Reduction of infinite dimensional equations

• Main Authors: Zhongding Li, Taixi Xu
• Format: Article
• Language: English
• Published: Texas State University 2006-02-01
• Series: Electronic Journal of Differential Equations
• Subjects: Soliton equations; Hamiltonian equation; Euler-Lagrange equation; integrable systems; Legendre transformation; involutive system; symmetries of equations; invariant manifold; Poisson bracket; symplectic space.
• Online Access: http://ejde.math.txstate.edu/Volumes/2006/17/abstr.html

In this paper, we use the general Legendre transformation to show the infinite dimensional integrable equations can be reduced to a finite dimensional integrable Hamiltonian system on an invariant set under the flow of the integrable equations. Then we obtain the periodic or quasi-periodic solution of the equation. This generalizes the results of Lax and Novikov regarding the periodic or quasi-periodic solution of the KdV equation to the general case of isospectral Hamiltonian integrable equation. And finally, we discuss the AKNS hierarchy as a special example.