76 research outputs found
From recursion operators to Hamiltonian structures. The factorization method
We describe a simple algorithmic method of constructing Hamiltonian structures for nonlinear PDE. Our approach is based on the geometrical theory of nonlinear differential equations and is in a sense inverse to the well-known Magri scheme. As an illustrative example, we take the KdV equation and the Boussinesq equation. Further applications, including construction of previously unknown Hamiltonian structures, are in preparation
The Monge-Ampère equation: Hamiltonian and symplectic structures, recursions, and hierarchies
Using methods of geometry and cohomology developed recently, we study the Monge-Ampère equation, arising as the first nontrivial equation in the associativity equations, or WDVV equations. We describe Hamiltonian and symplectic structures as well as recursion operators for this equation in its orginal form, thus treating the independent variables on an equal footing. Besides this we present nonlocal symmetries and generating functions (cosymmetries)
Nonlocal conservation laws of PDEs possessing differential coverings
In his 1892 paper [L. Bianchi, Sulla trasformazione di B\"{a}cklund per le
superfici pseudosferiche, Rend. Mat. Acc. Lincei, s. 5, v. 1 (1892) 2, 3--12],
L. Bianchi noticed, among other things, that quite simple transformations of
the formulas that describe the B\"{a}cklund transformation of the sine-Gordon
equation lead to what is called a nonlocal conservation law in modern language.
Using the techniques of differential coverings [I.S. Krasil'shchik, A.M.
Vinogradov, Nonlocal trends in the geometry of differential equations:
symmetries, conservation laws, and B\"{a}cklund transformations, Acta Appl.
Math. v. 15 (1989) 1-2, 161--209], we show that this observation is of a quite
general nature. We describe the procedures to construct such conservation laws
and present a number of illustrative examples
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