一类四阶特征值问题及与之联系的发展方程族

A FOURTH-ORDER EIGENVALUE PROBLEM AND RELATED EVOLUTION EQUATION HIERARCHY

本文研究了下列四阶特征值问题 Lφ=λφ,L=其中表示对x的导数,q、p、r是当x→∞时都快速趋于零的光滑函数 我们通过向量场的换位表示,给出一组Lenard对,由此定义了保谱发展方程族并给出了方程族中前两个方程的Darboux变换。通过这个变换得到这两个方程的一些重要解。

In this paper, we investigate the following fourth-order eigenvalue problemwhere q ,p.r are smooth functions.their boundary conditions are decaying to zero at ∞.Through the commutative representation of vector field, we obtain a Lenard pairs.by which, we define the nonlinear isospcctral evolution equations and discover their Lax pairs for each equation of this hierarchy.Besides, the Darboux transformation of (E1). (E2) are obtained from which some kinds of important solutions are found.