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PP-test for integrability of some evolution differential equations. (English) Zbl 0961.37024
Proceedings of the third international conference on symmetry in nonlinear mathematical physics, Kyiv, Ukraine, July 12-18, 1999. Part 2. Transl. from the Ukrainian. Kyiv: Institute of Mathematics of NAS of Ukraine. Proc. Inst. Math. Natl. Acad. Sci. Ukr., Math. Appl. 30(2), 387-391 (2000).
The author discusses the problem of connection between Painlevé transcendent and integrability of nonlinear PDE. This discussion is based on the Ablowitz-Ramani-Segur conjecture that every ODE obtained by similarity reduction from a PDE solvable with the inverse scattering method possesses the Painlevé property, i.e., all its removable singularities are poles. The author gives a survey of corresponding results and suggests a procedure for constructing the Painlevé transcendents. This procedure includes an algorithm of isolation of poles proposed by the author and uses the method of generalized power series developed by P. F. Filtschakow [Numerische und graphische Methoden der angewandten Mathematik. Braunschweig: Friedr. Vieweg (1975; Zbl 0333.65001)].
For the entire collection see [Zbl 0937.00046].
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions