TY - GEN A1 - Wolf, Thomas T1 - Applications of CRACK in the Classification of Integrable Systems N2 - The talk given by the author at the CRM workshop on Superintegrability in Sep.\ 2002 and this related paper report on work in two subjects. One is the collaboration with Vladimir Sokolov and Takayuki Tsuchida in an effort to classify polynomial integrable vector evolution equations. The other is the computer algebra package {\sc Crack} which did the main computations in solving large bi-linear algebraic systems. Although originally designed to solve over-determined systems of partial differential equations a number of extensions made {\sc Crack} a powerful tool for solving systems of bi-linear algebraic equations. Such systems turn up in many different classification problems some of which were investigated by other participants of this workshop. Two additional applications are outlined. In the talk on which this article is based a method to reduce the length of equations was presented which proved to be useful in solving the bi-linear algebraic systems. Due to numerous asked questions about the computer program, a more complete overview is given in the appendix. T3 - ZIB-Report - 02-48 KW - Integrability KW - evolution equations KW - quadratic Hamiltonians KW - Computer algebra KW - Bi-linear algebraic systems KW - conservation laws Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7159 ER - TY - GEN A1 - Anco, Stephen A1 - Bluman, George A1 - Wolf, Thomas T1 - Invertible Mappings of Nonlinear PDEs to Linear PDEs Through Admitted Conservation Laws N2 - An algorithmic method using conservation law multipliers is introduced that yields necessary and sufficient conditions to find invertible mappings of a given nonlinear PDE to some linear PDE and to construct such a mapping when it exists. Previous methods yielded such conditions from admitted point or contact symmetries of the nonlinear PDE. Through examples, these two linearization approaches are contrasted. T3 - ZIB-Report - 08-11 KW - conservation laws KW - linearization KW - symmetries Y1 - 2008 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10652 SN - 1438-0064 ER - TY - GEN A1 - Wolf, Thomas T1 - The integration of systems of linear PDEs using conservation laws of syzygies N2 - The purpose of the paper is to formulate and use syzygies for systems of linear PDEs. The computation of an equivalent of a GCD for linear partial differential operators will save us their factorization which is otherwise only possible algorithmically in special cases. After showing the computation with the new and the traditional method and comparing both in the next three sections, the algorithm is explained in general and an overview is given. T3 - ZIB-Report - 02-08 KW - PDE KW - syzygies KW - conservation laws Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6751 ER -