Geodesic
Reduction of infinite dimensional equations
Electronic journal of differential equations, Tome 2006 (2006)
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.
Classification : 37K15, 37K40
Keywords: soliton equations, Hamiltonian equation, Euler-Lagrange equation, integrable systems, Legendre transformation, involutive system, symmetries of equations, invariant manifold, Poisson bracket, symplectic space 
@article{EJDE_2006__2006__a77,
     author = {Li,  Zhongding and Xu,  Taixi},
     title = {Reduction of infinite dimensional equations},
     journal = {Electronic journal of differential equations},
     year = {2006},
     volume = {2006},
     zbl = {1086.37039},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a77/}
}
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Li,  Zhongding; Xu,  Taixi. Reduction of infinite dimensional equations. Electronic journal of differential equations, Tome 2006 (2006). http://geodesic.mathdoc.fr/item/EJDE_2006__2006__a77/