
This article is cited in 8 scientific papers (total in 8 papers)
Integration of weekly nonlinear semihamiltonian 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 semiHamiltonian 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 socalled 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 semiHamiltonian systems and Dupin cyclideshypersurfaces 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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Mathematics of the USSRSbornik, 1992, 71:1, 65–79
Bibliographic databases:
UDC:
514.763.8
MSC: Primary 58F05, 53A60; Secondary 83C55 Received: 26.09.1989
Citation:
E. V. Ferapontov, “Integration of weekly nonlinear semihamiltonian systems of hydrodynamic type by methods of the theory of webs”, Mat. Sb., 181:9 (1990), 1220–1235; Math. USSRSb., 71:1 (1992), 65–79
Citation in format AMSBIB
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\jour Math. USSRSb.
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http://mi.mathnet.ru/eng/msb1220 http://mi.mathnet.ru/eng/msb/v181/i9/p1220
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This publication is cited in the following articles:

Ferapontov E., “Integration of Weakly Nonlinear Hydrodynamic Systems in Riemann Invariants”, Phys. Lett. A, 158:34 (1991), 112–118

V. G. Mikhalev, “On the Hamiltonian formalism for Korteweg?de Vries type hierarchies”, Funct. Anal. Appl., 26:2 (1992), 140–142

Tsarev S., “Integrability of Equations of Hydrodynamic Type From the End of the 19th to the End of the 20th Century”, Integrability: the SeibergWitten and Whitham Equations, eds. Braden H., Krichever I., Gordon and Breach Science Publ, 2000, 251–265

Ferapontov E., “Invariant Description of Solutions of HydrodynamicType Systems in Hodograph Space: Hydrodynamic Surfaces”, J. Phys. AMath. Gen., 35:32 (2002), 6883–6892

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

S.I.. Agafonov, “Local classification of singular hexagonal 3webs with holomorphic Chern connection form and infinitesimal symmetries”, Geom Dedicata, 2014

A. M. Shelekhov, E. A. Onoprienko, “Bol threewebs with covariant constant curvature tensor”, Russian Math. (Iz. VUZ), 60:3 (2016), 72–81

E. A. Onoprienko, A. M. Shelekhov, “Bol threewebs $B_m^{\triangledown}$ with torsion tensor of rank $\rho$”, Sb. Math., 209:8 (2018), 1164–1210

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