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A. V. Tsiganov

Publications and source records attributed to A. V. Tsiganov.

At least 73 records · Page 4Linked to original sources
arXiv2007

New Lax pair for restricted multiple three wave interaction system, quasiperiodic solutions and bi-hamiltonian structure

We study restricted multiple three wave interaction system by the inverse scattering method. We develop the algebraic approach in terms of classical $r$-matrix and give an interpretation of the Poisson brackets as linear $r$-matrix algebra. The solutions are expressed in terms of polynomials of theta functions. In particular case for $n=1$ in terms of Weierstrass functions.

nlin.SI↗
arXiv2007

A family of the Poisson brackets compatible with the Sklyanin bracket

We introduce a family of compatible Poisson brackets on the space of $2\times 2$ polynomial matrices, which contains the Sklyanin bracket, and use it to derive a multi-Hamiltonian structure for a set of integrable systems that includes $XXX$ Heisenberg magnet, the open and periodic Toda lattices, the discrete self-trapping model and the Goryachev-Chaplygin gyrostat.

nlin.SI↗
arXiv2006

A note on elliptic coordinates on the Lie algebra e(3)

We introduce elliptic coordinates on the dual space to the Lie algebra e(3$ and discuss the separability of the Clebsch system in these variables. The proposed Darboux coordinates on e^*(3) coincide with the usual elliptic coordinates on the cotangent bundle of the two-dimensional sphere at the zero value of the corresponding Casimir function.

nlin.SI↗
arXiv2006

Towards a classification of natural bi-hamiltonian systems

For construction and classification of the natural integrable systems we propose to use a criterion of separability in Darboux--Nijenhuis coordinates, which can be tested without an a priori explicit knowledge of these coordinates.

nlin.SI↗
arXiv2004

On a family of integrable systems on $S^2$ with a cubic integral of motion

We discuss a family of integrable systems on the sphere $S^2$ with an additional integral of third order in momenta. This family contains the Coryachev-Chaplygin top, the Goryachev system, the system recently discovered by Dullin and Matveev and two new integrable systems. On the non-physical sphere with zero radius all these systems are isomorphic to each other.

nlin.SI↗
arXiv2004

On the Steklov-Lyapunov case of the rigid body motion

We construct a Poisson map between manifolds with linear Poisson brackets corresponding to the two samples of Lie algebra $e(3)$. Using this map we establish equivalence of the Steklov-Lyapunov system and the motion of a particle on the surface of the sphere under the influence of the fourth order potential. To study separation of variables for the Steklov case on the Lie algebra $so(4)$ we use the twisted Poisson map between the bi-Hamiltonian manifolds $e(3)$ and $so(4)$.

nlin.SI↗
arXiv2003

Poisson maps and integrable deformations of Kowalevski top

We construct a Poisson map between manifolds with linear Poisson brackets corresponding to the Lie algebras $e(3)$ and $so(4)$. Using this map we establish a connection between the deformed Kowalevski top on $e(3)$ proposed by Sokolov and the Kowalevski top on $so(4)$. The connection between these systems leads to the separation of variables for the deformed system on $e(3)$ and yields the natural $5\times 5$ Lax pair for the Kowalevski top on $so(4)$.

nlin.SI↗
arXiv2001

Commutative Poisson subalgebras for the Sklyanin bracket and deformations of known integrable models

A hierarchy of commutative Poisson subalgebras for the Sklyanin bracket is proposed. Each of the subalgebras provides a complete set of integrals in involution with respect to the Sklyanin bracket. Using different representations of the bracket, we find some integrable models and a separation of variables for them. The models obtained are deformations of known integrable systems like the Goryachev-Chaplygin top, the Toda lattice and the Heisenberg model.

nlin.SI↗
arXiv2000

On the Drach superintegrable systems

Cubic invariants for two-dimensional degenerate Hamiltonian systems are considered by using variables of separation of the associated Stäckel problems with quadratic integrals of motion. For the superintegrable Stäckel systems the cubic invariant is shown to admit new algebro-geometric representation that is far more elementary than the all the known representations in physical variables. A complete list of all known systems on the plane which admit a cubic invariant is discussed.

nlin.SI↗
arXiv1998

The Lax pairs for the Holt system

By using non-canonical transformation between the Holt system and the Henon-Heiles system the Lax pairs for all the integrable cases of the Holt system are constructed from the known Lax representations for the Henon-Heiles system.

solv-int↗
arXiv1997

The Stackel systems and algebraic curves

We show how the Abel-Jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves. We prove that derivative of the Abel-Jacobi map is just the Stäckel matrix, which determines $n$-orthogonal curvilinear coordinate systems in a flat space. The Lax pairs, $r$-matrix algebras and explicit form of the flat coordinates are constructed. An application of the Weierstrass reduction theory allows to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space.

solv-int↗
arXiv1997

On superintegrable systems closed to geodesic motion

In this work we consider superintegrable systems in the classical $r$-matrix method. By using other authomorphisms of the loop algebras we construct new superintegrable systems with rational potentials from geodesic motion on $R^{2n}$.

solv-int↗
arXiv1996

Automorphisms of sl(2) and dynamical r-matrices

Two outer automorphisms of infinite-dimensional representations of $sl(2)$ algebra are considered. The similar constructions for the loop algebras and yangians are presented. The corresponding linear and quadratic $R$-brackets include the dynamical $r$-matrices.

solv-int↗
arXiv1996

The Kowalewski top: a new Lax representation

The 2x2 monodromy matrices for the Kowalewski top on the Lie algebras e(3), so(4) and so(3,1) are presented. The corresponding quadratic R-matrix structure is the dynamical deformation of the standard R-matrix algebras. Some tops and Toda lattices related to the Kowalewski top are discussed.

solv-int↗