On Classification of Symmetry Reductions for the Eikonal Equation

Abstract

Abstract:We study the relationship between the classification of three-dimensional nonconjugate subalgebras of the Lie algebra of the Poincaré group P(1, 4) and the types of symmetry reduction of the eikonal equation to ordinary differential equations (ODEs).

On the Classification of Symmetry Reductions and Invariant Solutions for the Euler–Lagrange–Born–Infeld Equation

“…Grundland, Harnad, and Winternitz [4] were the first who pointed out and investigated the similar phenomenon. The details on this theme can be found in [5][6][7][8][9][10][11][12][13] (see also the references therein).…”

“…To try to explain some differences in the properties of the reduced equations and invariant solutions, which are obtained by using nonconjugate subalgebras of the same ranks of the Lie algebras of the symmetry groups of the PDEs under consideration, we recently suggested to use the structural properties of those nonconjugate subalgebras. The details on this theme can be found in [10,12,13] (see also the references therein).…”

On Symmetry Reduction of the (1 + 3)-Dimensional Inhomogeneous Monge-Ampère Equation to the First-Order ODEs

“…They also investigated the similar phenomenon. Some details on this theme can be found in [7][8][9][10][11][12][13][14] (see, also, the references therein).…”

“…At the present time, the relationship has been studied between the structural properties of the low-dimensional (dimL ≤ 3) nonconjugate subalgebras of the same rank of the Lie algebra of the Poincaré group P (1,4) and the properties of the reduced equations for the Eikonal and Euler-Lagrange-Born-Infeld equations. The details on this theme can be found in [12][13][14][15][16].…”

Symmetry Analysis and Conservation Laws of the Zoomeron Equation

“…These are then used to perform symmetry reductions and determine new group-invariant solutions of (1). It should be noted that such approach was previously used for examination of a wide range of nonlinear PDEs [23][24][25][26][27][28][29][30][31]. Furthermore, we derive the conservation laws of (1) using the multiplier method [32,33].…”