Cohomological approach to poisson structures on nonlinear evolution equations

I. S. Krasil'shchik

Lobachevskii journal of mathematics, Tome 3 (1999), pp. 127-145.

Résumé

Let $\mathcal E$ be a differential equation, and let $\mathcal F=\mathcal F(\mathcal E)$ be the function algebra on the infinite prolongation $\mathcal E^\infty$. Consider the algebra $\mathcal A=\Lambda^*(\mathcal F)$ of differential forms on $\mathcal F$ endowed with the horizontal differential $d_h\colon\mathcal A\to\mathcal A$. A Poisson structure $\mathsf P$ on $\mathcal E$ is understood as the homotopy equivalence class (with respect to $d_h$) of a skew-symmetric super bidifferential operator $\mathsf P$ in $\mathcal A$ satisfying the condition $[\![\mathsf P,\mathsf P]\!]^s=0$, $[\![\bullet,\bullet]\!]^s$ 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 $P\circ\widehat\ell_{\mathcal E}+\ell_{\mathcal E}\circ P=0$. We demonstrate that known structures for some evolution equations (e.g., the KdV equation) are special cases of those considered here.

Détail

Keywords: nonlinear evolution differential equations, Hamiltonian formalism.
Mots-clés : Poisson structures