Investigating DEs with CRACK and related programs

Abstract

The aim of the package to be described is to try modularizing the investigation of differential equations for which there are no complete algorithms available yet. All, that is available for such problems are algorithms for special situations, e.g. when first integrals with a simple structure exist (e.g. polynomial in first derivatives) or when the problem has infinitesimal symmetries. In all such cases, finally a system of differential equations has to be solved which is overdetermined in the sense that more conditions have to be satisfied than there are unknown functions. To do a variety of such investigations efficiently, like a symmetry analysis, application of symmetries, determination of first integrals, differential factors, equivalent Lagrangians, the strategy is to have one package (CRACK) for simplifying DEs and solving simple DEs as effective as possible and to use this program as the main tool for all the above mentioned investigations. For each investigation there is then only a short program necessary to just formulate the necessary conditions in form of an overdetermined DE-system and to call CRACK to solve this, possibly in a number of successive calls. The examples below shall indicate the range of possible applications.