On the Classification of Subalgebras of the Galilei Algebras

Abstract

We investigate the structure of certain types of subalgebras of Galilei algebras and the relationship between the conjugacies of these subalgebras under different groups of automorphisms.

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

“…The optimization of the first step of the Lie reduction procedure is standard for the Burgers system (1) and is to construct optimal lists of one-and two-dimensional subalgebras of the maximal Lie invariance algebra g of the system (1), which is the so-called reduced (i.e., centerless) special Galilei algebra with space dimension two. Although subalgebras of this and other Galilei algebras had been classified, e.g., in [4,5,20], this step was not properly implemented in the previous papers on symmetry analysis of the system (1). We re-classified one-and two-dimensional subalgebras of g, additionally taking into account the external automorphisms of g induced by discrete symmetries of (1).…”

“…In the present paper, Hopf-Cole-type transformations are also derived for certain reduced systems of (1). Under the other constraint v = 0, the second equation of the system (1) is satisfied identically, and its first equation reduces to a (1+2)-dimensional generalization of the Burgers equation, u t + uu x − u xx − u yy = 0, (5) which was derived in [40] as an equation for the wave phase of two-dimensional sound simple waves in weakly dissipative flows. Therein, symmetry analysis of this equation was carried out, which included the first exhaustive study of its Lie reductions in an optimized way and the construction of several new families of its exact solutions.…”

“…As a result, we get the following solution family of the (1+2)-dimensional Burgers equation (5) (resp. of the Burgers system (1) if additionally v = 0):…”

“…Therein, symmetry analysis of this equation was carried out, which included the first exhaustive study of its Lie reductions in an optimized way and the construction of several new families of its exact solutions. For the first time, the equation (5) had formally appeared in [16], where its maximal Lie invariance algebra had been computed and its two-step Lie reductions to ODEs had preliminarily been considered. The equation (5) was also singled out in [14] in the course of the group classification of (1+2)-dimensional diffusionconvection equations.…”

“…For the first time, the equation (5) had formally appeared in [16], where its maximal Lie invariance algebra had been computed and its two-step Lie reductions to ODEs had preliminarily been considered. The equation (5) was also singled out in [14] in the course of the group classification of (1+2)-dimensional diffusionconvection equations. Transformation properties of evolution equations and systems have been widely studied because of many practical benefits that such knowledge gives and also because of the variety of physical applications that are modeled by these equations.…”