Kostyantyn Zheltukhin

Dynamical systems and Poisson structures

We first consider the Hamiltonian formulation of n=3 systems, in general, and show that all dynamical systems in R3 are locally bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. The construction of the Poisson structures is based on solving an associated first order linear partial differential equations. We find the Poisson structures of a dynamical system recently given by Bender et al. [J. Phys. A: Math. Theor. 40, F793 (2007)]. Secondly, we show that all dynamical systems in Rn are locally (n−1)-Hamiltonian. We give also an algorithm, similar to the case in R3, to construct a rank two Poisson structure of dynamical systems in Rn. We give a classification of the dynamical systems with respect to the invariant functions of the vector field X and show that all autonomous dynamical systems in Rn are superintegrable.

Integrable boundary value problems for elliptic type Toda lattice in a disk

The concept of integrable boundary value problems for soliton equations on

Cartan matrices and integrable lattice Toda field equations

\mathbb{R}

ON THE INTEGRABILITY OF A CLASS OF MONGE–AMPÈRE EQUATIONS

and

Bi-presymplectic chains of co-rank 1 and related Liouville integrable systems

\mathbb{R}_+

On existence of an x-integral for a semi-discrete chain of hyperbolic type

is extended to bounded regions enclosed by smooth curves. Classes of integrable boundary conditions on a circle for the Toda lattice and its reductions are found.The concept of integrable boundary value problems for soliton equations on R and R+ is extended to regions enclosed by smooth curves. Classes of integrable boundary conditions in a disk for the Toda lattice and its reductions are found.

Semi-discrete hyperbolic equations admitting five dimensional characteristic x-ring

Differential-difference integrable exponential-type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras A2, B2, C2, G2, the complete sets of integrals in both directions are found. For the simple Lie algebras of the classical series AN, BN, CN and affine algebras of series D(2)N, the corresponding systems are supplied with the Lax representation.

On a transformation between hierarchies of integrable equations

We give the Lax representations for the elliptic, hyperbolic and homogeneous second order Monge–Ampere equations. The connection between these equations and the equations of hydrodynamical type give us a scalar dispersionless Lax representation. A matrix dispersive Lax representation follows from the correspondence between sigma models, a two parameter equation for minimal surfaces and Monge–Ampere equations. Local as well nonlocal conserved densities are obtained.

On the discretization of Laine equations

Bi-presymplectic chains of 1-forms of co-rank 1 are considered. The conditions under which such chains represent some Liouville integrable systems and the conditions under which there exist related bi-Hamiltonian chains of vector fields are derived. To present the construction of bi-presymplectic chains, the notion of a dual Poisson-presymplectic pair is used, and the concept of d-compatibility of Poisson bivectors and d-compatibility of presymplectic forms is introduced. It is shown that bi-presymplectic representation of a related flow leads directly to the construction of separation coordinates in a purely algorithmic way. As an illustration, bi-presymplectic and bi-Hamiltonian chains in are considered in detail.

Hydrodynamic type integrable equations on a segment and a half-line

A class of semi-discrete chains of the form