10 research outputs found
A geometric study of the dispersionless Boussinesq type equation
We discuss the dispersionless Boussinesq type equation, which is equivalent
to the Benney-Lax equation, being a system of equations of hydrodynamical type.
This equation was discussed in
. The results include: a
description of local and nonlocal Hamiltonian and symplectic structures,
hierarchies of symmetries, hierarchies of conservation laws, recursion
operators for symmetries and generating functions of conservation laws
(cosymmetries). Highly interesting are the appearances of operators that send
conservation laws and symmetries to each other but are neither Hamiltonian, nor
symplectic. These operators give rise to a noncommutative infinite-dimensional
algebra of recursion operators
The BV-algebra structure of W_3 cohomology
We summarize some recent results obtained in collaboration with J. McCarthy
on the spectrum of physical states in gravity coupled to matter. We
show that the space of physical states, defined as a semi-infinite (or BRST)
cohomology of the algebra, carries the structure of a BV-algebra. This
BV-algebra has a quotient which is isomorphic to the BV-algebra of polyvector
fields on the base affine space of . Details have appeared elsewhere.
[Published in the proceedings of "Gursey Memorial Conference I: Strings and
Symmetries," Istanbul, June 1994, eds. G. Aktas et al., Lect. Notes in Phys.
447, (Springer Verlag, Berlin, 1995)]Comment: 8 pages; uses macros tables.tex and amssym.def (version 2.1 or later
The Robinson-Trautman Type III Prolongation Structure Contains K
The minimal prolongation structure for the Robinson-Trautman equations of
Petrov type III is shown to always include the infinite-dimensional,
contragredient algebra, K, which is of infinite growth. Knowledge of
faithful representations of this algebra would allow the determination of
B\"acklund transformations to evolve new solutions.Comment: 20 pages, plain TeX, no figures, submitted to Commun. Math. Phy
Hamiltonian structures for general PDEs
We sketch out a new geometric framework to construct Hamiltonian operators for generic, non-evolutionary partial differential equations. Examples on how the formalism works are provided for the KdV equation, Camassa--Holm equation, and Kupershmidt's deformation of a bi-Hamiltonian system
Integrability of Kupershmidt Deformations
We prove that the Kupershmidt deformation of a bi-Hamiltonian system is itself bi-Hamiltonian. Moreover, Magri hierarchies of the initial system give rise to Magri hierarchies of Kupershmidt deformations as well. Since Kupershmidt deformations are not written in evolution form, we start with an outline a geometric framework to study Hamiltonian properties of general non-evolution differential equations, developed in Igonin et al. (to appear, 2009) (see also Kersten et al., In: Differential Equations: Geometry, Symmetries and Integrability, Springer, Berlin, 2009)