Reduction of infinite dimensional equations

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.

Keywords

• Soliton equations
• Hamiltonian equation
• Euler-Lagrange equation
• integrable systems
• Legendre transformation
• involutive system
• symmetries of equations
• invariant manifold
• Poisson bracket
• symplectic space.