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**Singularity analysis and its relation to complete, partial and non- integrability.**
*(English)*
Zbl 0715.58017

Partially integrable evolution equations in physics, Proc. NATO/ASI, Les Houches/Fr. 1989, NATO ASI Ser., Ser. C 310, 321-372 (1990).

[For the entire collection see Zbl 0703.00017.]

This work contains a very useful survey of the various problems of the integrability of evolution equations. In the abstract the authors say: “The aim of this course is to present and illustrate the connection between integrability and the singularity structure of the solutions of nonlinear dynamical systems. We start by reviewing the various aspects of integrability and then introduce the notions of partial and constrained integrability. Next we present the methodology of the singularity (“Painlevé”) analysis and apply it to the study of various systems. Finally we present a detailed review of the recent progress in the domain of nonintegrability. Based on Ziglin’s theorem, we prove rigorously the nonexistence of integrals for several systems of physical interest. As a non-linear extension of Ziglin’s approach we present the “poly- Painlevé” criterion of (non-) integrability, illustrate it through some example, and propose a practical method for its implementation.” I think that the authors accomplished their aims very successfully.

This work contains a very useful survey of the various problems of the integrability of evolution equations. In the abstract the authors say: “The aim of this course is to present and illustrate the connection between integrability and the singularity structure of the solutions of nonlinear dynamical systems. We start by reviewing the various aspects of integrability and then introduce the notions of partial and constrained integrability. Next we present the methodology of the singularity (“Painlevé”) analysis and apply it to the study of various systems. Finally we present a detailed review of the recent progress in the domain of nonintegrability. Based on Ziglin’s theorem, we prove rigorously the nonexistence of integrals for several systems of physical interest. As a non-linear extension of Ziglin’s approach we present the “poly- Painlevé” criterion of (non-) integrability, illustrate it through some example, and propose a practical method for its implementation.” I think that the authors accomplished their aims very successfully.

Reviewer: A.Klič

### MSC:

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

35J60 | Nonlinear elliptic equations |