25 research outputs found
Algebraic properties of Gardner's deformations for integrable systems
An algebraic definition of Gardner's deformations for completely integrable
bi-Hamiltonian evolutionary systems is formulated. The proposed approach
extends the class of deformable equations and yields new integrable
evolutionary and hyperbolic Liouville-type systems. An exactly solvable
two-component extension of the Liouville equation is found.Comment: Proc. conf. "Nonlinear Physics: Theory and Experiment IV" (Gallipoli,
2006); Theor. Math. Phys. (2007) 151:3/152:1-2, 16p. (to appear
Homological evolutionary vector fields in Korteweg-de Vries, Liouville, Maxwell, and several other models
We review the construction of homological evolutionary vector fields on
infinite jet spaces and partial differential equations. We describe the
applications of this concept in three tightly inter-related domains: the
variational Poisson formalism (e.g., for equations of Korteweg-de Vries type),
geometry of Liouville-type hyperbolic systems (including the 2D Toda chains),
and Euler-Lagrange gauge theories (such as the Yang-Mills theories, gravity, or
the Poisson sigma-models). Also, we formulate several open problems.Comment: Proc. 7th International Workshop "Quantum Theory and Symmetries-7"
(August 7-13, 2011; CVUT Prague, Czech Republic), 20 page