具有不依赖于时间的不变量的三维常微分方程组的Hamilton结构

本文证明了具有不依赖于时间的不变量的三维常微分方程组所描述的动力系统相对于一广义Poisson括号可以改写为Hamilton系统,并且这些不变量就是Hamilton量。作为例子,我们讨论了Kermack-Mckendrick传染病模型,所得结果推广了Y.Nutku的结果。     

Reduction and realization in toda and volterra

We construct a new symplectic, bi-Hamiltonian realization of the KM-system by reducing the corresponding one for the Toda lattice. The bi-Hamiltonian pair is constructed using a reduction theorem of Fernandes and Vanhaecke. In this paper we also review the important work of Moser on the Toda and KM-systems.      

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.      

Integrable generalizations of Schrödinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)$SO\left(3\right)$-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy. A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2+1 dimensional integrable SO(3)$SO\left(3\right)$-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, and constants of motion, describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and mKdV soliton equations. Based on the well-known equivalence between the Heisenberg model and the Schrödinger map equation in 1+1 dimensions, a geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is given in terms of dynamical maps into the 2-sphere. In particular, this formulation yields a new integrable generalization of the Schrödinger map equation in 2+1 dimensions as well as a mKdV analog of this map equation corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.     

A Bi-Hamiltonian Lattice System of Rational Type and Its Discrete Integrable Couplings

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.     

PERIODIC SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS WITH TWO DELAY VIA BI-HAMILTONIAN SYSTEMS

1 Illtroduction and Statement of the Main ResultIn this paper, we shall study the existence of periodic solutions for the twofOllowing differential delay equationsX'(t) = --f(x(t -- r1))g(x(t -- r2)) -- f(x(t -- r2))g(x(t -- r1)), (1)andX'(t) = f(x(t -- rl))g(x(t -- r2)) f(x(t -- r2))g(x(t -- rl)), (2)where ri (i = l,2) are positive constants. When the function g(x) = 1,equations (1) and (2) become respectivelyIn 1974, Kaplan and Yorke (see [101) proved the existence of periodic so1utions…     

A Difference Hamiltonian Operator and a Hierarchy of Generalized Toda Lattice Equations

A difference Hamiltonian operator with three arbitrary constants is presented. When the arbitrary constants in the Hamiltonian operator are suitably chosen, a pair of Hamiltonian operators are given. The resulting Hamiltonian pair yields a difference hereditary operator. Using Magri scheme of bi-Hamiltonian formulations a hierarchy of the generalized Toda lattice equations is constructed. Finally, the discrete zero curvature representation is given for the resulting hierarchy.     

Darboux-Nijenhuis variables for open generalized Toda chains

We consider the possibility of using the Sklyanin method to construct Darboux-Nijenhuis variables of special form in the example of generalized open Toda chains associated with classical root systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 440–456, September, 2007.     

Compatible Lie-Poisson brackets on the Lie algebras <Emphasis Type="Italic">e</Emphasis>(3) and <Emphasis Type="Italic">so</Emphasis>(4)

We completely classify the compatible Lie-Poisson brackets on the dual spaces of the Lie algebras e(3) and so(4). The corresponding bi-Hamiltonian systems are the spinning tops corresponding to the classical cases of integrability of the Euler equations, the Kirchhoff equations, and the Poincaré-Zhukovskii equations. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 26–43, April, 2007.