Reduced 4th-Order Eigenvalue Problem

摘要

The technique of the so-called nonlinearization of Lax pairs has been developed and applied to various soliton hierarchies, and this method also was generalized to discuss the nonlinearization of Lax pairs and adjoint Lax pairs of soliton hierarchies. In this paper, by use of the nonlinearization method, the reduced 4th-order eigenvalue problem is discussed and a Lax representation was deduced for the system. By means of Euler-Lagrange equations and Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system has been found, and the Bargmann system have been given. Then, the infinite-dimensional motion system described by Lagrange mechaics is changed into the Hamilton cannonical coordinate system.