Extensions of the MapDE algorithm for mappings relating differential equations

Abstract

This paper is a sequel of our previous work in which we introduced the MapDE algorithm to determine the existence of analytic invertible mappings of an input (source) differential polynomial system (DPS) to a specific target DPS, and sometimes by heuristic integration an explicit form of the mapping. A particular feature was to exploit the Lie symmetry invariance algebra of the source, without integrating its equations, to facilitate MapDE, making algorithmic an approach initiated by Bluman and Kumei. In applications, however, the explicit form of a target DPS is not available, and a more important question is, can the source be mapped to a more tractable class. This aspect was illustrated by giving an algorithm to determine the existence of a mapping of a linear differential equation to the class of constant coefficient linear differential equations, again algorithmically realizing a method of Bluman and Kumei. Key for this application was the exploitation of a commutative sub-algebra of symmetries corresponding to translations of the independent variables in the target. In this paper, we extend MapDE to determine if a source nonlinear DPS can be mapped to a linear differential system. The methods combine aspects of the Bluman-Kumei mapping approach, together with techniques introduced by Lyakhov et al,for the determination of exact linearizations of ODE. The Bluman-Kumei approach which is applied to PDE, focuses on the fact that such linearizable systems must admit an infinite Lie subpseudogroup corresponding to the linear superposition of solutions in the target. In contrast, Lyakhov et al., focus on ODE, and properties of the so-called derived sub-algebra of the (finite) dimensional Lie algebra of symmetries of the ODE. We also illustrate the powerful maximal symmetry groups facility as a natural tool to be used in conjunction with MapDE.