Ferapontov, E. V.; Pavlov, M. V. Quasiclassical limit of coupled KdV equations. Riemann invariants and multi-Hamiltonian structure. (English) Zbl 0742.35055 Physica D 52, No. 2-3, 211-219 (1991). Summary: We consider the quasiclassical limit of the first nontrivial flow in coupled KdV hierarchy. Written down in Riemann invariants, it assumes the extremely simple form \(R^ i_ t=\left(\sum^ n_{k=1}R^ k+2R^ i\right)R^ i_ x\), \(i=1,\ldots,n\). Up to certain natural equivalence this turns out to be the only hydrodynamic system with \(n+1\) compatible purely local Hamiltonian structures. Cited in 13 Documents MSC: 35Q53 KdV equations (Korteweg-de Vries equations) 35P05 General topics in linear spectral theory for PDEs 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 35Q35 PDEs in connection with fluid mechanics 35Q55 NLS equations (nonlinear Schrödinger equations) Keywords:first nontrivial flow; KdV hierarchy PDFBibTeX XMLCite \textit{E. V. Ferapontov} and \textit{M. V. Pavlov}, Physica D 52, No. 2--3, 211--219 (1991; Zbl 0742.35055) Full Text: DOI References: [1] Antonovicz, M.; Fordy, A. P., A family of completely integrable multi-Hamiltonian systems, Phys. Lett. A, 122, 95-99 (1987) [2] Antonovicz, M.; Fordy, A. P., Coupled KDV equations with multi-Hamiltonian structures, Physica D, 28, 345-357 (1987) · Zbl 0638.35079 [3] Antonovicz, M.; Fordy, A. P., Coupled KDV equations associated with a novel Schrödinger spectral problem, (Leon, J. P., Proceedings of the 4th Workshop on Nonlinear Evolution Equations and Dynamical Systems. Proceedings of the 4th Workshop on Nonlinear Evolution Equations and Dynamical Systems, France. Proceedings of the 4th Workshop on Nonlinear Evolution Equations and Dynamical Systems. Proceedings of the 4th Workshop on Nonlinear Evolution Equations and Dynamical Systems, France, Nonlinear Evolution (1987)), 145-159 [4] Fordy, A. P.; Reyman, A. G.; Semenov-Tian-Shansky, M. A., Classical \(R\)-matrices and compatible Poisson brackets for coupled KDV systems, Lett. Math. Phys., 17, 25-29 (1989) · Zbl 0699.58043 [5] Tsarev, S. P., On Poisson brackets and one-dimensional systems of hydrodynamic type, Sov. Math. Dokl., 282, 534-537 (1985) · Zbl 0605.35075 [6] Mokhov, O. I.; Ferapontov, E. V., Nonlocal Hamiltonian operators of hydrodynamic type and constant curvature metric, Usp. Math. Nauk., 45, 191-192 (1990) · Zbl 0712.35080 [7] Zakharov, V. E., On the Benney equations, Physica D, 3, 193-202 (1981) · Zbl 1194.35338 [8] Gibbons, J., Collisionless Boltzmann equations and integrable moment equations, Physica D, 3, 503-511 (1981) · Zbl 1194.35298 [9] Dubrovin, B. A.; Novikov, S. P., Hamiltonian formalism of one-dimensional hydrodynamic systems and averaging method of Bogoliybov-Whitham, Sov. Math. Dokl., 270, 781-785 (1983) · Zbl 0553.35011 [10] Ferapontov, E. V., Hamiltonian systems of hydrodynamic type and their realization on hypersurfaces of pseudoeuclidean space, Geomet. Sbornik, 22, 59-96 (1990), [in Russian] · Zbl 0741.58016 [11] Pavlov, M. V., Hamiltonian formalism of equations of electrophoresis. Integrable equations of hydrodynamics (1987), [in Russian] · Zbl 0632.76001 [12] Sokolov, V. V., ON hamiltonian property of Krichever-Novikov equation, Sov. Math. Dokl., 277, 48-50 (1984) · Zbl 0592.35008 [13] Nutku, J., On a new class of completely integrable nonlinear wave equations, II. Multi-Hamiltonian structure, J. Math. Phys., 28, 2579-2585 (1987) · Zbl 0662.35084 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.