论文标题
对称动作和伴随伴随的括号。 I：主要结果和应用
Symmetry actions and brackets for adjoint-symmetries. I: Main results and applications
论文作者
论文摘要
Infinitesimal symmetries of a partial differential equation (PDE) can be defined algebraically as the solutions of the linearization (Frechet derivative) equation holding on the space of solutions to the PDE, and they are well-known to comprise a linear space having the structure of a Lie algebra.Solutions of the adjoint linearization equation holding on the space of solutions to the PDE are called伴随的对称。本文研究了它们针对一般PDE系统的代数结构。 This is motivated by the correspondence between variational symmetries and conservation laws arising from Noether's theorem, which has a well-known generalization to non-variational PDEs, where infinitesimal symmetries are replaced by adjoint-symmetries, and variational symmetries are replaced by multipliers (adjoint-symmetries satisfying a certain Euler-Lagrange condition).获得了几个主要结果。对称性显示在伴随对称的线性空间上具有三种不同的线性动作。这些线性动作用于构建双线性伴随对称括号，其中之一是对称换向器支架的背包，具有谎言支架的属性。支架不使用或不需要任何局部变异结构（哈密顿量或拉格朗日），因此适用于一般的PDE系统。表明其中一种对称作用可以编码预选前（Noeth）操作员和符号2形式，从而导致用于进化系统的Symblectic 2-Form和Poisson支架的构建。电势形式的广义KDV方程用于说明所有结果。
Infinitesimal symmetries of a partial differential equation (PDE) can be defined algebraically as the solutions of the linearization (Frechet derivative) equation holding on the space of solutions to the PDE, and they are well-known to comprise a linear space having the structure of a Lie algebra.Solutions of the adjoint linearization equation holding on the space of solutions to the PDE are called adjoint-symmetries. Their algebraic structure for general PDE systems is studied herein. This is motivated by the correspondence between variational symmetries and conservation laws arising from Noether's theorem, which has a well-known generalization to non-variational PDEs, where infinitesimal symmetries are replaced by adjoint-symmetries, and variational symmetries are replaced by multipliers (adjoint-symmetries satisfying a certain Euler-Lagrange condition). Several main results are obtained. Symmetries are shown to have three different linear actions on the linear space of adjoint-symmetries. These linear actions are used to construct bilinear adjoint-symmetry brackets, one of which is a pull-back of the symmetry commutator bracket and has the properties of a Lie bracket. The brackets do not use or require the existence of any local variational structure (Hamiltonian or Lagrangian) and thus apply to general PDE systems. One of the symmetry actions is shown to encode a pre-sympletic (Noether) operator and a symplectic 2-form, which lead to the construction of a symplectic 2-form and Poisson bracket for evolution systems. The generalized KdV equation in potential form is used to illustrate all of the results.
