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 TMF, 1987, Volume 73, Number 2, Pages 316–320 (Mi tmf5634)

Hamiltonian formalism of weakly nonlinear hydrodynamic systems

M. V. Pavlov

Abstract: Systems of quasilinear equations are considered which are diagonalizable and Hamiltonian, with the condition $\partial_iv^i=0$ where $u_t^i=v^i(u)u_x^i$, $i=1,…,N$. Conservation laws of such systems are found as well as metrics and Poisson brackets. By concrete examples the procedure of finding the solutions is demonstrated. Conditions of the existence of solutions and continuity of commuting flows are pointed out.

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English version:
Theoretical and Mathematical Physics, 1987, 73:2, 1242–1245

Bibliographic databases:

Citation: M. V. Pavlov, “Hamiltonian formalism of weakly nonlinear hydrodynamic systems”, TMF, 73:2 (1987), 316–320; Theoret. and Math. Phys., 73:2 (1987), 1242–1245

Citation in format AMSBIB
\Bibitem{Pav87} \by M.~V.~Pavlov \paper Hamiltonian formalism of weakly nonlinear hydrodynamic systems \jour TMF \yr 1987 \vol 73 \issue 2 \pages 316--320 \mathnet{http://mi.mathnet.ru/tmf5634} \zmath{https://zbmath.org/?q=an:0653.76005|0632.76001} \transl \jour Theoret. and Math. Phys. \yr 1987 \vol 73 \issue 2 \pages 1242--1245 \crossref{https://doi.org/10.1007/BF01017597} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1987P005000016} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. B. A. Dubrovin, S. P. Novikov, “Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory”, Russian Math. Surveys, 44:6 (1989), 35–124
2. E. V. Ferapontov, “Integration of weekly nonlinear semi-hamiltonian systems of hydrodynamic type by methods of the theory of webs”, Math. USSR-Sb., 71:1 (1992), 65–79
3. V. R. Kudashev, S. E. Sharapov, “Generalized hodograph method from the group-theoretical point of view”, Theoret. and Math. Phys., 85:2 (1990), 1155–1159
4. Ferapontov, EV, “Reciprocal transformations of Hamiltonian operators of hydrodynamic type: Nonlocal Hamiltonian formalism for linearly degenerate systems”, Journal of Mathematical Physics, 44:3 (2003), 1150
5. Blaszak, M, “A COORDINATE-FREE CONSTRUCTION OF CONSERVATION LAWS AND RECIPROCAL TRANSFORMATIONS FOR A CLASS OF INTEGRABLE HYDRODYNAMIC-TYPE SYSTEMS”, Reports on Mathematical Physics, 64:1–2 (2009), 341
6. El G.A., Kamchatnov A.M., Pavlov M.V., Zykov S.A., “Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions”, J Nonlinear Sci, 21:2 (2011), 151–191
7. Lorenzoni P., Pedroni M., “Natural Connections for Semi-Hamiltonian Systems: The Case of the epsilon-System”, Lett Math Phys, 97:1 (2011), 85–108
8. M. V. Pavlov, V. B. Taranov, G. A. El, “Generalized hydrodynamic reductions of the kinetic equation for a soliton gas”, Theoret. and Math. Phys., 171:2 (2012), 675–682
9. Pavlov M.V., “Integrable Dispersive Chains and Energy Dependent Schrodinger Operator”, J. Phys. A-Math. Theor., 47:29 (2014), 295204
10. Maxim V. Pavlov, “Integrability of exceptional hydrodynamic-type systems”, Proc. Steklov Inst. Math., 302 (2018), 325–335
11. Marvan M. Pavlov M.V., “Integrable Dispersive Chains and Their Multi-Phase Solutions”, Lett. Math. Phys., 109:5 (2019), 1219–1245
12. R. Ch. Kulaev, A. B. Shabat, “Darboux system and separation of variables in the Goursat problem for a third order equation in $\mathbb {R}^3$”, Russian Math. (Iz. VUZ), 64:4 (2020), 35–43
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