Jacobi structures of evolutionary partial differential equations
Jacobi structures of evolutionary partial differential equations
复制标题
演化偏微分方程的雅可比结构
DOI：
10.1016/j.aim.2011.01.015
复制
发表时间：
2011-05-01
影响因子：
1.7
通讯作者：
Zhang, Youjin
中科院分区：
文献类型：
--
作者：
Liu, Si-Qi;Zhang, Youjin
本文引入无穷维Jacobi结构的概念来描述一类非局部Hamilton系统的几何结构，当对Hamilton演化偏微分方程进行倒易变换时，这种几何结构自然出现.证明了无穷维Jacobi结构在只改变空间变量的互逆变换作用下是不变的。主要的技术工具是将经典的Schouten-Nijenhuis括号适当地推广到所谓的准局部多向量空间，并在超流形的框架中简单地实现这种结构。这些结构被用来计算Lichnerowicz-Jacobi上同调，并证明了一个Darboux定理的Jacobi结构与流体动力学的领导条款。我们还介绍了双雅可比结构的概念，并考虑系统的进化偏微分方程，具有双雅可比结构的可积性。(C)2011 Elsevier Inc. All rights reserved.
In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structures is invariant under the action of reciprocal transformations that only change the spatial variable. The main technical tool is in a suitable generalization of the classical Schouten-Nijenhuis bracket to the space of the so called quasi-local multi-vectors, and a simple realization of this structure in the framework of supermanifolds. These constructions are used to compute the Lichnerowicz-Jacobi cohomologies and to prove a Darboux theorem for Jacobi structures with hydrodynamic leading terms. We also introduce the notion of bi-Jacobi structures, and consider the integrability of a system of evolutionary PDEs that possesses a bi-Jacobi structure. (C) 2011 Elsevier Inc. All rights reserved.