Two new completely integrable systems related to the KdVequation hierarchy

摘要

In this paper, two new completely integrable systems related tothe spectral problem --4_(xx)+u4 = iA4_x and the constrained flows of the KdVequations are generated. According to the viewpoint of Hamiltonian mechanics, theEuler-Lagrange equations and the Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system is obtained. Moreover, by means of the constrainedcondition between the potential u and the eigenfunction co, the involutive solutionsof the KdV equation hierarchy are given.