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Applications of CRACK in the Classification of Integrable Systems

Please always quote using this URN: urn:nbn:de:0297-zib-7159
  • 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.

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Metadaten
Author:Thomas Wolf
Document Type:ZIB-Report
Tag:Bi-linear algebraic systems; Computer algebra; Integrability; conservation laws; evolution equations; quadratic Hamiltonians
MSC-Classification:08-XX GENERAL ALGEBRAIC SYSTEMS / 08-04 Explicit machine computation and programs (not the theory of computation or programming)
34-XX ORDINARY DIFFERENTIAL EQUATIONS / 34Cxx Qualitative theory [See also 37-XX] / 34C14 Symmetries, invariants
35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Axx General topics / 35A30 Geometric theory, characteristics, transformations [See also 58J70, 58J72]
58-XX GLOBAL ANALYSIS, ANALYSIS ON MANIFOLDS [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx](For geometric integration theory, see 49Q15) / 58Jxx Partial differential equations on manifolds; differential operators [See also 32Wxx, 35-XX, 53Cxx] / 58J65 Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]
Date of first Publication:2002/12/17
Series (Serial Number):ZIB-Report (02-48)
ZIB-Reportnumber:02-48
Published in:Appeared in: CRM Proceedings and Lecture Notes, vol. 37 (2004) 283-300
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