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 Izv. RAN. Ser. Mat., 1996, Volume 60, Issue 6, Pages 3–30 (Mi izv93)

Systems of conservation laws in the context of the projective theory of congruences

S. I. Agafonova, E. V. Ferapontov

a Loughborough University

Abstract: We associate to a system of $n$ conservation laws
$$u_t^i=f^i(u)_x, \qquad i=1,…,n,$$
an $n$-parameter family of lines in $(n+1)$-dimensional space $A^{n+1}$ given by the equations
$$y^i=u^iy^0-f^i(u), \qquad i=1,…,n.$$
Thereby we establish a correspondence between the reciprocal transformations of the system of conservation laws and the projective transformations of the space $A^{n+1}$, the rarefaction curves of the system of conservation laws and the developable surfaces of the associated family of lines, the Temple class of systems of conservation laws and the class of families of lines whose developable surfaces are either flat or conic. In the particular case $n=2$ the systems of the Temple class are explicitly described in terms of the theory of congruences.

DOI: https://doi.org/10.4213/im93

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English version:
Izvestiya: Mathematics, 1996, 60:6, 1097–1122

Bibliographic databases:

MSC: 35L65, 53A20

Citation: S. I. Agafonov, E. V. Ferapontov, “Systems of conservation laws in the context of the projective theory of congruences”, Izv. RAN. Ser. Mat., 60:6 (1996), 3–30; Izv. Math., 60:6 (1996), 1097–1122

Citation in format AMSBIB
\Bibitem{AgaFer96} \by S.~I.~Agafonov, E.~V.~Ferapontov \paper Systems of conservation laws in the context of the projective theory of congruences \jour Izv. RAN. Ser. Mat. \yr 1996 \vol 60 \issue 6 \pages 3--30 \mathnet{http://mi.mathnet.ru/izv93} \crossref{https://doi.org/10.4213/im93} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1438880} \zmath{https://zbmath.org/?q=an:0889.35063} \transl \jour Izv. Math. \yr 1996 \vol 60 \issue 6 \pages 1097--1122 \crossref{https://doi.org/10.1070/IM1996v060n06ABEH000093} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1996XF63000001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33747002253} 

• http://mi.mathnet.ru/eng/izv93
• https://doi.org/10.4213/im93
• http://mi.mathnet.ru/eng/izv/v60/i6/p3

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. Tsarev S.P., “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, 2000, 251–265
2. Andreichenko D.K., Andreichenko K.P., Petrova T.Y., “Dynamic modelling of a non-conservative discrete-continuous system”, Pmm Journal of Applied Mathematics and Mechanics, 68:5 (2004), 691–698
3. Konopelchenko B.G., Ortenzi G., “Algebraic varieties in the Birkhoff strata of the Grassmannian Gr((2)): Harrison cohomology and integrable systems”, Journal of Physics a-Mathematical and Theoretical, 44:46 (2011), 465201
4. Ferapontov E.V., Pavlov M.V., Vitolo R.F., “Systems of Conservation Laws With Third-Order Hamiltonian Structures”, Lett. Math. Phys., 108:6 (2018), 1525–1550
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