标题： 弱非局部辛结构，Whitham方法，以及流体动力学类型的弱非局部辛结构 显示英文标题

标题： Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type

摘要： 我们考虑一种称为弱非局部的场论辛结构的特殊类型。 这种类型的结构对于可积系统如KdV或NLS来说尤其常见。 我们在这里引入一种特殊的弱非局部辛结构类，我们称之为流体动力学类型的弱非局部辛结构。 然后我们研究这种结构与Whitham平均方法之间的联系，并提出对弱非局部辛结构进行“平均”的过程。 平均过程为相应的Whitham系统给出了流体动力学类型的弱非局部辛结构。 该过程还给出了对应于初始系统$m$-相解的波数的“作用变量”，这些变量为Whitham系统提供了额外的守恒定律。 显示英文摘要

摘要： We consider the special type of the field-theoretical Symplectic structures called weakly nonlocal. The structures of this type are in particular very common for the integrable systems like KdV or NLS. We introduce here the special class of the weakly nonlocal Symplectic structures which we call the weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then the connection of such structures with the Whitham averaging method and propose the procedure of "averaging" of the weakly nonlocal Symplectic structures. The averaging procedure gives the weakly nonlocal Symplectic Structure of Hydrodynamic Type for the corresponding Whitham system. The procedure gives also the "action variables" corresponding to the wave numbers of $m$-phase solutions of initial system which give the additional conservation laws for the Whitham system.