Second-Order Differential Invariants for Some Extensions of the Poincaré Group and Invariant Equations

Abstract

It is well-known that symmetry properties are extremely important for choosing differential equations which can be suitable for description of real physical processes.We present functional bases of second-order differential invariants for various representations of some extensions of the Poincaré group and for a set of m scalar functions (e.g., for similarity and conformal groups). These results enable us to describe new classes of nonlinear multi-dimensional invariant equations and to simplify the problem of symmetry classification of second-order scalar partial differential equations.

Extensions of realisations for low-dimensional Lie algebras

“…Many extensions were studied for many famous algebras of the mathematical physics without limitations of linearity and without any relation to finding RDIs (see e.g. [14] for the Poincaré algebra P(1, 2), and [11] for P(1, 1)). These nonlinear realisations were used to find their differential invariants and whence new equations invariant under these algebras.…”

“…Operators (14) with ε = 0 give from its functional basis of ADIs R −1 exp t, R −1 exp x a set of RDIs exp t, exp x.…”

Invariants for sets of vectors and rank 2 tensors, and differential invariants for vector functions

“…Description of invariants for nonlinear representations of the rotation group, see e.g. [13] These sets of invariants can be useful tools in other problems in studying of partial differential equations.…”

The Tresse theorem and differential invariants for the nonlinear Schrödinger equation

“…Differential invariants and their applications for other important equations in physics, e.g. KdV, KP and Monge-Ampère equations, have been investigated by Olver, Chen, Pohjanpelto, Yehorchenko and Nutku, Sheftel in [5][6][7][8]. The differential invariants and the so-called semiinvariants of the generalized Schrödinger equation by using equivalence transformations have been described by Senthilvelan, Torrisi and Valenti in [9].…”