Algorithmic symmetry classification with invariance

Abstract

Symmetry classification for a system of differential equations can be achieved algorithmically by applying a differential reduction and completion algorithm to the infinitesimal determining equations of the system. The branches of the classification should be invariant under the action of the equivalence group. We show that such invariance can be tested algorithmically knowing only the determining equations of the equivalence group. The method relies on computing the prolongation of a group operator reduced modulo these determining equations. The method is implemented in Maple: a novel pivot selection strategy is able to guide the rifsimp command towards more favourable branchings.

Symbolic computation of equivalence transformations and parameter reduction for nonlinear physical models

“…A symbolic software package SymmetryClassification for Maple developed by Huang and Lisle includes a detEqsForEquiv routine which computes rather a restricted class of point equivalence transformations, namely, ones where components for each variable or arbitrary element depend only on that specific variable or arbitrary element. A reduced simplified version of the rifsimp-generated symmetry classification tree is consequently obtained [9,21]. The work contains detailed examples of symmetry classification for a family of nonlinear heat equations, and a family of nonlinear convection-diffusion (Richards) equations.…”

“…In particular, equivalence transformations may be used to invertibly map any PDE within (2.19) into one with A r = 1 and A 1 = 0. Moreover, in [28], it is shown that when A r = 1 and A 1 = 0, the family (2.19) admits the equivalence transformations with the transformed variables given by 21) and the transformations for A k , B, C are given by complex formulas found in [28]. Importantly, the transformation of the dependent variable u(x, t) (2.21) involves the arbitrary element C(x, t).…”

Symbolic Computation of Local Symmetries of Nonlinear and Linear Partial and Ordinary Differential Equations

“…An excellent attempt to correct this minor deficiency using equivalence transformations was made in [26].…”

Similarity: generalizations, applications and open problems

“…In [89], symmetry classification for a system of differential equations is achieved algorithmically by applying a differential reduction and completion (DRC) algorithm (cf. [63]) to the linear infinitesimal determining equations of the system.…”