72 research outputs found
A class of integrable lattices and KP hierarchy
We introduce a class of integrable -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 -field () 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 with appropriate
homogeneous polynomial discrete function .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 , 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 and VII)
with a perfect fluid and a linear barotropic -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 models. We prove the
important result that for non-inflationary Bianchi type VII models vacuum
plane-wave solutions are the only future attracting equilibrium points in the
Bianchi type VII 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 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 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 VI Universes
We study the asymptotic behaviour of the Bianchi type VI universes with a
tilted -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, , 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 .Comment: 22 pages, 4 figures, to appear in CQ
- …