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Eikonal equation for anisotropic media. (English. Russian original) Zbl 1301.35164

J. Math. Sci., New York 197, No. 2, 248-289 (2014); translation from Tr. Semin. Im. I. G. Petrovskogo 29, No. 1, 162-229 (2013).
Summary: The methods of group analysis are applied to establish a classification of eikonal equations for anisotropic stationary media, \(g^{ij}(x)\psi_{i}\psi_{j} = 1\). The equivalence group and the groups of symmetries are described. The classification is based on the fact that the Riemannian space (with the metric \(ds^{2} = g_{ij}(x) dx^{i}dx^{j})\) associated with the equation has a special structure, namely, that of a semi-homogeneous space: the metric form can be represented as \(ds^2=g_{\hat\imath \hat\jmath}(\hat x)\,dx^{\hat\imath}\,dx^{\hat\jmath}+ G^2(\hat x) g_{\tilde \imath\tilde \jmath}(x)\,dx^{\tilde \imath}\,dx^{\tilde \jmath}\), where the principal part \(g_{\hat\imath \hat\jmath}(\hat x)\,dx^{\hat\imath}\,dx^{\hat\jmath}\) is the metric of a Riemannian space of constant curvature.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
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