Abstract:
The equations of one-dimensional gas dynamics in Lagrange coordinates are connected with the inhomogeneous Monge–Ampère equation by means of a differential substitution. A classification of Monge–Ampère equations based on point and contact transformations is carried out. In the case of an infinite group various linearizations of the equations of gas dynamics are presented. New conservation laws are constructed on the basis of Nöther's theorem. Examples of invariant solutions with variable entropy are considered, and some boundary value problems with curved shock waves are also solved.
Citation:
S. V. Khabirov, “Nonisentropic one-dimensional gas motions constructed by means of the contact group of the nonhomogeneous Monge–Ampère equation”, Math. USSR-Sb., 71:2 (1992), 447–462
\Bibitem{Kha90}
\by S.~V.~Khabirov
\paper Nonisentropic one-dimensional gas motions constructed by means of the contact group of the nonhomogeneous Monge--Amp\`ere equation
\jour Math. USSR-Sb.
\yr 1992
\vol 71
\issue 2
\pages 447--462
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\crossref{https://doi.org/10.1070/SM1992v071n02ABEH001405}
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\zmath{https://zbmath.org/?q=an:0776.76079|0713.76085}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?1992SbMat..71..447K}
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Linking options:
https://www.mathnet.ru/eng/sm1250
https://doi.org/10.1070/SM1992v071n02ABEH001405
https://www.mathnet.ru/eng/sm/v181/i12/p1607
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I. V. Rakhmelevich, “Sistema dvumernykh uravnenii Monzha — Ampera: reduktsii i tochnye resheniya”, Vladikavk. matem. zhurn., 26:4 (2024), 121–136
I. V. Rakhmelevich, “Non-autonomous evolutionary equation of Monge–Ampere type with two space variables”, Russian Math. (Iz. VUZ), 67:2 (2023), 52–64
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E. Sh. Gutshabash, “Preobrazovanie Lezhandra v modeli Borna–Infelda, uravnenie Monzha–Ampera i tochnye resheniya”, Voprosy kvantovoi teorii polya i statisticheskoi fiziki. 29, Zap. nauchn. sem. POMI, 520, POMI, SPb., 2023, 151–161
I. V. Rakhmelevich, “O mnogomernykh determinantnykh differentsialno-operatornykh
uravneniyakh”, Vladikavk. matem. zhurn., 22:2 (2020), 53–69
I. V. Rakhmelevich, “O resheniyakh dvumernogo uravneniya Monzha–Ampera so stepennoi nelineinostyu po pervym proizvodnym”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2016, no. 4(42), 33–43
O. N. Shablovskii, “Parametricheskie resheniya uravneniya Monzha–Ampera i techeniya gaza s peremennoi entropiei”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2015, no. 1(33), 105–118
E.A. Levchenko, A.V. Shapovalov, A.Yu. Trifonov, “Symmetries of the Fisher–Kolmogorov–Petrovskii–Piskunov equation with a nonlocal nonlinearity in a semiclassical approximation”, Journal of Mathematical Analysis and Applications, 2012
Polyanin A.D., Zhurov A.I., Vyazmina E.A., “Exact Solutions to Nonlinear Equations and Systems of Equations of General Form in Mathematical Physics”, Applications of Mathematics in Engineering and Economics `34, AIP Conference Proceedings, 1067, ed. Todorov M., Amer Inst Physics, 2008, 64–86
D.J. Arrigo, J.M. Hill, “On a Class of Linearizable Monge-Ampère Equations”, Journal of Nonlinear Mathematical Physics, 5:2 (1998), 115
Nutku Y., Sheftel M., Malykh A., “Gravitational Instantons”, Class. Quantum Gravity, 14:3 (1997), L59–L63
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