The two-dimensional eikonal equation

Abstract

We study the two-dimensional eikonal equation ψ 2 x + ψ 2 y = 1/v 2 (x, y). We carry out the group analysis of the equation, establish a connection between the group properties and geometric characteristics of the Riemannian space with the metric ds 2 = [dx 2 + dy 2 ]/v 2 (x, y). We select the most important classes of equations and derive some conditions for reducibility of a given equation to an equation of one of those classes. We find a condition for two equations to be equivalent (the theorem of seven invariants). For the equations corresponding to Riemannian spaces of constant curvature, we obtain explicit formulas for the solutions describing the wave front for a point source and also the ray equations.

On reserve and double covering problems for the sets with non-Euclidean metrics

“…Recall that the radius R here means the time of moving from center to the boundary of the circle. It is easy to observe that the results correspond to the theoretical ones obtained by A. Borovskikh [7]; he proved that covering objects are circles with displaced centers for the given metric.…”

On Classification of Symmetry Reductions for the Eikonal Equation

“…The group P(1, 4) is a group of rotations and translations of the five-dimensional Minkowski space M (1,4). This group has many applications in the theoretical and mathematical physics (see, for example, [29,30]).…”

“…The complete group classification of the eikonal equations for a two-and three-dimensional nonhomogeneous medium is carried out in [1][2][3][4]. In those papers the exact solutions for equations under investigation are also obtained.…”

“…The Lie algebra of the group C(1, 4) contains, as subalgebras, the Lie algebras of the following groups: Poincaré group P(1, 4), extended Poincaré group P (1,4), and the optical group Opt(3) [13,14].…”

Complete group classification of a class of nonlinear wave equations

“…In other words, for each γ ∈ Γ one should explicitly describe the subsetS γ of S which is the union of G ∼ -orbits of elements from S γ . Although this step is usually neglected, it may lead to nontrivial results (see, e.g., [9]).…”