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Group theory and exact solutions of partially integrable differential systems. (English) Zbl 0726.35127

Partially integrable evolution equations in physics, Proc. NATO/ASI, Les Houches/Fr. 1989, NATO ASI Ser., Ser. C 310, 515-567 (1990).
[For the entire collection see Zbl 0703.00017.]
This paper is a nice review of the applications of the Lie point transformation method to differential equations. The author reviews how this method can be used to obtain large classes of exact analytic solutions. He emphasizes the generality of the method by explicitly dealing with non-integrable, nonlinear partial differential equations. Infinite dimensional Lie groups figure prominently in this review. Algorithms for classifying finite dimensional subgroups are also discussed. The interplay between Lie theory and Painlevé analysis is also discussed. The author also emphasizes the role of computer algebra in the explicit determination of the symmetry group of an equation, its Lie algebra, the finding of its subgroups, and in performing the Painlevé analysis. Specific equations are also discussed in detail.
Reviewer: T.Ratiu (Tucson)

MSC:

35Q58 Other completely integrable PDE (MSC2000)
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
35C05 Solutions to PDEs in closed form
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

Citations:

Zbl 0703.00017

Software:

SPLIT; CANONIK