##
*I.S. Krasil'shchik *

Cohomological approach to Poisson structures

on nonlinear evolution equations

*(Lobachevskii Journal of Mathematics,
Vol.3, pp.127-145)*

Let $\CE$ be a differential equation, and let $\CF=\CF(\CE)$ be the
function algebra on the infinite prolongation $\Ei$. Consider the algebra
$\CA= \La^*(\CF)$ of differential forms on $\CF$ endowed with the horizontal
differential $d_h\colon \CA\ra\CA$. A Poisson structure $\bP$ on $\CE$
is understood as the homotopy equivalence class (with respect to $d_h$)
of a skew\h symmetric super bidifferential operator $\bP$ in $\CA$ satisfying
the condition $\ldb\bP,\bP\rdb^s=0$, $\ldb\bullet,\bullet\rdb^s$ being
the super Schouten bracket. \par 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 $P\circ\wh{\ell}_{\CE}+ \ell_{\CE} \circ P=0$. 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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