On Classification of Symmetry Reductions for Partial Differential Equations

Abstract

Symmetry reduction is one of the most powerful tools for the investigation of partial differential equations. In particular, for this purpose, we can use a classical Lie method. To try to explain some of the differences in the properties of the reduced equations, we suggest investigating the relationship between the structural properties of nonconjugate subalgebras of the Lie algebras belonging to the symmetry groups of the equations under consideration and the properties of the reduced equations corresponding to them.In our talk, we plan to present some of the results concerning the relationship between the structural properties of low-dimensional (dim L ≤ 3) nonconjugate subalgebras of the Lie algebra of the generalized Poincaré group P(1,4) and the types of symmetry reduction for some P(1,4)-invariant equations in the space M(1,3) × R(u). More details on this theme can be found in:Fedorchuk V

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).…”

“…To try to explain some of the differences in the properties of the reduced equations for PDEs with nontrivial symmetry groups, we suggested to investigate the relationship between the structural properties of nonconjugate subalgebras of the same rank of the Lie algebras of the symmetry groups of those PDEs and the properties of the reduced equations corresponding with them [13].…”

“…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].…”