On some invariant solutions for the Euler–Lagrange–Born–Infeld equation

Abstract:
Let us consider the Euler–Lagrange–Born–Infeld equation of the form $$ \Box u\left(1-u_{\nu}u^{\nu}\right)+u^{\mu}u^{\nu}u_{\mu\nu}=0, $$ where $u=u(x), x=(x_0, x_1, x_2, x_3)\in M(1,3),$ $u_{\mu}\equiv\dfrac{\partial u}{\partial x^{\mu}},$ $u_{\mu\nu}\equiv\dfrac{\partial^2 u}{\partial x^{\mu}\partial x^{\nu}},$ $u^{\mu}=g^{\mu\nu}u_{\nu}, g_{\mu\nu}=(1,-1,-1,-1)\delta_{\mu\nu},$ $\mu,\nu=0,1,2,3,$ $\Box$ is the d'Alembert operator.

A connection between structural properties of low-dimensional ($\dim L \le3$) nonconjugate subalgebras of the Lie argebra of the generalized Poincaré group $P(1,4)$ and results of symmetry reduction for the Euler–Lagrange–Born–Infeld equation is studied.

In my talk I plan to present some invariant solutions for the Euler–Lagrange–Born–Infeld equation.

References:
[1] Born M., On the quantum theory of electromagnetic field, Proc. R. Soc. Lond. Ser. A, 143 (1934) 410–437.
[2] Born M., Infeld L., Foundations of the new field theory, Proc. R. Soc. Lond. Ser. A, 144 (1934) 425–451.
[3] Lie S., Zur Allgemeinen Theorie der Partiellen Differentialgleichungen Beliebiger Ordnung, Leipz. Berichte, I. 53. (Reprinted in Lie, S., Gesammelte Abhandlungen, Vol. 4, Paper IX.), 1895.
[4] Ovsiannikov L.V., Group Analysis of Differential Equations, Academic Press, New York, 1982.
[5] Olver P.J., Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1986.
[6] Fushchich W.I., Barannik L.F. and Barannik A.F., Subgroup Analysis of Galilei and Poincare Groups and Reduction of Nonlinear Equations, Naukova Dumka, Kiev, 1991.
[7] Fushchych W.I., Shtelen W.M. and Serov N.I., Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical Physics, Kluver Academic Publishers, Dordrecht, 1993.
[8] Grundland A.M., Harnad J. and Winternitz P., Symmetry reduction for nonlinear relativistically invariant equations, J. Math. Phys., 25 (1984) 791–806.
[9] Fedorchuk V.M., Symmetry reduction and exact solutions of the Euler–Lagrange–Born–Infeld, multidimensional Monge–Ampere and Eikonal equations, Symmetry Nonlin. Math. Phys., Vol. 1 (Kiev, 1995). J. Nonlinear Math. Phys., 2 (1995) 329–333.
[10] Fedorchuk V.M., Symmetry reduction and some exact solutions of a nonlinear five-dimensional wave equation (In Ukrainian), Ukrain. Mat. Zh., 48 (1996) 573–576; translation in Ukrainian Math. J., 48 (1997) 636–640.
[11] Nikitin A.G. and Kuriksha O., Group analysis of equations of axion electrodynamics, Group Analysis Diff. Equat. Integrable Sys., 152–163, Department of Mathematics and Statistics, University of Cyprus, Nicosia, 2011.
[12] Nikitin A.G. and Kuriksha O., Invariant solutions for equations of axion electrodynamics, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012) 4585–4601.
[13] Grundland A.M. and Hariton A., Algebraic aspects of the supersymmetric minimal surface equation, Symmetry, 2017, 9(12), 318; doi:10.3390/sym9120318.
[14] Fedorchuk V.M. and Fedorchuk V.I., On classification of the low-dimensional non-conjugate subalgebras of the Lie algebra of the Poincare group $P(1,4)$ (In Ukrainian), Proceedings of Institute of Mathematics of NAS of Ukraine, 3 (2006) 302–308.
[15] Fedorchuk V.M. and Fedorchuk V.I., On classification of symmetry reductions for the Eikonal equation, Symmetry, 2016, 8(6), 51; doi:10.3390/sym8060051.
[16] Fedorchuk V.M. and Fedorchuk V.I., On classification of symmetry reductions for partial differential equations, Collection of the works dedicated to 80th of anniversary of B.J. Ptashnyk, 241–255, Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine, Lviv, 2017.
[17] Fedorchuk V.M. and Fedorchuk V.I., 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, 2018. 176 pp.