# Introduction of the MapDE algorithm for determination of mappings relating differential equations

## TL;DR

The paper introduces MapDE, an algorithm implemented in Maple, for determining mappings between differential equations by exploiting Lie algebra symmetries, enabling the decision and construction of such mappings.

## Contribution

It presents the first algorithmic approach for deciding and computing mappings of differential equations using Lie symmetries, including implementation details and example applications.

## Key findings

- MapDE can decide the existence of mappings between differential systems.
- The algorithm can determine features of the mappings, sometimes explicitly.
- Examples demonstrate the effectiveness of the approach.

## 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 ^R by exploiting the Lie algebra of vector fields leaving R invariant. For systems of exact differential polynomials R, ^R our algorithm MapDE can decide, under certain assumptions, if there exists a local invertible mapping Psi(x; u) = (^x; ^u) that maps the Source system R to the Target ^R. We use a result of Bluman and Kumei who have shown that the mapping satisfies infinitesimal (linearized) mapping equations that map the infinitesimals of the Lie invariance algebra for R to those for ^R. MapDE involves applying the differential-elimination algorithm to the defining systems for infinitesimal symmetries of R, ^R, and also to the nonlinear mapping equations (including the Bluman-Kumei mapping subsystem); returning 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 ^R. For example, we give an algorithmic approach that avoids the integrations of the Bluman-Kumei approach where MapDE can determine if a linear differential equation R can be mapped to a linear constant coefficient differential equation.