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Integrable bi-Hamiltonian systems of hydrodynamic type. (English. Russian original) Zbl 1044.37047

Russ. Math. Surv. 57, No. 1, 153-154 (2002); translation from Usp. Mat. Nauk 57, No. 1, 157-158 (2002).
The author considers the problem of constructing local bi-Hamiltonian systems of hydrodynamic type. Starting from a set of Hamiltonian operators of hydrodynamic type, the author presents a class of local Hamiltonian pairs to yield recursion operators. Applying each of the resulting recursion operators to the system \(v_t^i=v_x^i\) leads to a hierarchy of bi-Hamiltonian systems of hydrodynamic type.
The paper is clearly written. The basis of the work is a principle broadly used in the KdV theory. There are also many concrete examples in the literature [see, for example, G. Z. Tu and W. X. Ma, J. Partial Differ. Equations 5, 43–56 (1992; Zbl 0751.58016) and W. X. Ma, J. Phys. A, Math. Gen. 31, 7279–7289 (1998; Zbl 0929.35147)].
Reviewer: Ma Wen-Xiu (Tampa)

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
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