标题： 偏微分方程的不变约化。 III：泊松括号 显示英文标题

标题： Invariant Reduction for Partial Differential Equations. III: Poisson brackets

摘要： 我们证明，在适当条件下，描述偏微分方程（PDEs）的不变解的有限维系统通过不变约化机制继承局部哈密顿算子，该机制适用于点对称、接触对称和高阶对称。 继承的算子赋予约化系统以泊松双矢量，这些双矢量将运动常数与对称性联系起来。 将同样的机制应用于不变守恒律，我们进一步证明，诱导的泊松括号与原系统的泊松括号一致，仅相差一个符号。 这是通过两个例子来说明的，在这些例子中，继承的泊松括号和继承的运动常数导致约化系统的可积性。 显示英文摘要

摘要： We show that, under suitable conditions, finite-dimensional systems describing invariant solutions of partial differential equations (PDEs) inherit local Hamiltonian operators through the mechanism of invariant reduction, which applies uniformly to point, contact, and higher symmetries. The inherited operators endow the reduced systems with Poisson bivectors that relate constants of motion to symmetries. Applying the same mechanism to invariant conservation laws, we further show that the induced Poisson brackets agree with those of the original systems, up to sign. This is illustrated by two examples in which the inherited Poisson brackets and inherited constants of motion yield integrability of the reduced systems.