On Classification of Symmetry Reductions for the Eikonal Equation

The eikonal equations in spaces of different dimensions and different types have many applications in the geometric optics, acoustics of inhomogeneous media, theoretical physics, etc. The details on this theme can be found in [1–7] (see also the references therein). Sheng, Guha and Gonzalez (see [8,9] and the references therein) proposed new effective methods for solving many problems of computational optics. The complete group classification of the eikonal equations for a twoand three-dimensional nonhomogeneous medium is carried out in [1–4]. In those papers the exact solutions for equations under investigation are also obtained. In [5], a group classification of generalized eikonal equations was performed. The paper contains a list of all non-equivalent (with respect to equivalence group) equations with symmetry extensions of the kernel. New nonlinear equations of first-order with wide symmetry groups are found. In [6], the method of propagating waves in multidimensional media is studied in order to find cases of integrability in an explicit form of the eikonal equation. Many important results conserning symmetry, reduction, and exact solutions of the eikonal equations can be found in the monograph of Fushchych, Shtelen, and Serov [10]. In this paper, we consider the eikonal equation of the form as follows: ( ∂u ∂x0 )2 − ( ∂u ∂x1 )2 − ( ∂u ∂x2 )2 − ( ∂u ∂x3 )2 = 1 (1)