Introduction of the MapDE Algorithm for Determination of Mappings Relating Differential Equations

Abstract

This paper is the first of a series in which we develop exact and approximate algorithms for mappings of systems of differential equations. Here we introduce the MapDE algorithm and its implementation in Maple, for mappings relating differential equations. We consider the problem of how to algorithmically characterize, and then to compute mappings of less tractable (Source) systems R to more tractable (Target) systems \hatR by exploiting the Lie algebra of vector fields leaving R invariant. Suppose that R is a (Source) system of (partial or ordinary) differential equations with independent variables x = (x^1, x^2, ldots , x^n ) \in \mathbbC ^n and dependent variables u = (u^1, ldots , u^m)\in \mathbbC ^m. Similarly suppose \hatR is a (Target) system in the variables (\hatx, \hatu ) \in \mathbbC ^n+m . For systems of exact differential polynomials \Sys, \Syshat our algorithm MapDE can decide, under certain assumptions, if there exists a local invertible mapping ¶si(x,u) = (\hatx, \hatu ) that maps the Source system \Sys to the Target \hatR . We use a result of Bluman and Kumei who have shown that the mapping ¶si satisfies infinitesimal (linearized) mapping equations that map the infinitesimals of the Lie invariance algebra for R to those for \hatR . MapDE applies a differential-elimination algorithm to the defining systems for infinitesimal symmetries of R, \hatR , and also to the nonlinear mapping equations (including the Bluman-Kumei mapping subsystem); giving them in a form which includes its integrability conditions and for which an existence uniqueness theorem is available. Once existence is established, a second stage can determine features of the map, and some times by integration, explicit forms of the mapping. Examples are given to illustrate the algorithm. Algorithm MapDE also allows users to enter broad target classes instead of a specific system \hatR . For example avoiding the integrations of the Bluman-Kumei approach MapDE can determine if a linear differential equation R can be mapped to a linear constant coefficient differential equation.