symplectic geometry, we establish some results on stable and unstable manifolds for hyperbolic invariant sets, which are used in the theory of adiabatic invariants of slowly perturbed integrable Hamiltonian systems. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 675–691, May, 2006.     

Connections on manifolds and new characteristic classes

We give an extension of Maslov-Arnold classes to a certain class of symplectic manifolds. It is proved that any such generalized class of minimal surfaces is equal to zero for a large class of stable minimal surfaces. We describe some applications to pseudo-Riemannian geometry and to the investigation of completely integrable Hamiltonian systems.     

Generalized de Rham-Hodge complexes,the related characteristic Chern classes,and some applications to integrable multidimensional differential systems on Riemannian manifolds

We study the differential-geometric aspects of generalized de Rham-Hodge complexes naturally related to integrable multidimensional differential systems of the M. Gromov type, as well as the geometric structure of the Chern characteristic classes. Special differential invariants of the Chern type are constructed, their importance for the integrability of multidimensional nonlinear differential systems on Riemannian manifolds is discussed. An example of the three-dimensional Davey-Stewartson-type nonlinear integrable differential system is considered, its Cartan type connection mapping, and related Chern-type differential invariants are analyzed. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 327–344, March, 2007.     

A description of Lax type integrable dynamical systems via the Marsden–Weinstein reduction method

A Lie-algebraic approach to constructing nonlinear Lax type integrable dynamical systems of modern mathematical and theoretical physics, based on the Marsden–Weinstein reduction method on canonically symplectic manifolds with group symmetry, is proposed. Its natural relationship with the well known Adler–Kostant–Souriau–Berezin–Kirillov method and the associated R-matrix approach is analyzed.     

Specific features of families of invariant manifolds of conservative systems

In this paper we demonstrate themethod of the enveloping first integral via an example of a completely integrable system of differential equations. This method allows a researcher to find and investigate singular invariant manifolds for a given family of invariant manifolds of steady motions represented by an initial system of equations. We describe specific properties of branching of the obtained families of singular invariant manifolds.