Tri-hamiltonian Toda lattice and a canonical bracket for closed discrete curves

作为多孔介质方程的Toda流

The Toda flow as a porous medium equation

本文揭示了连续Toda流与不可压缩多孔介质（IPM）方程的内在联系，证明Toda流可视为特殊的IPM方程，并指出双括号性质为所有IPM型方程共有，同时探讨了QR对角化算法的修正及一般李群下的双括号流。

Harmonic spinors and metrics of positive curvature via the Gromoll filtration and Toda brackets

We construct non-trivial elements of order 2 in the homotopy groups $π_{8j+1+*} Diff(D^6,\partial)$, for * congruent 1 or 2 modulo 8, which are detected by the "assembling homomorphism" (giving rise to the Gromoll filtration), followed by the alpha-invariant in $KO_*=Z/2$. These elements are constructed by means of Morlet's homotopy equivalence between $Diff(D^6,\partial)$ and $Ω^7(PL_6/O_6)$, and Toda brackets in $PL_6/O_6$. We also construct non-trivial elements of order 2 in $π_* PL_m$ for every m greater or equal to 6 and * congruent to 1 or 2 modulo 8, which are detected by the alpha-inva…

Webs, Lenard schemes, and the local geometry of bihamiltonian Toda and Lax structures

We introduce a criterion that a given bihamiltonian structure allows a local coordinate system where both brackets have constant coefficients. This criterion is applied to the bihamiltonian open Toda lattice in a generic point, which is shown to be locally isomorphic to a Kronecker odd-dimensional pair of brackets with constant coefficients. This shows that the open Toda lattice cannot be locally represented as a product of two bihamiltonian structures. In a generic point the bihamiltonian periodic Toda lattice is shown to be isomorphic to a product of two open Toda lattices (one of which is a…

Manin Triples for Lie Bialgebroids

In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi identity except on certain subspaces. In this paper we systematize the properties of this bracket in the definition of a Courant algebroid. This structure on a vector bundle $E\rightarrow M$, consists of an antisymmetric bracket on the sections of $E$ whose ``Jacobi anomaly'' has an explicit expression in terms of a bundle map $E\rightarrow TM$ and a field of symmetric bilin…

Courant algebroids, derived brackets and even symplectic supermanifolds

In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant algebroids to generalize the notion of the Drinfeld double to Lie bialgebroids. As a first step towards understanding the complicated properties of Courant algebroids, we interpret them by associating to each Courant algebroid a strongly homotopy Lie algebra in a natural way. Next, we propose an alternative construction of the double of a Lie bialgebroid as a homological hamiltonian vector field on an even…

Dispersive deformations of the Hamiltonian structure of Euler's equations

Euler's equations for a two-dimensional system can be written in Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated to the Lie algebra of divergence free vector fields. We show how to derive the Poisson brackets of 2d hydrodynamics of ideal fluids as a reduction from the one associated to the full algebra of vector fields. Motivated by some recent results about the deformations of Lie-Poisson brackets of vector fields, we study the dispersive deformations of the Poisson brackets of Euler's equation and show that, up to the second order, they are trivial.