无穷维谱不变可积发展方程在有限维不变子流形上的约化——谱不变发展方程的周期解

The Constraining Infinite-dimensional Isospectral Evolution Equations on Finite-dimensional Invariant Submanifold——The Periodic Solutions of the Isospectral Evolutin Equations

利用广义 Legendrge变换 ,证明了无穷维的可积方程 utm=JδHmδu可约化为在一个不变子流形 S上有限维可积的 Hamiltonian系统 ,即证明了在非奇异条件下 Flaschka[1]和Ablowitz所提出的无穷维可积系统的约化原理 ,从而求得了方程 utm=JδHmδu( m=0 ,1 ,2 ,… )的周期或拟周期解 ,这一结果将 P.D.L[2 ,3] 、Novi Kov[4] 的关于 Kdv方程的周期或拟周期解的结果推广到了一般的谱不变 Hamiltonian可积方程上去。作为特例 ,讨论了 AKNS族。

In this paper,making use of general Legendrge transformation,it is proved to produce finitedimensional Hamiltonians integrable systems by constraining the infinite dimensional integrable equations u t m =δH mδu on an invariant submanifold S.Namelg,and it is proved that Flaschka [1] and Ablowitz point out an important principle to constrain infinite dimensional integrable systems.For example,we give the constraining AKNS hierarchy.