The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies

We reduce an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature (compatible Mokhov–Ferapontov brackets) to a canonical form, find an integrable system describing all such pairs, and, for an arbitrary solution of this integrable system, i.e., for any pair of compatible Poisson brackets in question, construct (in closed form) integrable bi-Hamiltonian systems of hydrodynamic type possessing this pair of compatible Poisson brackets of hydrodynamic type. The corresponding special canonical forms of metrics of constant Riemannian curvature are considered. A theory of special Liouville coordinates for Poisson brackets is developed. We prove that the classification of these compatible Poisson brackets is equivalent to the classification of special Liouville coordinates for Mokhov–Ferapontov brackets.     

Multi‐Hamiltonian structure for the finite defocusing Ablowitz‐Ladik equation

We study the Poisson structure associated to the defocusing Ablowitz‐Ladik equation from a functional‐analytical point of view by reexpressing the Poisson bracket in terms of the associated Carathéodory function. Using this expression, we are able to introduce a family of compatible Poisson brackets that form a multi‐Hamiltonian structure for the Ablowitz‐Ladik equation. Furthermore, we show using some of these new Poisson brackets that the Geronimus relations between orthogonal polynomials on the unit circle and those on the interval define an algebraic and symplectic mapping between the Ablowitz‐Ladik and Toda hierarchies. © 2008 Wiley Periodicals, Inc.     

Analog of the Korteweg-de Vries equation for the superconformal algebra

Group-theoretic approach to the construction of integrable nonlinear evolution equations is applied to the superconformal algebra. Poisson brackets of even and odd functions give a functional realization of Lie brackets of this algebra. The superschwartzian coincides with the Miura supertransformation.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 142–149, 1986.     

Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them

We construct integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type and solve the problem of the canonical form for a pair of compatible nonlocal Poisson brackets of hydrodynamic type. A system of equations describing compatible nonlocal Poisson brackets of hydrodynamic type is derived. This system can be integrated by the inverse scattering problem method. Any solution of this integrable system generates integrable bi-Hamiltonian systems of hydrodynamic type according to explicit formulas. We construct a theory of Poisson brackets of the special Liouville type. This theory plays an important role in the construction of integrable hierarchies.     

Necessary optimality conditions for different phase portraits in a neighborhood of a singular arc

An optimal control problem with scalar control is characterized by two Hamiltonians related to boundary values of the control parameter. Intermediate (internal) values of the control and the corresponding singular trajectories (arcs) can be constructed in terms of these two Hamiltonians using Poisson brackets. All multiple Poisson brackets using these Hamiltonians two, three, and four times vanish on a singular arc of the second order and the brackets with five Hamiltonians in general differ from zero. There exist six different multiple Poisson brackets in which Hamiltonians are used five times. A regular arc in the optimal phase portrait is linked with a singular arc after one, several, or infinitely many (Fuller phenomenon) switchings. In the paper it is shown that various collections of the signs for these six quantities—multiple Poisson brackets—correspond to the above-mentioned cases. There exist four different collections of the signs for the set consisting of six Poisson brackets. The singularity including a universal surface is investigated for the general case, whereas two other types of singularities are studied in particular examples.     

The classification of nonsingular multidimensional Dubrovin-Novikov brackets

In this paper, the well-known Dubrovin-Novikov problem posed as long ago as in 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. Generally speaking, such Poisson brackets are generated by nontrivial canonical special infinite-dimensional Lie algebras. In this paper, we obtain a classification of all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic type for any number N of components and for any dimension n by differential-geometric methods. A key role in the solution of this problem is played by the theory of compatible metrics earlier constructed by the present author.     

Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations

We solve the problem of describing compatible nonlocal Poisson brackets of hydrodynamic type. We prove that for nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type, there exist special local coordinates such that the metrics and the Weingarten operators of both brackets are diagonal. The nonlinear evolution equations describing all nonsingular pairs of compatible nonlocal Poisson brackets of hydrodynamic type are derived in these special coordinates, and the integrability of these equations is proved using the inverse scattering transform. The Lax pairs with a spectral parameter for these equations are found. We construct various classes of integrable reductions of the derived equations. These classes of reductions are of an independent differential-geometric and applied interest. In particular, if one of the compatible Poisson brackets is local, we obtain integrable reductions of the classical Lamé equations describing all orthogonal curvilinear coordinate systems in a flat space; if one of the compatible brackets is generated by a constant-curvature metric, the corresponding equations describe integrable reductions of the equations for orthogonal curvilinear coordinate systems in a space of constant curvature.     

The zero-curvature representation for the nonlinear O(3) sigma-model

We consider the O(3) sigma-model as a reduction of the principal chiral field. This approach allows us to introduce currents with ultralocal Poisson brackets and to obtain the zero-curvature equation which admits the fundamental Poisson bracket. Bibliography: 5 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 100–114. Translated by A. G. Bytsko.     

Poisson brackets of mappings obtained as (<Emphasis Type="Italic">q</Emphasis>,−<Emphasis Type="Italic">p</Emphasis>) reductions of lattice equations

In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson brackets are derived from a Lagrangian, using the so-called Ostrogradsky transformation. The (q,?p) reductions are (p + q)-dimensional maps and explicit Poisson brackets for such reductions of the discrete KdV equation, the discrete Lotka–Volterra equation, and the discrete Liouville equation are included. Lax representations of these equations can be used to construct sufficiently many integrals for the reductions. As examples we show that the (3,?2) reductions of the integrable partial difference equations are Liouville integrable in their own right.     

Commutative Poisson Subalgebras for Sklyanin Brackets and Deformations of Some Known Integrable Models

We construct hierarchies of commutative Poisson subalgebras for Sklyanin brackets. Each of the subalgebras is generated by a complete set of integrals in involution. Some new integrable systems and schemes for separation of variables for them are elaborated using various well-known representations of the brackets. The constructed models include deformations for the Goryachev–Chaplygin top, the Toda chain, and the Heisenberg model.     

Poisson Brackets after Jacobi and Plücker

We construct a symplectic realization and a bi-Hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We generalize this system and the related Poisson brackets to higher dimensions. These more general systems are parametrized by lines in projective space. For these rank 2 Poisson brackets the Jacobi identity is satisfied only when the Plücker relations hold. Two of these Poisson brackets are compatible if and only if the corresponding lines in projective space intersect. We present several examples of such systems.     

Algebras of Jordan brackets and generalized Poisson algebras

We construct bases for free unital generalized Poisson superalgebras and for free unital superalgebras of Jordan brackets. Also, we prove an analogue of Farkas’ theorem for PI unital generalized Poisson algebras and PI unital algebras of Jordan brackets. Relations between generic Poisson superalgebras and superalgebras of Jordan brackets are studied.     

Dispersive deformations of the Hamiltonian structure of Euler’s equations

Euler’s equations for a two-dimensional fluid can be written in the Hamiltonian form, where the Poisson bracket is the Lie–Poisson bracket associated with the Lie algebra of divergence-free vector fields. For the two-dimensional hydrodynamics of ideal fluids, we propose a derivation of the Poisson brackets using a reduction from the bracket associated with the full algebra of vector fields. Taking the results of some recent studies of the deformations of Lie–Poisson brackets of vector fields into account, we investigate the dispersive deformations of the Poisson brackets of Euler’s equation: we show that they are trivial up to the second order.     

Classification of constant solutions of the associative Yang-Baxter equation on Mat3

We find all nonequivalent constant solutions of the classical associative Yang-Baxter equation for Mat ₃ . New examples found in the classification yield the corresponding Poisson brackets on traces, double Poisson brackets on a free associative algebra with three generators, and anti-Frobenius associative algebras.     

Poisson geometry of directed networks in a disk

We investigate Poisson properties of Postnikov’s map from the space of edge weights of a planar directed network into the Grassmannian. We show that this map is Poisson if the space of edge weights is equipped with a representative of a 6-parameter family of universal quadratic Poisson brackets and the Grassmannian is viewed as a Poisson homogeneous space of the general linear group equipped with an appropriate R-matrix Poisson–Lie structure. We also prove that the Poisson brackets on the Grassmannian arising in this way are compatible with the natural cluster algebra structure.      

Hamiltonian formulation of generalized quantum dynamics

The Hamiltonian formulation of the usual complex quantum mechanics in the theory of generalized quantum dynamics is discussed. After the total trace Lagrangian, total trace Hamiltonian and two kinds of Poisson brackets are introduced, both the equations of motion of some total trace functionals which are expressed by total trace Poisson brackets and the equations of motion of some operators which are expressed by the without-total-trace Poisson brackets are obtained. Then a set of basic equations of motion of the usual complex quantum mechanics are obtained, which are also expressed by the Poisson brackets and total trace Hamiltonian in the generalized quantum dynamics. The set of equations of motion are consistent with the corresponding Heisenberg equations. Project supported by Prof. T.D. Lee’s NNSC Grant, the National Natural Science Foundation of China, the Foundation of Ph. D. Directing Programme of Chinese University, and the Chinese Academy of Sciences.     

Monodromy map and classicalr-matrices

A regularization of the Poisson brackets of the monodromy matrices based on the method of zero range potentials is proposed. Classification of the regularized brackets is reduced to the classical Yang-Baxter identity for the square of the initial Lie algebra. Bibliography: 11 titles. Respectfully dedicated to O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 156–166. Translated by M. A. Semenov-Tian-Shansky.     

Higher-Order Dispersive Deformations of Multidimensional Poisson Brackets of Hydrodynamic Type

The theory of multidimensional Poisson vertex algebras provides a completely algebraic formalism for studying the Hamiltonian structure of partial differential equations for any number of dependent and independent variables. We compute the cohomology of the Poisson vertex algebras associated with twodimensional, two-component Poisson brackets of hydrodynamic type at the third differential degree. This allows obtaining their corresponding Poisson–Lichnerowicz cohomology, which is the main building block of the theory of their deformations. Such a cohomology is trivial neither in the second group, corresponding to the existence of a class of nonequivalent infinitesimal deformations, nor in the third group, corresponding to the obstructions to extending such deformations.     

Examples of Poisson modules,I

We sketch some differential calculus on Poisson algebras and introduce a concept of module and representation on a Poisson algebras; we give examples and consider cohomologies connecting these constructions to the algebra of Poisson brackets.     

一种构造可积Hamilton系统的新方法

建议了一种新的构造可积Hamilton系统的方法。对于给定的Poisson流形,本文利用Dirac-Poisson结构构造其上的新Poisson括号[1],进而获得了新的可积Hamilton系统。构造的Poisson括号一般是非线的,并且这种方法也不同于通常的方法[2～4]。本文还给出了两个实例。