I.S. Krasil'shchik

Cohomological approach to Poisson structures on nonlinear evolution equations

(Lobachevskii Journal of Mathematics, Volume III)

Let E be a differential equation, and let F=F(E) be the function algebra on the infinite prolongation E. Consider the algebra A= Λ(F) of differential forms on F endowed with the horizontal differential dh: A→A. A Poisson structure P on E is understood as the homotopy equivalence class (with respect to dh) of a skew symmetric super bidifferential operator P in A satisfying the condition [[P,P]]s=0, [[•,•]]s being the super Schouten bracket.

A description of Poisson structures for an evolution equation with an arbitrary number of space variables is given. It is shown that the computations, in essence, reduce to solving the operator equation PºlE+ lE º P=0. We demonstrate that known structures for some evolution equations (e.g., the KdV equation) are special cases of those considered here.
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