The Painlevé property for partial differential equations. (English) Zbl 0514.35083

Summary: In this paper we define the Painlevé property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Bäcklund transforms, the linearizing transforms, and the Lax pairs of three well-known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painlevé property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.


35Q53 KdV equations (Korteweg-de Vries equations)
35A30 Geometric theory, characteristics, transformations in context of PDEs
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI


[1] DOI: 10.1007/BF02413316 · JFM 22.0921.02 · doi:10.1007/BF02413316
[2] DOI: 10.1063/1.525389 · Zbl 0492.70019 · doi:10.1063/1.525389
[3] DOI: 10.1103/PhysRevA.25.1257 · doi:10.1103/PhysRevA.25.1257
[4] DOI: 10.1063/1.524491 · Zbl 0445.35056 · doi:10.1063/1.524491
[5] DOI: 10.1002/cpa.3160210503 · Zbl 0162.41103 · doi:10.1002/cpa.3160210503
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