Electronic Journal of Differential Equations,
Vol. 2006(2006), No. 17, pp. 1-15.
Title: Reduction of infinite dimensional equations
Authors: Zhongding Li (Shijiazhuang Railway Institute, Hebei, China)
Taixi Xu (Southern Polytechnic State Univ., Marietta, GA, USA)
Abstract:
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.
Submitted February 11, 2005. Published February 2, 2006.
Math Subject Classifications: 37K15, 37K40.
Key Words: Soliton equations; Hamiltonian equation;
Euler-Lagrange equation; integrable systems; Legendre transformation;
involutive system; symmetries of equations;
invariant manifold; Poisson bracket; symplectic space