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 Mat. Sb., 1990, Volume 181, Number 9, Pages 1220–1235 (Mi msb1220)

Integration of weekly nonlinear semi-hamiltonian systems of hydrodynamic type by methods of the theory of webs

E. V. Ferapontov

Dorodnitsyn Computing Centre of the Russian Academy of Sciences

Abstract: Weakly nonlinear semi-Hamiltonian systems of $n$ differential equations of hydrodynamic type in Riemann invariants are considered, and the geometry of the $(n+2)$-web formed by the characteristics and the level lines of the independent variables are studied. It is shown that the rank of this web on the general solution of the system is equal to $n$. This result is used to obtain formulas for the general integral of the systems under consideration, with the necessary arbitrariness in $n$ functions of a single argument.
Separate consideration is given to the cases $n=3$ and $n=4$, for which it is possible not only to integrate the corresponding systems, but also to give a complete classification of them to within so-called transformations via a solution (reciprocal transformations). It turns out that for $n=3$ they can all be linearized (and are thus equivalent), while for $n=4$ there exist exactly five mutually nonequivalent systems, and any other system can be reduced to one of them by a transformation via a solution.
There is a discussion of the connection between weakly nonlinear semi-Hamiltonian systems and Dupin cyclides-hypersurfaces of Euclidean space whose principal curvatures are constant along the corresponding principal directions.
Some unsolved problems are formulated at the end of the paper.

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English version:
Mathematics of the USSR-Sbornik, 1992, 71:1, 65–79

Bibliographic databases:

UDC: 514.763.8
MSC: Primary 58F05, 53A60; Secondary 83C55

Citation: E. V. Ferapontov, “Integration of weekly nonlinear semi-hamiltonian systems of hydrodynamic type by methods of the theory of webs”, Mat. Sb., 181:9 (1990), 1220–1235; Math. USSR-Sb., 71:1 (1992), 65–79

Citation in format AMSBIB
\Bibitem{Fer90} \by E.~V.~Ferapontov \paper Integration of weekly nonlinear semi-hamiltonian systems of hydrodynamic type by methods of the theory of~webs \jour Mat. Sb. \yr 1990 \vol 181 \issue 9 \pages 1220--1235 \mathnet{http://mi.mathnet.ru/msb1220} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1085152} \zmath{https://zbmath.org/?q=an:0802.58029|0798.76003} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1992SbMat..71...65F} \transl \jour Math. USSR-Sb. \yr 1992 \vol 71 \issue 1 \pages 65--79 \crossref{https://doi.org/10.1070/SM1992v071n01ABEH001391} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1992HJ82500006} 

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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. Ferapontov E., “Integration of Weakly Nonlinear Hydrodynamic Systems in Riemann Invariants”, Phys. Lett. A, 158:3-4 (1991), 112–118
2. V. G. Mikhalev, “On the Hamiltonian formalism for Korteweg?de Vries type hierarchies”, Funct. Anal. Appl., 26:2 (1992), 140–142
3. Tsarev S., “Integrability of Equations of Hydrodynamic Type From the End of the 19th to the End of the 20th Century”, Integrability: the Seiberg-Witten and Whitham Equations, eds. Braden H., Krichever I., Gordon and Breach Science Publ, 2000, 251–265
4. Ferapontov E., “Invariant Description of Solutions of Hydrodynamic-Type Systems in Hodograph Space: Hydrodynamic Surfaces”, J. Phys. A-Math. Gen., 35:32 (2002), 6883–6892
5. E. V. Ferapontov, M. V. Pavlov, “Reciprocal transformations of Hamiltonian operators of hydrodynamic type: Nonlocal Hamiltonian formalism for linearly degenerate systems”, J Math Phys (N Y ), 44:3 (2003), 1150
6. S.I.. Agafonov, “Local classification of singular hexagonal 3-webs with holomorphic Chern connection form and infinitesimal symmetries”, Geom Dedicata, 2014
7. A. M. Shelekhov, E. A. Onoprienko, “Bol three-webs with covariant constant curvature tensor”, Russian Math. (Iz. VUZ), 60:3 (2016), 72–81
8. E. A. Onoprienko, A. M. Shelekhov, “Bol three-webs $B_m^{\triangledown}$ with torsion tensor of rank $\rho$”, Sb. Math., 209:8 (2018), 1164–1210
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