Non-autonomous evolutionary equation of Monge–Ampere type with two space variables

I. V. Rakhmelevich

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2023), pp. 66-80

Résumé

There are investigated the exact solutions of non-autonomous evolutionary equation with two space variables, the right side of which contains Monge–Ampere operator. The solutions with additive and multiplicative separation of variables are founded. There are considered the reductions of the given equation to ordinary differential equations (ODE). The classic and generalized self-similar solutions, and the solutions with functional separation of variables are received. In particular, it is shown that the given equation can be reduced to ODE, if the coefficient at the time derivative can be represented in the form of production of the functions depend on time, space variables, and unknown function. Also the reductions of the given equation to two-dimensional PDE are founded.

Détail

Keywords: evolutionary equation, Monge–Ampere equation, reduction, separation of variables, self-similar solution.

Bibliographie

[1] Polyanin A.D., Zaitsev V.F., Handbook of Nonlinear Partial Differential Equations, 2nd edition, Chapman and Hall-CRC Press, Boca Raton–London, 2012

[2] Rozhdestvenskii B.L., Yanenko N.N., Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978

[3] Khabirov S.V., “Neizentropicheskie odnomernye dvizheniya gaza, postroennye s pomoschyu kontaktnoi gruppy neodnorodnogo uravneniya Monzha–Ampera”, Matem. sb., 181:12 (1990), 1607–1622

[4] Shablovskii O.N., “Parametricheskie resheniya uravneniya Monzha–Ampera i techeniya gaza s peremennoi entropiei”, Vestn. Tomsk. gos. un-ta. Matem. i mekhan., 1:33 (2015), 105–118

[5] Pogorelov A.V., Mnogomernoe uravnenie Monzha–Ampera, Nauka, M., 1988

[6] Ibragimov N.H. (ed.), CRC Handbook of Lie Groups to Differential Equations, v. 1, Symmetries, Exact Solutions and Conservation Laws, CRC Press, Boca Raton, 1994

[7] Rakhmelevich I.V., “O resheniyakh dvumernogo uravneniya Monzha–Ampera so stepennoi nelineinostyu po pervym proizvodnym”, Vestn. Tomsk. gos. un-ta. Matem. i mekhan., 4:42 (2016), 33–43

[8] Rakhmelevich I.V., “Modifitsirovannoe dvumernoe uravnenie Monzh–Ampera”, Vestn. Voronezhsk. gos. un-ta. Ser. Fizika. Matem., 2017, no. 3, 159–168

[9] Rakhmelevich I.V., “Mnogomernoe uravnenie Monzha–Ampera so stepennymi nelineinostyami po pervym proizvodnym”, Vestn. Voronezhsk. gos. un-ta. Ser. Fizika. Matem., 2020, no. 2, 86–98

[10] Galaktionov V. A., Svirshchevskii S.R., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman and Hall/CRC Press, Boca Raton, 2006

[11] Gutiérres C.E., The Monge–Ampère Equation, Birkhäuser, Boston, 2001

[12] Taylor M.E., Partial Differential Equations, v. III, Nonlinear Equations, Springer-Verlag, New York, 1996

[13] Leibov O., “Reduction and Exact Solutions of the Monge-Ampere Equation”, Nonlinear Math. Phys., 4:1–2 (1997), 146–148

[14] Fedorchuk V.M., Leibov O.S., “Symmetry reduction and exact solutions of the multidimensional Monge-Ampere equation”, Ukr. Math. J., 48:5 (1996), 775–783

[15] Fushchych W.I., “The symmetry and exact solutions of some multidimensional nonlinear equations of mathematical physics”, Sci. Works, 3 (2001), 175–179

[16] Chou K.-S., Wang X.-J., “A logarithmic Gauss curvature flow and the Minkowski Problem”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17:6 (2000), 733–751

[17] Polyanin A.D., Zhurov A.I., Metody razdeleniya peremennykh i tochnye resheniya nelineinykh uravnenii matematicheskoi fiziki, Izd-vo IPMekh RAN, M., 2020