Geodesic
Group classification of the eikonal equation for
Sbornik. Mathematics, Tome 195 (2004) no. 4, pp. 479-520 
A. V. Borovskikh. Group classification of the eikonal equation for. Sbornik. Mathematics, Tome 195 (2004) no. 4, pp. 479-520. http://geodesic.mathdoc.fr/item/SM_2004_195_4_a1/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The equation $(\nabla\psi)^2=1/v^2(x,y,z)$, known as the eikonal equation, is studied. This is the characteristic equation for the wave equations in an inhomogeneous medium, which plays a central role in the description of the geometry of the rays and the wave fronts. A full geometric classification of the family of eikonal equations is carried out (an equation is determined by the function $v(x,y,z)$, which has the meaning of the propagation velocity of a perturbation in the medium). In the cases of equations with linear or quadratic velocity function $v(x,y,z)$, explicit solutions – point source eikonals – are presented and the geometry of the rays is completely described.

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