Про класифікацію симетрійних редукцій (1+3)-вимірного рівняння Монжа–Ампера

V. M. Fedorchuk, V. I. Fedorchuk

Анотація


Здійснено класифікацію симетрійних редукцій рівняння Монжа–Ампера в просторі M(1,3)×R(u). Наведено деякі результати, отримані з використанням класифікації тривимірних неспряжених підалгебр алгебри Лі групи Пуанкаре P(1,4).

 

Зразок для цитування: В. М. Федорчук, В. І. Федорчук, “Про класифікацію симетрійних редукцій (1+3)-вимірного рівняння Монжа–Ампера,” Мат. методи та фіз.-мех. поля математики, 63, № 2, 7–16 (2020), https://doi.org/10.15407/mmpmf2020.63.2.7-16

Translation: V. М. Fedorchuk, V. I. Fedorchuk, “On the classification of symmetry reductions for the (1+3)-dimensional Monge–Ampère equation,” J. Math. Sci., 272, No. 1, 1–13 (2023), https://doi.org/10.1007/s10958-023-06395-0


Ключові слова


класифікація симетрійних редукцій, рівняння Монжа–Ампера, класифікація алгебр Лі, неспряжені підалгебри алгебр Лі, група Пуанкаре P(1,4).

Посилання


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, “Symmetry reduction and some exact solutions of a nonlinear five-dimensional wave equation,” Ukr. Mat. Zh., 48, No. 4, 573–577 (1996); English translation: Ukr. Math. J., 48, No. 4, 636–640 (1996), https://doi.org/10.1007/BF02390625

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: Zb. Prats’ Inst. Mat. Nats. Akad. Nauk Ukr., Symmetry and integrability of the equations of mathematical physics, 3, No. 2, 302–308 (2006).

W. I. Fushchich, A. G. Nikitin, Symmetry of Equations of Quantum Mechanics, [in Russian], Nauka, Moscow (1990); [in English] Allerton Press, New York (1994).

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).

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

M. J. P. Cullen, R. J. Douglas, “Applications of the Monge–Ampère equation and Monge transport problem to meteorology and oceanography,” in: Proc. Conf. Monge–Ampère equation: Applications to geometry and optimization, Contemp. Math., Vol. 226, Amer. Math. Soc., Providence, RI (1999), p. 33–53.

V. Fedorchuk,” Symmetry reduction and exact solutions of the Euler–Lagrange–Born–Infeld, multidimensional Monge–Ampère and eikonal equations,” J. Nonlinear Math. Phys., 2, No. 3-4, 329–333 (2013), https://doi.org/10.2991/jnmp.1995.2.3-4.13

V. Fedorchuk, V. Fedorchuk, Classification of Symmetry Reductions for the Eikonal Equation, Pidstryhach Inst. Appl. Probl. Mech. Mat., Nat. Acad. Sci. Ukr, Lviv (2018).

V. Fedorchuk, V. Fedorchuk, “On classification of symmetry reductions for partial differential equations,” in: Non-classical Problems of Theory of Differential Equations, Collection of the works dedicated to 80th of anniversary of B. Yo. Ptashnyk, Pidstryhach Inst. Appl. Probl. Mech. Math., Nat. Acad. Sci. Ukr., Lviv (2017), p. 241-255.

V. Fedorchuk, V. Fedorchuk, “On classification of symmetry reductions for the eikonal equation,” Symmetry, 8, No. 6, Art. 51, 32 p. (2016), https://doi.org/10.3390/sym8060051

V. M. Fedorchuk, V. I. Fedorchuk, “On the classification of symmetry reductions and invariant solutions for the Euler–Lagrange–Born–Infeld equation,” Ukr. Fiz. Zh., 64, No. 12, 1103–1107 (2019), https://doi.org/10.15407/ujpe64.12.1103

A. M. Grundland, J. Harnad, P. Winternitz, “Symmetry reduction for nonlinear relativistically invariant equations,” J. Math. Phys., 25, No. 4, 791–806 (1984), https://doi.org/10.1063/1.526224

C. E. Gutierrez, 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

Jia Xiaobiao, Li Dongsheng, Li Zhisu, “Asymptotic behavior at infinity of solutions of Monge–Ampère equations in half spaces,” J. Differ. Equat., 269, No. 1, 326–348 (2020), https://doi.org/10.1016/j.jde.2019.12.007

A. Kushner, V. 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), Chapt. 2, p. 53–94.

Q. Le Nam, “Global Holder estimates for 2D linearized Monge–Ampère equations with right-hand side in divergence form,” J. Math. Anal. Appl., 485, No. 2, Art. 123865, 13 p. (2020), https://doi.org/10.1016/j.jmaa.2020.123865

Li Dongsheng, Li Zhisu, Yuan Yu, “A Bernstein problem for special Lagrangian equations in exterior domains,” Adv. Math., 361, Art. 106927, 29 p., (2020), https://doi.org/10.1016/j.aim.2019.106927

S. Lie, Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebiger Ordnung, Berichte, Leipzig (1895), S. 53–128.

A. G. Nikitin, O. Kuriksha, “Invariant solutions for equations of axion electrodynamics,” Commun. Nonlinear Sci. Numer. Simulat., 17, No. 12, 4585–4601 (2012), https://doi.org/10.1016/j.cnsns.2012.04.009

P. J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York (1986).

V. P. Pingali, “A vector bundle version of the Monge–Ampère equation,” Adv. Math., 360, Art. 106921, 40 p. (2020), https://doi.org/10.1016/j.aim.2019.106921

M. Sroka, “The C0 estimate for the quaternionic Calabi conjecture,” Adv. Math., 370, Art. 107237, 15 p. (2020), https://doi.org/10.1016/j.aim.2020.107237

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

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

E. Witten, “Superstring perturbation theory via super Riemann surfaces: an overview,” Pure Appl. Math. Quart., 15, No. 1, 517–607 (2019), https://doi.org/10.4310/PAMQ.2019.v15.n1.a4

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).


Повний текст: PDF

Посилання

  • Поки немає зовнішніх посилань.


Creative Commons License
Ця робота ліцензована Creative Commons Attribution 3.0 License.