Lie symmetries and form-preserving transformations of reaction–diffusion–convection equations

Abstract

New Lie symmetry classification of the known class of reaction-diffusion-convection equations is presented. The classification method is based on combining the standard group classification method and the form-preserving transformation approach.

Lie Symmetry of the Diffusive Lotka–Volterra System with Time-Dependent Coefficients

“…To essentially reduce the number of inequivalent systems, we use the algorithm based on so called form-preserving (admissible) transformations [21][22][23] (local substitutions, which can map some systems from a given class to other those belonging to the same class). During recent years, the application of admissible transformations to the Lie symmetry classification problems becomes more common because it enables one to decrease substantially the number of obtained cases [24][25][26][27] (see also an extensive discussion on this matter in the very recent monograph [28]). Here, this approach will be essentially used because the Lie-Ovsiannikov approach leads to many locally-equivalent systems of the form (5).…”

“…(27) and (28) Since c 2 1 + b 2 2 = 0 (otherwise the DLV system (5) contains two independent equations, which are excluded from consideration) and taking into account (26), one arrives at…”

Lie Symmetry Classification of the Generalized Nonlinear Beam Equation

“…Ovsiannikov's method (also referred as the Lie-Ovsiannikov method) of Lie symmetry classification of differential equations [21] is based on the classical Lie scheme and a set of equivalence transformations of a given equation [44]. The formal application of this method to equations containing several arbitrary functions (Equation (1) contains three arbitrary functions) usually leads to a large number of equations admitting nontrivial Lie algebras of invariance [44].…”

“…The formal application of this method to equations containing several arbitrary functions (Equation (1) contains three arbitrary functions) usually leads to a large number of equations admitting nontrivial Lie algebras of invariance [44].…”

“…Recent years, this method has been extended by many authors, in which they proposed a numbers of novel techniques, such as algebraic methods based on subgroup analysis of the equivalence group [45][46][47][48] and their generalizations [49][50][51][52][53][54], local transformations and form-preserving transformations [44,[55][56][57][58], to solve group classification problem for numerous nonlinear partial differential equations. In this paper we extend the classical Lie-Ovsiannikov method based on equivalence transformations to the generalized nonlinear beam equation.…”