Differential geometric computations and computer algebra. (English) Zbl 0893.53017
The authors discuss the use of computer algebra (CA) in the field of differential geometry (DG) and its applications to geometric structures of partial differential equations. A number of CA packages allow to do DG computations: computations for differential forms and vector fields, such as differentiation, inner products, Lie derivatives, and so on.
Section 2 contains some historical development; the importance of “literate programming style”, i.e., programs are developed as “readable documents”, is pointed out. Section 3 contains geometrical algebraic settings. Due to influences of Russian mathematicians A. M. Vinogradov and I. S. Krasil’shchik, the authors’ point of view towards symmetries, conservation laws, prolongations, and other geometrical objects gradually changed towards a vector field approach, the features of which are described compactly here. Section 5 contains “NOWEB” programming example. Using a “WEB” means that one develops at the same time a document of one’s programming efforts as well as the program itself. “NOWEB” is a “WEB” which is suitable for any CA language. Section 6 contains a rough description of some software CA packages (“Liesurfer”, super vector fields, total derivative operators). In Section 7 an application of the construction of supersymmetric extensions to the Korteweg-de Vries equation is demonstrated.
Section 2 contains some historical development; the importance of “literate programming style”, i.e., programs are developed as “readable documents”, is pointed out. Section 3 contains geometrical algebraic settings. Due to influences of Russian mathematicians A. M. Vinogradov and I. S. Krasil’shchik, the authors’ point of view towards symmetries, conservation laws, prolongations, and other geometrical objects gradually changed towards a vector field approach, the features of which are described compactly here. Section 5 contains “NOWEB” programming example. Using a “WEB” means that one develops at the same time a document of one’s programming efforts as well as the program itself. “NOWEB” is a “WEB” which is suitable for any CA language. Section 6 contains a rough description of some software CA packages (“Liesurfer”, super vector fields, total derivative operators). In Section 7 an application of the construction of supersymmetric extensions to the Korteweg-de Vries equation is demonstrated.
Reviewer: V.Yu.Rovenskij (Krasnoyarsk)
MSC:
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 68W30 | Symbolic computation and algebraic computation |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
Software:
SYMMGRPReferences:
| [1] | Estabrook, F. B.; Wahlquist, H. D., Prolongation structures of nonlinear evolution equations, J. Math. Phys., 16, 1293-1297 (1976) · Zbl 0333.35064 |
| [2] | Harrison, B. K.; Estabrook, F. B., Geometric approach to invariance groups and solutions of partial differential systems, J. Math. Phys., 12, 653-666 (1971) · Zbl 0216.45702 |
| [3] | W. Hereman, Review of symbolic software for Lie symmetry analysis, Mathl. Comput. Modelling; W. Hereman, Review of symbolic software for Lie symmetry analysis, Mathl. Comput. Modelling · Zbl 0898.34002 |
| [4] | Vinogradov, A. M., Local symmetries and conservation laws, Acta Appl. Math., 2, 21-78 (1984) · Zbl 0534.58005 |
| [5] | Krasil’shchik, I. S.; Vinogradov, A. M., Nonlocal symmetries and the theory of coverings, Acta Appl. Math., 2, 79-96 (1984) · Zbl 0547.58043 |
| [6] | Olver, P. J., Applications of Lie groups to differential equations, (Graduate Texts in Mathematics 107 (1986), Springer-Verlag: Springer-Verlag New York) · Zbl 0591.73024 |
| [7] | Khor’hova, N. G., Conservation laws and nonlocal symmetries, Mat. Zametki, 44, 134-144 (1988), (in Russian) · Zbl 0656.58038 |
| [8] | Krasil’chshik, I. S., Some new cohomological invariants for nonlinear differential equations, Diff. Geom. Appl., 2, 307-350 (1992) · Zbl 0765.58028 |
| [9] | Krasil’shchik, I. S.; Kersten, P. H.M., Deformations of differential equations and recursion operators, (Prastaro, A.; Rassias, Th. M., Geometry in Partial Differential Equations (1986), Gordon and Breach: Gordon and Breach New York) · Zbl 0878.35106 |
| [10] | Kersten, P. H.M.; Krasil’chshik, I. S., Graded Frölicher-Nijenhuis brackets and the theory of recursion operators for super differential equations, (Lychagin, V. V., The Interplay between Differential Geometry and Differential Equations. The Interplay between Differential Geometry and Differential Equations, Advances in the Mathematical Sciences, 24 (1995), American Mathematical Society: American Mathematical Society Providence, RI, U.S.A), 91-130 · Zbl 0841.58032 |
| [11] | Kersten, P. H.M., Software to compute infinitesimal symmetries of exterior differential systems, with applications, Acta Appl. Math., 16, 207-229 (1989) · Zbl 0702.35003 |
| [12] | [GR].; [GR]. |
| [13] | Gragert, P. K.H.; Martini, R., Presentation for the Virasoro algebra, Internat. J. Algebra Comput., 5, 251-257 (1995) · Zbl 0838.17026 |
| [14] | Gragert, P. K.H.; Kersten, P. H.M., Graded differential geometry in REDUCE: Super-symmetry structures of the modified KdV equation, Acta Appl. Math., 24, 211-231 (1991) · Zbl 0745.58007 |
| [15] | Roelofs, G. H.M., Prolongation structures of super-symmetric systems, (Ph.D. Thesis (1993), University of Twente: University of Twente Enschede, The Netherlands) · Zbl 0754.35162 |
| [16] | Matthieu, P., Super-symmetric extension of the Korteweg-de Vries equation, J. Math. Phys., 29, 2499-2506 (1988) · Zbl 0665.35076 |
| [17] | Kostant, B., Graded manifolds, graded Lie theory and prequantization, (Proc. Differential Geometrical Methods in Mathematical Physics. Proc. Differential Geometrical Methods in Mathematical Physics, Bonn, 1975. Proc. Differential Geometrical Methods in Mathematical Physics. Proc. Differential Geometrical Methods in Mathematical Physics, Bonn, 1975, Lecture Notes in Mathematics 570 (1975), Springer: Springer Heidelberg), 177-306 · Zbl 0358.53024 |
| [18] | van Bemmelen, Th., Symmetry computations with total derivative operators, (Ph.D. Thesis (1993), University of Twente: University of Twente Enschede, The Netherlands) |
| [19] | van Bemmelen, Th., The symmetry equation package for REDUCE, (Intern. Report No. 1053 (1991), Faculty of Applied Mathematics, University of Twente: Faculty of Applied Mathematics, University of Twente Enschede, The Netherlands) |
| [20] | Roelofs, G. H.M.; Kersten, P. H.M., Super-symmetric extensions of the nonlinear Schrödinger equation: Symmetries and coverings, J. Math. Phys., 33, 2185-2211 (1992) · Zbl 0761.35103 |
| [21] | P.H.M. Kersten and P.K.H. Gragert, \(n\); P.H.M. Kersten and P.K.H. Gragert, \(n\) |
| [22] | Gragert, P. K.H., Symbolic computations in prolongation theory, (Ph.D. Thesis (1981), University of Twente: University of Twente Enschede, The Netherlands) · Zbl 0695.17013 |
| [23] | Roelofs, G. H.M.; Gragert, P. K.H., Implementation of multilinear operators in REDUCE and applications in mathematics, (Proceedings ISSAC ’91. Proceedings ISSAC ’91, Bonn (1991), ACM: ACM New York) · Zbl 0800.93224 |
| [24] | Kersten, P. H.M., Infinitesimal symmetries, a computational approach, (Ph.D. Thesis (1985), University of Twente: University of Twente Enschede, The Netherlands) · Zbl 0648.68052 |
| [25] | Knuth, D. E., Literate programming, Computer Journal, 27, 97-111 (1984) · Zbl 0533.68005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
