Про часткову попередню групову класифікацію певного класу (1+3)-вимірних рівнянь Монжа–Ампера. І. Одновимірні алгебри Лі

V. M. Fedorchuk, V. I. Fedorchuk

Анотація


Вивчається часткова попередня групова класифікація певного класу (1+3)-вимірних рівнянь Монжа–Ампера. Наведено результати, отримані з використанням одновимірних неспряжених підалгебр алгебри Лі групи Пуанкаре P(1,4) та нееквівалентних функціональних базисів диференціальних інваріантів першого порядку цих підалгебр.

 

Зразок для цитування: В. М. Федорчук, В. І. Федорчук, “Про часткову попередню групову класифікацію певного класу (1+3)-вимірних рівнянь Монжа–Ампера. І. Одновимірні алгебри Лі,” Мат. методи та фіз.-мех. поля, 66, No. 1-2, 40–47 (2023), https://doi.org/


Ключові слова


попередня групова класифікація, рівняння Монжа–Ампера, неспряжені підалгебри алгебр Лі, диференціальнi інваріанти, група Пуанкаре P(1,4)

Посилання


N. H. Ibragimov, “On the group classification of differential equations of second order,” Dokl. AN SSSR, 183, No. 2, 274–277 (1968) (in Russian); English translation: Sov. Math. Dokl., 9, No. 6, 1365–1369 (1968).

V. I. Lahno, S. V. Spichak, V. I. Stohnii, Symmetry Analysis of Equations of Evolution Type [in Ukrainian], Mathematics and its Applications, Vol. 45, Institute of Mathematics of the Ukrainian National Academy of Sciences, Kyiv (2002).

L. V. Ovsiannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978); [in English] Acad. Press, New York (1982).

A. V. Pogorelov, The Minkowski Multidimensional Problem [in Russian], Nauka, Moscow (1975).

V. M. Fedorchuk, V. I. Fedorchuk, “On classification of the low-dimensional non-conjugated subalgebras of the Lie algebra of the Poincare group P(1,4),” in: Symmetry and integrability of the equations of mathematical physics, Zb. Prats’ Inst. Mat. Nats. Akad. Nauk Ukr., 3, No. 2, 301–308 (2006) (in Ukrainian).

W. I. Fushchich, N. I. Serov, “Symmetry and some exact solutions of the multidimensional Monge–Ampère equation,” Dokl. Akad. Nauk SSSR, 273, No. 3, 543–546 (1983) (in Russian); English translation: Sov. Math. Dokl., 28, No. 3, 679–682 (1983).

S. V. Khabirov, “Application of contact transformations of the inhomogeneous Monge–Ampère equation in one-dimensional gas dynamics,” Dokl. Akad. Nauk SSSR, 310, No. 2, 333–336 (1990) (in Russian); English translation: Sov. Phys. Dokl., 35, No. 1, 29–30 (1990).

D. J. Arrigo, “Group properties of a Monge–Ampère type equation,” in: W. F. Ames, P. J. van der Houwen (eds), Computational and applied mathematics. II. Differential equations: Selected and revised papers from the IMACS World Congress (Dublin, Ireland, July 1991), North-Holland, Amsterdam (1992), pp. 107–115.

I. J. Bakelman, Convex Analysis and Nonlinear Geometric Elliptic Equations, Springer, Berlin (1994), https://doi.org/10.1007/978-3-642-69881-1

B. Banos, “On symplectic classification of effective 3-forms and Monge–Ampère equations,” Differ. Geom. Appl., 19, No. 2, 147–166 (2003), https://doi.org/10.1016/S0926-2245(03)00017-2

P. Basarab-Horwath, V. Lahno, R. Zhdanov, “The structure of Lie algebras and the classification problem for partial differential equations,” Acta Appl. Math., 69, No. 1, 43–94 (2001), https://doi.org/10.1023/A:1012667617936

V. M. Boyko, O. V. Lokaziuk, R. O. Popovych, “Realizations of Lie algebras on the line and the new group classification of (1+1)-dimensional generalized nonlinear Klein–Gordon equations,” Anal. Math. Phys., 11, No. 3, Art. 127, 38 p. (2021), https://doi.org/10.1007/s13324-021-00550-z

Cui Fan, Jian Huaiyu, “Symmetry of solutions to a class of Monge–Ampère equations,” Commun. Pure Appl. Anal., 18, No. 3, 1247–1259 (2019), https://doi.org/10.3934/cpaa.2019060

M. J. P. Cullen, The Mathematics of Large-Scale Atmosphere and Ocean, World Sci. Publ., Hackensack (2021), https://doi.org/10.1142/12367

A. De Paris, A. M. Vinogradov, “Scalar differential invariants of symplectic Monge–Ampère equations,” Centr. Eur. J. Math., 9, No. 4, 731–751 (2011), https://doi.org/10.2478/s11533-011-0046-7

V. M. Fedorchuk, V. I. Fedorchuk, “First-order differential invariants of the splitting subgroups of the Poincaré group P(1,4),” Univ. Iagel. Acta Math., 1290, No. 44, 21–30 (2006).

V. M. Fedorchuk, V. I. Fedorchuk, “On non-equivalent functional bases of first-order differential invariants of the nonconjugate subgroups of the Poincaré group P(1,4),” Acta Phys. Debrecina, 42 (XLII), 122–132 (2008).

V. I. Fushchich, A. G. Nikitin, “Reduction of the representations of the generalised Poincare algebra by the Galilei algebra,” J. Phys. A: Math. Gen., 13, No. 7, 2319–2330 (1980), https://doi.org/10.1088/0305-4470/13/7/015

C. E. Gutiérrez, T. van Nguyen, “On Monge–Ampère type equations arising in optimal transportation problems,” Calcul. Var. Partial Differ. Equat., 28, No. 3, 275–316 (2007), https://doi.org/10.1007/s00526-006-0045-x

F. Jiang, N. S. Trudinger, “On the second boundary value problem for Monge–Ampère type equations and geometric optics,” Arch. Ration. Mech. Anal., 229, No. 2, 547–567 (2018), https://doi.org/10.1007/s00205-018-1222-8

A. Kushner, V. Lychagin, J. Slovak, “Lectures on geometry of Monge–Ampère equations with Maple,” in: Nonlinear PDEs, their Geometry, and Applications, R. A. Kycia, M. Ulan, E. Schneider (Eds.), Birkhäuser, Basel (2019), Chapter 2, pp. 53–94, https://doi.org/10.1007/978-3-030-17031-8_2

S. Lie, “Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen,“ Arch. Math., 6, No. 3, 328–368 (1881).

V. V. Lychagin, V. N. Rubtsov, I. V. Chekalov, “A classification of Monge–Ampère equations,” Ann. Sci. Ècole Norm. Sup. Ser. 4, 26, No. 3, 281–308 (1993), https://doi.org/10.24033/asens.1673

A. G. Nikitin, “Symmetries of Schrödinger–Pauli equations for charged particles and quasirelativistic Schrödinger equations,” J. Phys. A. Math. Theor., 55, No. 11, Art. 115202, 24 p. (2022), https://doi.org/10.1088/1751-8121/ac515d

R. O. Popovych, “Point and contact equivalence groupoids of two-dimensional quasilinear hyperbolic equations,” Appl. Math. Lett., 116, Art. 107068, 8 p. (2021), https://doi.org/10.1016/j.aml.2021.107068

Stępień Ł. T. On some exact solutions of heavenly equations in four dimensions // AIP Adv. – 2020. – 10. – Art. 065105. – https://doi.org/10.1063/1.5144327.

Ł. T. Stępień, “On some exact solutions of heavenly equations in four dimensions,” AIP Advances, 10, Art. 065105 (2020), https://doi.org/10.1063/1.5144327

D. Tseluiko, “On classification of hyperbolic Monge–Ampère equations on 2-dimensional manifolds,” Rend. Sem. Mat. Messina Ser. II, 8(23), 139–150 (2001).

C. Udrişte, N. Bilă, “Symmetry group of Ţiţeica surfaces PDE,” Balkan J. Geom. Appl., 4, No. 2, 123–140 (1999).

C. Udrişte, N. Bîlă, “Symmetry Lie group of the Monge–Ampère equation,” Balkan J. Geom. Appl., 3, No. 2, 121–134 (1998).

O. O. Vaneeva, R. O. Popovych, C. Sophocleous, “Extended symmetry analysis of two-dimensional degenerate Burgers equation,” J. Geom. Phys., 169, Art. 104336, 21 p. (2021), https://doi.org/10.1016/j.geomphys.2021.104336

Yau Shing-Tung, Nadis Steve, The Shape of a Life. One Mathematician’s Search for the Universe’s Hidden Geometry, Yale Univ. Press, New Haven (2019).


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