Electron. J. Diff. Eqns., Vol. 2006(2006), No. 17, pp. 1-15.

Reduction of infinite dimensional equations

Zhongding Li, Taixi Xu

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

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  Zhongding Li
Department of Mathematics, Shijiazhuang Railway Institute, Hebei, China
Taixi Xu
Department of Mathematics
Southern Polytechnic State University
1100 South Marietta Parkway
Marietta, GA 30060, USA
email: txu@spsu.edu

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