arXiv subjects
Publications and source records attributed to Kostyantyn Zheltukhin.
The negative integrable hierarchies of shallow water waves and dispersionless Toda lattice equations are considered. The integrability is shown by explicit construction of an infinite set of conservation laws.
The problem of discretization of Darboux integrable equations is considered. Given a Darboux integrable continuous equation, one can obtain a Darboux integrable differential-discrete equation, using the integrals of the continuous equation. In the present paper, the discretization of the differential-discrete equations is done using the corresponding characteristic algebras. New examples of integrable discrete equations are obtained.
We study discretization of Darboux integrable systems. The discretization is done by using $x$- or $y$-integrals of the considered systems. New examples of semi-discrete Darboux integrable systems are obtained.
We study the discretization of Darboux integrable systems. The discretization is done using $x$-, $y$-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.
In the present paper we consider a discretization of hyperbolic systems of exponential type. We proved that, in the case of $2\times 2$ systems, the resulting semi-discrete system is Darboux integrable only if it corresponds to a Cartan matrix of a semi-simple Lie algebra.
The necessary and sufficient conditions for a hyperbolic semi-discrete equation to have five dimensional characteristic {\it x}-ring are derived. For any given chain, the derived conditions are easily verifiable by straightforward calculations.
A class of semi-discrete chains of the form $t_{1x} = f(x, t, t_1, t_x)$ is considered. For the given chains easily verifiable conditions for existence of x-integral of minimal order 4 are obtained.
A recursion operator is constructed for a hydrodynamic type system admitting dispersionless Lax representation with non-rational Lax function.
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 $A_2$, $B_2$, $C_2$, $G_2$ the complete sets of integrals in both directions are found. For the simple Lie algebras of the classical series $A_N$, $B_N$, $C_N$ and affine algebras of series $D^{(2)}_N$ the corresponding systems are supplied with the Lax representation.
Bi-presymplectic chains of one-forms of co-rank one are considered. The conditions in which such chains represent some Liouville integrable systems and the conditions in which there exist related bi-Hamiltonian chains of vector fields are derived. To present the construction of bi-presymplectic chains, the notion of 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 related flow leads directly to the construction of separation coordinates in purely algorithmic way. As an illustration bi-presymplectic and bi-Hamiltonian chains in ${\mathbb R}^3$ are considered in detail.
We first consider the Hamiltonian formulation of $n=3$ systems in general and show that all dynamical systems in ${\mathbb R}^3$ are bi-Hamiltonian. An algorithm is introduced to obtain Poisson structures of a given dynamical system. We find the Poisson structures of a dynamical system recently given by Bender et al. Secondly, we show that all dynamical systems in ${\mathbb R}^n$ are $(n-1)$-Hamiltonian. We give also an algorithm, similar to the case in ${\mathbb R}^3$, to construct a rank two Poisson structure of dynamical systems in ${\mathbb R}^n$. We give a classification of the dynamical systems with respect to the invariant functions of the vector field $\vec{X}$ and show that all autonomous dynamical systems in ${\mathbb R}^n$ are super-integrable.
The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable reductions of multi-field systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamiltonian integrable hydrodynamic type systems on a segment and a semi-line are presented.
The concept of integrable boundary value problems for soliton equations on $\mathbb{R}$ and $\mathbb{R}_+$ 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.
A transforation between a hierarchy of integrable equations arising from the standard $R$-matrix construction on the algebra of differential operators and a hierarchy of integrable equations arising from a deformation of the standard $R$-matrix is given.
Hamiltonian formulation of N=3 systems is considered in general. The Jacobi equation is solved in three classes. Compatible Poisson structures in these classes are determined and explicitly given. The corresponding bi-Hamiltonian systems are constructed and some explicit examples are given.
We consider equations arising from rational Lax representations. A general method to construct recursion operators for such equations is given. Several examples are given, including a degenerate bi-Hamiltonian system with a recursion operator.