Geodesic
A new finite-dimensional Hamiltonian systems with a~mixed Poisson structure for the~KdV equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 3, pp. 361-374

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A Lax pair for the KdV equation is derived by a transformation of the eigenfunction. By a polynomial expansion of the eigenfunction for the resulting Lax pair, finite-dimensional integrable systems can be obtained from the Lax pair. These integrable systems are proved to be the Hamiltonian and are shown to have a new Poisson structure such that the entries of its structure matrix are a mixture of linear and quadratic functions of coordinates. The odd and even functions of the spectral parameter are introduced to build a generating function for conserved integrals. Based on the generating function, the integrability of these Hamiltonian systems is shown.
Mots-clés : polynomial expansion, Poisson structure
Keywords: Hamiltonian system, conserved integrals. 
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     author = {Dianlou Du and Xue Wang},
     title = {A new finite-dimensional {Hamiltonian} systems with a~mixed {Poisson} structure for {the~KdV} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {361--374},
     publisher = {mathdoc},
     volume = {211},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2022_211_3_a0/}
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Dianlou Du; Xue Wang. A new finite-dimensional Hamiltonian systems with a~mixed Poisson structure for the~KdV equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 211 (2022) no. 3, pp. 361-374. http://geodesic.mathdoc.fr/item/TMF_2022_211_3_a0/