Geodesic
Nonisentropic one-dimensional gas motions constructed by means of the contact group of the nonhomogeneous Monge--Amp\`ere equation
Sbornik. Mathematics, Tome 71 (1992) no. 2, pp. 447-462

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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. 
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     author = {S. V. Khabirov},
     title = {Nonisentropic one-dimensional gas motions constructed by means of the contact group of the nonhomogeneous {Monge--Amp\`ere} equation},
     journal = {Sbornik. Mathematics},
     pages = {447--462},
     publisher = {mathdoc},
     volume = {71},
     number = {2},
     year = {1992},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1992_71_2_a11/}
}
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S. V. Khabirov. Nonisentropic one-dimensional gas motions constructed by means of the contact group of the nonhomogeneous Monge--Amp\`ere equation. Sbornik. Mathematics, Tome 71 (1992) no. 2, pp. 447-462. http://geodesic.mathdoc.fr/item/SM_1992_71_2_a11/