On Symmetry Reduction of the (1 + 3)-Dimensional Inhomogeneous Monge-Ampère Equation to the First-Order ODEs

DOI: 10.4236/am.2020.1111080   PDF   HTML     74 Downloads   195 Views  

Abstract

We present the results obtained concerning the classification of symmetry reduction of the (1 + 3)-dimensional inhomogeneous Monge-Ampère equation to first-order ODEs. Some classes of the invariant solutions are constructed.

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Fedorchuk, V. and Fedorchuk, V. (2020) On Symmetry Reduction of the (1 + 3)-Dimensional Inhomogeneous Monge-Ampère Equation to the First-Order ODEs. Applied Mathematics, 11, 1178-1195. doi: 10.4236/am.2020.1111080.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References

[1] Lie, S. (1895) Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebiger Ordnung. Berichte Sächs. Ges., 2, 53-128.
[2] Ovsiannikov, L.V. (1978) Group Analysis of Differential Equations. In: Mathematics in Science and Engineering, Nauka, Moscow. (In Russian) (English translation, Academic Press, New York, 1982)
[3] Olver, P.J. (1986) Applications of Lie Groups to Differential Equations. Springer-Verlag, New York.
https://doi.org/10.1007/978-1-4684-0274-2
[4] Oliveri, F. (2010) Lie Symmetries of Differential Equations: Classical Results and Recent Contributions. Symmetry, 2, 658-706.
https://doi.org/10.3390/sym2020658
[5] Patera, J., Winternitz, P. and Zassenhaus, H. (1975) Continuous Sub-Groups of the Fundamental Groups of Physics. I. General Method and the Poincaré Group. Journal of Mathematical Physics, 16, 1597-1614.
https://doi.org/10.1063/1.522729
[6] Grundland, A.M., Harnad, J. and Winternitz, P. (1984) Symmetry Reduction for Non-Linear Relativistically Invariant Equations. Journal of Mathematical Physics, 25, 791-806.
https://doi.org/10.1063/1.526224
[7] Fedorchuk, V.M., Fedorchuk, I.M. and Leibov, O.S. (1991) Reduction of the Born-Infeld, the Monge-Ampère and the Eikonal Equation to Linear Equations. Doklady Akademii Nauk Ukrainy, 11, 24-27. (In Ukrainian)
[8] Fedorchuk, V. (1995) Symmetry Reduction and Exact Solutions of the Euler-Lagrange-Born-Infeld, Multidimensional Monge-Ampere and Eikonal Equations. Journal of Nonlinear Mathematical Physics, 2, 329-333.
https://doi.org/10.2991/jnmp.1995.2.3-4.13
[9] Fedorchuk, V.M. (1996) Symmetry Reduction and Some Exact Solutions of a Nonlinear Five-Dimensional Wave Equation. Ukrains'kyi Matematychnyi Zhurnal, 48, 573-576. (In Ukrainian) (Translation in Ukrainian Mathematical Journal, 48, 636-640)
https://doi.org/10.1007/BF02390625
[10] Nikitin, A.G. and Kuriksha, O. (2012) Invariant Solutions for Equations of Axion Electrodynamics. Communications in Nonlinear Science and Numerical Simulation, 17, 4585-4601.
https://doi.org/10.1016/j.cnsns.2012.04.009
[11] Grundland, A.M. and Hariton, A. (2017) Algebraic Aspects of the Supersymmetric Minimal Surface Equation. Symmetry, 9, 318.
https://doi.org/10.3390/sym9120318
[12] Fedorchuk, V. and Fedorchuk, V. (2016) On Classification of Symmetry Reductions for the Eikonal Equation. Symmetry, 8, 51.
https://doi.org/10.3390/sym8060051
[13] Fedorchuk, V. and Fedorchuk, V. (2017) On Classification of Symmetry Reductions for Partial Differential Equations. Collection of the works dedicated to 80th of anniversary of B.J. Ptashnyk; Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine, Lviv, Ukraine, 241-255.
[14] Fedorchuk, V. and Fedorchuk, V. (2018) Classification of Symmetry Reductions for the Eikonal Equation. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences of Ukraine, Lviv, Ukraine.
[15] Fedorchuk, V.M. and Fedorchuk, V.I. (2019) On Symmetry Reduction of the Euler-Lagrange-Born-Infeld Equation to Linear ODEs. Symmetry and Integrability of Equations of Mathematical Physics, 16, 193-202.
[16] Fedorchuk, V.M. and Fedorchuk, V.I. (2019) On the Classification of Symmetry Reduction and Invariant Solutions for the Euler-Lagrange-Born-Infeld Equation. Ukrainian Journal of Physics, 64, 1103-1107.
https://doi.org/10.15407/ujpe64.12.1103
[17] Pogorelov, A.V. (1975) The Multidimensional Minkowski Problem. Nauka, Moscow. (In Russian)
[18] Cheng, S.Y. (1984) On the Real and Complex Monge-Ampère Equation and Its Geometric Applications. Proceedings of the International Congress of Mathematicians, 1-2, 533-539.
[19] Pogorelov, A.V. (1988) The Multidimensional Monge-Ampère Equation det||zij|| = φ(z1, ..., zn, z, x1, ..., xn). Nauka, Moscow. (In Russian)
[20] Khabirov, S.V. (1990) Application of Contact Transformations of the Inhomogeneous Monge-Ampère Equation in One-Dimensional Gas Dynamics. Doklady Akademii Nauk SSSR, 310, 333-336. (In Russian) (Translation in Soviet Physics-Doklady, 1990, 35, 29-30)
[21] Udrişte, C. and Bǐlǎ, N. (1999) Symmetry Group of Ţiţeica Surfaces PDE. Balkan Journal of Geometry and Its Applications, 4, 123-140.
[22] Cullen, M.J.P. and Douglas, R.J. (1999) Applications of the Monge-Ampère Equation and Monge Transport Problem to Meteorology and Oceanography. Monge Ampère Equation: Applications to Geometry and Optimization (Deerfield Beach, FL, 1997). Contemporary Mathematics, 226, 33-53.
https://doi.org/10.1090/conm/226/03234
[23] Wang, X.-J. (2017) Monge-Ampère Equation and Optimal Transportation. Proceedings of the Sixth International Congress of Chinese Mathematicians, 36, 153-172
[24] Figalli, A. (2017) The Monge-Ampère Equation and Its Applications. In: Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, Switzerland.
https://doi.org/10.4171/170
[25] Lewicka, M. and Mahadevan, L. and Pakzad, M.R. (2017) The Monge-Ampère Constraint: Matching of Isometries, Density and Regularity, and Elastic Theories of Shallow Shells. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 34, 45-67.
https://doi.org/10.1016/j.anihpc.2015.08.005
[26] Jiang, F. and Trudinger, N.S. (2018) On the Second Boundary Value Problem for Monge-Ampère Type Equations and Geometric Optics. Archive for Rational Mechanics and Analysis, 229, 547-567.
https://doi.org/10.1007/s00205-018-1222-8
[27] Kushner, A., Lychagin, V.V. and Slovák, J. (2019) Lectures on Geometry of Monge-Ampère Equations with Maple. In: Kycia, R.A., Uan, M. and Schneider, E., Eds., Nonlinear PDEs, Their Geometry, and Applications, Springer, Cham, 53-94.
https://doi.org/10.1007/978-3-030-17031-8_2
[28] Yau, S.-T. and Nadis, S. (2019) The Shape of a Life. One Mathematician’s Search for the Universe’s Hidden Geometry. Yale University Press, New Haven, CT.
https://doi.org/10.2307/j.ctvbnm3qt
[29] Le, N.Q. (2020) Global Hölder Estimates for 2D Linearized Monge-Ampère Equations with Right-Hand Side in Divergence Form. Journal of Mathematical Analysis and Applications, 485, Article ID: 123865.
https://doi.org/10.1016/j.jmaa.2020.123865
[30] Jia, X.B., Li, D.S. and Li, Z.S. (2020) Asymptotic Behavior at Infinity of Solutions of Monge-Ampère Equations in Half Spaces. Journal of Differential Equations, 269, 326-348.
https://doi.org/10.1016/j.jde.2019.12.007
[31] Stȩpień, Ł.T. (2020) On Some Exact Solutions of Heavenly Equations in Four Dimensions. AIP Advances, 10, Article ID: 065105.
https://doi.org/10.1063/1.5144327
[32] Sroka, M. (2020) The C0 Estimate for the Quaternionic Calabi Conjecture. Advances in Mathematics, 370, Article ID: 107237.
https://doi.org/10.1016/j.aim.2020.107237
[33] Li, D.S., Li, Z.S. and Yuan, Y. (2020) A Bernstein Problem for Special Lagrangian Equations in Exterior Domains. Advances in Mathematics, 361, Article ID: 106927.
https://doi.org/10.1016/j.aim.2019.106927
[34] Jordan, J. and Streets, J. (2020) On a Calabi-Type Estimate for Pluriclosed Flow. Advances in Mathematics, 366, Article ID: 107097.
https://doi.org/10.1016/j.aim.2020.107097
[35] Fushchich, W.I. and Nikitin, A.G. (1980) Reduction of the Representations of the Generalized Poincaré Algebra by the Galilei Algebra. Journal of Physics A: Mathematical and General, 13, 2319-2330.
https://doi.org/10.1088/0305-4470/13/7/015
[36] Fushchich, V.I. and Serov, N.I. (1983) Symmetry and Some Exact Solutions of the Multidimensional Monge-Ampère Equation. Doklady Akademii Nauk SSSR, 273, 543-546. (In Russian)
[37] Fedorchuk, V.M. (1979) Splitting Subalgebras of the Lie Algebra of the Generalized Poincaré Group P(1,4). Ukrains'kyi Matematychnyi Zhurnal, 31, 717-722. (In Russian). (Translation in Ukrainian Mathematical Journal, 1979, 31, 554-558)
https://doi.org/10.1007/BF01092537
[38] Fedorchuk, V.M. (1981) Nonsplitting Subalgebras of the Lie Algebra of the Generalized Poincaré Group P(1,4). Ukrains'kyi Matematychnyi Zhurnal, 33, 696-700. (In Russian) (Translation in Ukrainian Mathematical Journal, 1981, 33, 535-538)
https://doi.org/10.1007/BF01085898
[39] Fushchich, W.I., Barannik, A.F., Barannik, L.F. and Fedorchuk, V.M. (1985) Continuous Sub-Groups of the Poincaré Group P(1,4). Journal of Physics A: Mathematical and Genera, 18, 2893-2899.
https://doi.org/10.1088/0305-4470/18/15/017
[40] Fedorchuk, V.M. and Fedorchuk, V.I. (2006) On Classification of the Low-Dimensional Non-Conjugate Subalgebras of the Lie Algebra of the Poincaré Group P(1,4). Proceedings of Institute of Mathematics of NAS of Ukraine, 3, 302-308. (In Ukrainian)

  
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