Eur. Phys. J. Plus (2012) 127: 77
https://doi.org/10.1140/epjp/i2012-12077-y

Regular Article

The Neumann system for the 4th-order eigenvalue problem and constraint flows of the coupled KdV-type equations

Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, China

^* e-mail: guzq@stdu.edu.cn

Received: 23 April 2012
Revised: 25 May 2012
Accepted: 18 June 2012
Published online: 24 July 2012

Abstract

In this paper, the Neumann system for the 4th-order eigenvalue problem Ly = (∂⁴+q∂²+∂² q+ip∂+i∂p+y = Λy) has been given. By means of the Neumann constraint condition, the perfect constraint set Γ and the relations between the potentials {q, p, r} and the eigenvector y are obtained. Then, based on the Euler-Lagrange function and Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system has been found, which can be equal to the real Hamiltonian canonical coordinate system in R ^8N . Using Cao’s method and Moser’s constraint manifold, the Lax pairs of the evolution equation hierarchy with the 4th-order eigenvalue problems are nonlinearized. So a new finite-dimensional integrable Hamilton system on the constraint submanifold R ^8N−4 is generated. Moreover, the solutions of the evolution equations for the infinite-dimensional soliton systems are obtained by the involutive flow of the finite-dimensional completely integrable systems.