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Zbl 1153.34002
Conte, Robert M.; Musette, Micheline
The Painlevé handbook. (English)
[B] Dordrecht: Springer. xxiv, 256~p. EUR~79.95/net; SFR~133.00; \$~129.00; \sterling~60.00 (2008). ISBN 978-1-4020-8490-4/hbk

The new monograph of the leading experts in the singularity analysis of differential equations provides a profound introduction to the Painlevé property and related topics on the boundary between integrable and nonintegrable differential and difference models. Explaining their point of view, the authors follow the principal line: first specify equations which pass or partially pass the Painlevé test for conditions which are assumed to be necessary for integrability, and then, using a singular part transformation and its extensions, integrate the obtained equation, or prove its integrability, or at least construct particular solutions to the equation. In short but important Chapter 1, the authors compare two approaches to nonlinear ODEs: a perturbative approach based on the construction of a power series with respect to a formal parameter with a seed solution used as the initial term of the series, and a method based on a singularity analysis and a singular part transformation. For illustrative sample ODEs, both methods successfully produce the general solutions; however the authors explain why the perturbative approach, in general, is less efficient than the singularity analysis. In the remaining section of this chapter, the authors introduce and discuss the principal notion of the book, the Painlevé property (for ODE case) defined as the singlevaluedness of the general solution of the differential equation under consideration. Chapter 2 is devoted to the description of the modern version of the Painlevé test for ODEs, i.e.\ the algorithm aimed to verify that certain necessary for the Painlevé property conditions are satisfied. The method, explained in detail by its application to the Lorenz model, to the traveling wave reductions for Kuramoto-Sivashinsky (KS) and Ginzburg-Landau (GL) equations, the Duffing-van der Pol oscillator and the Hénon-Heiles (HH) system, consists in an attempt to build up a Laurent expansion for a general solution, i.e.\ a Laurent series containing as many arbitrary parameters as the order of the differential model is. The procedure relies on a study of a recursion relation for the Laurent series coefficients; moreover the coefficients with arbitrary values necessarily appear at the positions indicated by the so-called positive Fuchs indices. If some of the Fuchs indices are negative, or their number is not enough to get a Laurent representation of a general solution, which makes the test in the above Gambier-Kowalevski version inconclusive, the authors describe further perturbative procedures based on the Poincaré theorem and resembling in this respect the $\alpha$ test by Painlevé. In Chapter 3, for the parameter values extracted in the previous chapter, the general or particular solutions are constructed. Namely, for the Lorenz model in specified cases, the authors describe the first integrals which allow them to reduce the model to elliptic or Painlevé equations. After clarifying the appearance of the traveling wave solutions in the Korteweg-de Vries (KdV) and nonlinear Scrödinger (NLS) equations, the authors apply the singularity analysis to describe conditions which are necessary for existence of elliptic solutions for the partially integrable cases in the KS and GL traveling wave reductions. In the course of evaluation of the trigonometric traveling wave solutions admitted by the KS and GL traveling wave reductions, the authors introduce the truncation method. The latter relies on a representation of a possible solution of an ODE as a polynomial in one or two variables which in their turn satisfy a Riccati equation or a projective Riccati system. The idea of the truncation method gives rise to an algorithm of finding of all possible doubly periodic solutions for an ODE. This algorithm is illustrated by its application to the traveling wave reductions in KdV, KS and GL equations. In Chapter 4, the authors extend their program to the PDE case. The idea is really challenging since, for PDEs, there exist several definitions of integrability and no commonly accepted definition of the Painlevé property. The authors however succeed to elaborate certain working definitions to the both notions. Namely, a PDE is called integrable if either it can be solved in an explicit closed form, or it can be linearized, or it possesses an auto-Bäcklund transformation, or it can be transformed into another integrable PDE. The Painlevé property implies integrability in the above sense plus the absence of movable critical singularities near any noncharacteristic manifold. A practical Painlevé test for PDEs is aimed to check the latter condition. The algorithm presented in the book follows the steps of that for ODEs and is illustrated by application to the KdV and Kolmogorov-Petrovskii-Piskunov (KPP) equations. In Chapter 5, the information provided by the Painlevé test in the previous chapter is used to derive for integrable and partially integrable PDEs their Lax pairs (in various representations: scalar, matrix, zero-curvature, Sato and projective Riccati), Bäcklund transformations and solutions in closed form. In short, the procedure, called the singular manifold method, starts with the choice of the singular part operator implied by the previously obtained principal term of the Laurent expansion. Next, it introduces the singular part transformation which represents the unknown solution of the PDE as the sum of the singular part operator applied to an auxiliary function and of a regular part of the solution. Then identifying this representation with the previously obtained Laurent expansion and substituting it into the integrable PDE, it is possible to find equations for the introduced auxiliary functions. The equation for the regular part of the introduced representation satisfies an integrable PDE thus yielding an (auto)-Bäcklund transformation, while the system of equations for the auxiliary function in the singular part of the representation yields the corresponding Lax pair. The process of evaluation of the Lax pairs, Bäcklund transformations and nonlinear superposition formulae is illustrated by KdV, Boussinesq, sine-Gordon (SG), modified KdV (mKdV), Sawada-Kotera (SK) and Kaup-Kupershmidt (KK) equations. In the cases of partially integrable PDEs (Fisher and KPP equations), the Painlevé test nevertheless provides some constructive information which can be used to find particular solutions. In the remaining sections of this chapter, the authors, in a traditional way, describe the appearance of the Lax pairs and birational transformations for integrable ODEs, i.e.\ Painlevé equations, as the reductions of the Lax pairs and the Bäcklund transformations for relevant integrable PDEs. They also explain a singularity analysis procedure for evaluation of the birational transformations for the Painlevé equations. Chapter 6 discusses the integrability (in the sense of the explicit separation of variables) of the HH models whose parameter values were selected by the Painlevé test in Chapter 2. Chapter 7 considers discrete nonlinear systems. In spite of absence of any satisfactory definition of the discrete Painlevé property except for the existence of the discrete Lax pair, the authors describe the practical tests: the method of singularity confinement illustrated by its application to a rational map containing the well known Freud equation; the method of polynomial growth illustrated by integrable and chaotic models; and the method of perturbation of continuum limit which is similar to its continuous counterpart. \par Next the authors discuss the methods of finding interesting discrete equations, e.g.\ discretization of the continuous differential equations. Illustrative examples are the discretization of the Ermakov-Pinney equation, addition formulae for the elliptic functions and several known discretizations of the NLS equation. The nonautonomous discrete equations appear as extensions of the autonomous discrete equations, e.g. those obtained from addition formulae for elliptic functions. Finally the authors mention the geometric approach to the discrete Painlevé equations by Sakai based on a classification of rational surfaces. The shortest in the book Chapter 8 is really remarkable and useful for newcomers. It contains answers to several frequently asked questions (FAQ) which clarify the status of the Painlevé test and rebut various misrepresentations. Appendices A--E provide useful information on classical results in Painlevé equation theory including classification, irreducibility, transformation properties, classical solutions, as well as some basics in elliptic functions, Nevanlinna theory and bilinear formalism. In spite of the presentation sometimes seems too technical, and the reading is not always easy, no doubts, ``The Painlevé handbook" gives a new insight and is really useful for anyone interesting in the theory of integrable systems.
[Andrei A. Kapaev (St. Petersburg)]

MSC 2000:: *34-02 Research monographs (ordinary differential equations)
34A25 Analytical theory of ODE
34M05 Entire and meromorphic solutions
35B10 Periodic solutions of PDE
39A13 Difference equations, scaling ($q$-differences)

Keywords: Painlevé property; integrability; Painlevé test; singularity analysis; Lax pair; Bäcklund transformation

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