International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Issue 3, Pages 13479, 9 p.
doi:10.1155/IJMMS/2006/13479
A finite-dimensional integrable system associated with a polynomial eigenvalue problem
Taixi Xu1
, Weihua Mu2
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
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.