The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs

Abstract

We present the Maple package TDDS (Thomas Decomposition of Differential Systems). Given a polynomially nonlinear differential system, which in addition to equations may contain inequations, this package computes a decomposition of it into a finite set of differentially triangular and algebraically simple subsystems whose subsets of equations are involutive. Usually the decomposed system is substantially easier to investigate and solve both analytically and numerically. The distinctive property of a Thomas decomposition is disjointness of the solution sets of the output subsystems. Thereby, a solution of a well-posed initial problem belongs to one and only one output subsystem. The Thomas decomposition is fully algorithmic. It allows to perform important elements of algebraic analysis of an input differential system such as: verifying consistency, i.e., the existence of solutions; detecting the arbitrariness in the general analytic solution; given an additional equation, checking whether this equation is satisfied by all common solutions of the input system; eliminating a part of dependent variables from the system if such elimination is possible; revealing hidden constraints on dependent variables, etc. Examples illustrating the use of the package are given. Program summaryProgram Title: TDDSProgram Files doi:http://dx.doi.org/10.17632/twk8zjxgbz.1Licensing provisions: GNU LGPLProgramming language: MAPLE 11 to MAPLE 2017, available independently in MAPLE 2018Nature of problem: Systems of polynomially nonlinear partial differential equations are not given in a formally integrable form in general. In order to determine analytic solutions in terms of power series, symbolic manipulations are necessary to find a complete set of conditions for the unknown Taylor coefficients. A particular case of that problem is deciding consistency of a system of PDEs. Nonlinear PDEs require splitting into different cases in general. Deciding whether another PDE is a consequence of a given system depends on similar symbolic manipulations. Computing all consequences of a given system which involve only a subset of the unknown functions or a certain subset of their derivatives are instances of differential elimination problems, which arise, e.g., in detection of hidden constraints in singular dynamical systems and field theoretical models.Solution method: The solution method consists, in principle, of pseudo-division of differential polynomials, as in Euclid’s algorithm, with case distinctions according to vanishing or non-vanishing leading coefficients and discriminants, combined with completion to involution for partial differential equations. Since an enormous growth of expressions can be expected in general, efficient versions of these techniques need to be used, e.g., subresultants, Janet division, and need to be applied in an appropriate order. Factorization of polynomials, while not strictly necessary for the method, should be utilized to reduce the size of expressions whenever possible.