5 research outputs found
On the integrability conditions for some structures related to evolution differential equations
Using the result by D.Gessler (Differential Geom. Appl. 7 (1997) 303-324,
DIPS-9/98, http://diffiety.ac.ru/preprint/98/09_98abs.htm), we show that any
invariant variational bivector (resp., variational 2-form) on an evolution
equation with nondegenerate right-hand side is Hamiltonian (resp., symplectic).Comment: 5 pages, AMS-LaTeX. v2: minor correction
On integrability of the Camassa-Holm equation and its invariants. A geometrical approach
Using geometrical approach exposed in arXiv:math/0304245 and
arXiv:nlin/0511012, we explore the Camassa-Holm equation (both in its initial
scalar form, and in the form of 2x2-system). We describe Hamiltonian and
symplectic structures, recursion operators and infinite series of symmetries
and conservation laws (local and nonlocal).Comment: 24 page
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