A class of integrable lattices and KP hierarchy

We introduce a class of integrable $l$-field first-order lattices together with corresponding Lax equations. These lattices may be represented as consistency condition for auxiliary linear systems defined on sequences of formal dressing operators. This construction provides simple way to build lattice Miura transformations between one-field lattice and $l$-field ($l\ge 2$) ones. We show that the lattices pertained to above class is in some sense compatible with KP flows and define the chains of constrained KP Lax operators.Comment: LaTeX, 13 pages, accepted for publication in J. Phys. A: Math. Ge

Darboux Transformations, Infinitesimal Symmetries and Conservation Laws for Nonlocal Two-Dimensional Toda Lattice

The technique of Darboux transformation is applied to nonlocal partner of two-dimensional periodic A_{n-1} Toda lattice. This system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear systems with quantized spectral parameter. The generalization of the Darboux transformation technique on linear equations of such a kind is given. The connections between the solutions of overdetermined linear systems and their expansions in series at singular points neighborhood are presented. The solutions of the nonlocal Toda lattice and infinite hierarchies of the infinitesimal symmetries and conservation laws are obtained.Comment: 12 pages, infinitesimal symmetries and conservation laws are adde

On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice

We show a broad class of constraints compatible with Itoh-Narita-Bogoyavlenskii lattice hierarchy. All these constraints can be written in the form of discrete conservation law $I_{i+1}=I_i$ with appropriate homogeneous polynomial discrete function $I=I[a]$.Comment: 15 page

Matrix Model and Stationary Problem in Toda Chain

We analyze the stationary problem for the Toda chain, and show that arising geometric data exactly correspond to the multi-support solutions of one-matrix model with a polynomial potential. For the first nontrivial examples the Hamiltonians and symplectic forms are calculated explicitly, and the consistency checks are performed. The corresponding quantum problem is formulated and some its properties and perspectives are discussed.Comment: 11 pages, LaTeX; Based on talks at "Classical and quantum integrable systems", Dubna, January 2005 and "Selected topics of modern mathematical physics", St.Petersburg, June 2005, and a lecture for the minicourse: "Toda lattices: basics and perspectives", Fields Institute, Toronto, April 200

On some integrable lattice related by the Miura-type transformation to the Itoh-Narita-Bogoyavlenskii lattice

We show that by Miura-type transformation the Itoh-Narita-Bogoyavlenskii lattice, for any $n\geq 1$, is related to some differential-difference (modified) equation. We present corresponding integrable hierarchies in its explicit form. We study the elementary Darboux transformation for modified equations.Comment: Latex, 9 page

A dynamical systems approach to the tilted Bianchi models of solvable type

We use a dynamical systems approach to analyse the tilting spatially homogeneous Bianchi models of solvable type (e.g., types VI$_h$ and VII$_h$) with a perfect fluid and a linear barotropic $\gamma$-law equation of state. In particular, we study the late-time behaviour of tilted Bianchi models, with an emphasis on the existence of equilibrium points and their stability properties. We briefly discuss the tilting Bianchi type V models and the late-time asymptotic behaviour of irrotational Bianchi VII$_0$ models. We prove the important result that for non-inflationary Bianchi type VII$_h$ models vacuum plane-wave solutions are the only future attracting equilibrium points in the Bianchi type VII$_h$ invariant set. We then investigate the dynamics close to the plane-wave solutions in more detail, and discover some new features that arise in the dynamical behaviour of Bianchi cosmologies with the inclusion of tilt. We point out that in a tiny open set of parameter space in the type IV model (the loophole) there exists closed curves which act as attracting limit cycles. More interestingly, in the Bianchi type VII$_h$ models there is a bifurcation in which a set of equilibrium points turn into closed orbits. There is a region in which both sets of closed curves coexist, and it appears that for the type VII$_h$ models in this region the solution curves approach a compact surface which is topologically a torus.Comment: 29 page

On the integrability of stationary and restricted flows of the KdV hierarchy.

A bi--Hamiltonian formulation for stationary flows of the KdV hierarchy is derived in an extended phase space. A map between stationary flows and restricted flows is constructed: in a case it connects an integrable Henon--Heiles system and the Garnier system. Moreover a new integrability scheme for Hamiltonian systems is proposed, holding in the standard phase space.Comment: 25 pages, AMS-LATEX 2.09, no figures, to be published in J. Phys. A: Math. Gen.

Kovalevski exponents and integrability properties in class A homogeneous cosmological models

Qualitative approach to homogeneous anisotropic Bianchi class A models in terms of dynamical systems reveals a hierarchy of invariant manifolds. By calculating the Kovalevski Exponents according to Adler - van Moerbecke method we discuss how algebraic integrability property is distributed in this class of models. In particular we find that algebraic nonintegrability of vacuum Bianchi VII_0 model is inherited by more general Bianchi VIII and Bianchi IX vacuum types. Matter terms (cosmological constant, dust and radiation) in the Einstein equations typically generate irrational or complex Kovalevski exponents in class A homogeneous models thus introducing an element of nonintegrability even though the respective vacuum models are integrable.Comment: arxiv version is already officia

Integrable quadratic Hamiltonians on so(4) and so(3,1)

We investigate a special class of quadratic Hamiltonians on so(4) and so(3,1) and describe Hamiltonians that have additional polynomial integrals. One of the main results is a new integrable case with an integral of sixth degree.Comment: 16 page

The Asymptotic Behaviour of Tilted Bianchi type VI0_0 Universes

We study the asymptotic behaviour of the Bianchi type VI$_0$ universes with a tilted $\gamma$-law perfect fluid. The late-time attractors are found for the full 7-dimensional state space and for several interesting invariant subspaces. In particular, it is found that for the particular value of the equation of state parameter, $\gamma=6/5$, there exists a bifurcation line which signals a transition of stability between a non-tilted equilibrium point to an extremely tilted equilibrium point. The initial singular regime is also discussed and we argue that the initial behaviour is chaotic for $\gamma<2$.Comment: 22 pages, 4 figures, to appear in CQ