Pavlov, Maxim V. The Kupershmidt hydrodynamic chains and lattices. (English) Zbl 1115.37058 Int. Math. Res. Not. 2006, No. 20, Article ID 46987, 43 p. (2006). The author of this very interesting paper studies the Kupershmidt hydrodynamic chains of the form \[ B_{t}^{k}=B_{x}^{k+1}+(1/\beta )B^{0}B_{x}^{k}+(k+\gamma )B^{k}B_{x}^{0}, \quad k=0,1,2,..., \] introduced by B. A. Kupershmidt [Lie Groups, Hist. Front. Appl., Ser. B 2, 357–358 (1984; Zbl 0594.76003)]. These hydrodynamic chains constitute a simple one-parameter generalization of the Benney hydrodynamic chain [D. J. Benney, Stud. Appl. Math. 52, 45–50 (1973; Zbl 0259.35011)] . The main result is the existence of an infinite set of local Hamiltonian structures. It is shown that the Benney hydrodynamic chain \(A_{t}^{k}=A_{x}^{k+1}+kA^{k-1}A_{x}^{0}\) (\(k=0,1,2,...\)) (\(A^{k}\) are some potentials) has infinite series of local Hamiltonian structures. Three important features of the hydrodynamic chains are considered: (i) new explicit hydrodynamic reductions determined by the hypergeometric function; (ii) infinitely many local and nonlocal Hamiltonian structures; (iii) reciprocal transformations connecting Kupershmidt hydrodynamic chains with distinct parameters \(\beta \) to each other. The integrability of these hydrodynamic chains by the generalized hodograph method is discussed as well. Reviewer: Dimitar A. Kolev (Sofia) Cited in 17 Documents MSC: 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Keywords:Hamiltonian structure Citations:Zbl 0259.35011; Zbl 0594.76003 PDFBibTeX XMLCite \textit{M. V. Pavlov}, Int. Math. Res. Not. 2006, No. 20, Article ID 46987, 43 p. (2006; Zbl 1115.37058) Full Text: DOI arXiv