Poisson Structures on Nonlinear Evolution Equations

摘要

Let epsilon be a differential equation and F = F(epsilon) be the function algebraon the infinite prolongation epsilon(infinity). Consider the algebra A = lambda(star)(F) of differential forms on F endowed with the horizontal differential dh : A -> A. A Poisson structure P on epsilon is understood as the homotopy equivalence class (with respect to dh) of a skew super bidifferential operator P in A satisfying (P,P)s = 0,(0,0)s being the super Schouten bracket. Description of Poisson structures for evolution equation with arbitrary number of space variables is given. It is shown that computations, in essence, reduce to solving the operator equation P ol(epsilon) + l(epsilon)oP = 0. The authors demonstrate that known structures for some evolution equations (e.g. the KdV equation) are particular cases of those considered here.