Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
1997-10-25
Nonlinear Sciences
Exactly Solvable and Integrable Systems
40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer School on "Nonlinear Systems," Pondicherry, India, Janua
Scientific paper
10.1007/BFb0113696
The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable equations have the Painlev\'e property, that is, all solutions are single-valued around all movable singularities. In this expository article, we review methods for analysing such singularity structure. In particular, we describe well known techniques of nonlinear regular-singular-type analysis, i.e. the Painlev\'e tests for ordinary and partial differential equations. Then we discuss methods of obtaining sufficiency conditions for the Painlev\'e property. Recently, extensions of \textit{irregular} singularity analysis to nonlinear equations have been achieved. Also, new asymptotic limits of differential equations preserving the Painlev\'e property have been found. We discuss these also.
Halburd Rod
Joshi Nalini
Kruskal Martin D.
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