International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 13479, 9 pages
doi:10.1155/IJMMS/2006/13479

A finite-dimensional integrable system associated with a polynomial eigenvalue problem

Taixi Xu,1 Weihua Mu,2 and Zhijun Qiao3

1Department of Mathematics, Southern Polytechnic State University, 1100 South Marietta Parkway, Marietta 30060, GA, USA
2Department of Mathematics, Shijiazhuang Railway Institute, Hebei 050043, China
3Department of Mathematics, University of Texas – Pan American, 1201 W. University Drive Edinburg, 78541, TX, USA

Received 6 February 2006; Accepted 21 March 2006

Copyright © 2006 Taixi Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

M. Antonowicz and A. P. Fordy (1988) introduced the second-order polynomial eigenvalue problem Lφ=(∂2+∑i=1nviλi)φ=αφ(∂=∂/∂x,α=constant) and discussed its multi-Hamiltonian structures. For n=1 and n=2, the associated finite-dimensional integrable Hamiltonian systems (FDIHS) have been discussed by Xu and Mu (1990) using the nonlinearization method and Bargmann constraints. In this paper, we consider the general case, that is, n is arbitrary, provide the constrained Hamiltonian systems associated with the above-mentioned second-order polynomial ergenvalue problem, and prove them to be completely integrable.