1,721,052 research outputs found
On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian.
No full textIt is shown that a class of dynamical systems (encompassing the one recently considered by F. Calogero in J. Math. Phys. 37 (1996) 1735) is both quasi-bi-Hamiltonian and bi-Hamiltonian. The first formulation entails the separability of these systems; the second one is obtained trough a non canonical map whose form is directly suggested by the associated Nijenhuis tensor
Nijenhuis and bi-Hamiltonian manifolds with symmetries.
No full textThe reduction theory for Nijenhuis and bi-Hamiltonian manifolds with deformation and symmetries, presented in a previous paper of ours, is completed. Some applications to the integarbility structures of NLEE's in one and two space dimensions are given
Quasi-bi-Hamiltonian systems and separability.
No full textTwo quasi--biHamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved
Bi-Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation variables.
No full textWe discuss from a bi-Hamiltonian point of view the Hamilton-Jacobi separability of a few dynamical systems. They are shown to admit, in their natural phase space, a quasi-bi-Hamiltonian formulation of Pfaffian type. This property allows us to straightforwardly recover a set of separation variables for the corresponding Hamilton-Jacobi equation
Integrability structures and master symmetries.
No full textThe main properties and the existence of mastersymmetries for algebraic integrability structures are discussed. Moreover, it is shown how a hierarchy of integrable equations can be recursively obtained by a mastersymmetries approach that is independent of the realization of the abstract scheme, being consequently the same for both one and two spatial dimension
Some remarks on the bi-Hamiltonian structure of integral and discrete evolution equations.
No full textFrom the MR review by W.Oevel: "The authors investigate the relations between two abstract algebraic structures leading to a set of integrable evolution equations admitting a bi-Hamiltonian structure. The first structure, recently introduced by the authors, consists of an associative algebra with unit, endowed with a linear map solving the Yang-Baxter equation. A(1,1)-tensor field with vanishing Nijenhuis torsion (i.e., a hereditary recursion operator) can be built from these structures, leading to a bi-Hamiltonian scheme based on the Lie-Kirillov bracket. By suitable realisations of the abstract algebra integrable equations such as the ILW hierarchy in one and two spatial dimensions can be described. The second structure, recently introduced by Ragnisco and Santini, also consists of an abstract associative algebra, now endowed with an algebra homomorphism. Again, an abstract hereditary recursion operator can be defined in terms of the homomorphism, leading to another bi-Hamiltonian scheme. It is shown that both constructions lead to the same Nijenhuis operator by defining a suitable solution of the Yang-Baxter equation from the algebra homomorphism. Nevertheless, only for a special choice of parameters involved will the related Poisson brackets coincide. By a suitable realisation of the abstract scheme the infinite and the finite nonperiodic Toda lattices are considered. In "physical variables'' the abstract structure leads to a bi-Hamiltonian scheme with a hereditary operator. In "Flaschka variables'' the two basic Poisson brackets can be reduced, whereas the abstract hereditary operator becomes singular on the submanifold under consideration.
Separation of variables in multi-Hamiltonian systems. Application to the Lagrange top.
No full textThis paper completes the analysis, started in J.Phys.A 35 (2002) 1741-1750, of the Lagrange top (LT) as a quasi-bi-Hamiltonian system. Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the reduction of the vector field and of the Poisson tensors. We show explicitly that, after the reduction on each one of the symplectic leaves, the vector field of the Lagrange top is separable in the sense of Hamilton-Jacobi
Yang-Baxter equations and intermediate long wave hierarchies.
No full textFrom the MR review by T.S.Ratiu: "The authors investigate the algebraic structure of the intermediate long wave equation (ILW). The general framework is that of a manifold endowed with a Nijenhuis tensor and a group of symmetries keeping this tensor invariant (a GN manifold). It is shown that the ILW hierarchy can be obtained by different realizations of two abstract structures intimately connected with the Yang-Baxter equations. Two different GN manifolds corresponding to solutions of the modified Yang-Baxter and the Yang-Baxter equations are constructed. Two particular realizations of their reductions then yield equivalent realizations of the ILW hierarchy.
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