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The isomonodromic deformation method in the theory of Painlevé equations. (English) Zbl 0592.34001
Lecture Notes in Mathematics. 1191. Berlin etc.: Springer-Verlag. IV, 313 p. DM 45.00 (1986).
A linear differential equation system (1) $d\phi /d\lambda =A(\lambda,t)\phi$, where A is a rational matrix in $\lambda$, and t stands for some parameters, is given. One speaks of an isomonodromic deformation of the system (1) if a variation of t does not affect the whole monodromy group of (1). Recently this method arose in the investigation of certain nonlinear differential equations (e.g. the KdV- equation) in theoretical physics. In the main parts of the present text the differential equations $$(2)\quad d\phi /d\lambda =\{-(4i\lambda\sp 2+ix+iu\sp 2)\sigma\sb 3-4u\lambda \sigma\sb 2-2w\sigma\sb 1\}\phi$$ and $$(3)\quad d\phi /d\lambda =\{-i\frac{x\sp 2}{16}\sigma\sb 3- \frac{ixw}{2\lambda}\sigma\sb 1+\frac{i}{\lambda\sp 2}\cos u\sigma\sb 3- \frac{i}{\lambda\sp 2}\sin u\sigma\sb 2\}\phi$$ with the Pauli matrices $\sigma\sb 1=\left( \matrix 0\\ 1\endmatrix \matrix 1\\ 0\endmatrix \right)$, $\sigma\sb 2=\left( \matrix 0\\ i\endmatrix \matrix -i\\ 0\endmatrix \right)$, $\sigma\sb 3=\left( \matrix 1\\ 0\endmatrix \matrix 0\\ -1\endmatrix \right)$ are treated. Equation (2) is now connected via isomonodromic deformation with the second Painlevé equation (PII) $u''(x)=2u\sp 3(x)+xu(x)$ and equation (3) is connected in the same way with (PIII) $u''(x)=\frac{1}{u}(u\sp{'2}-u')+4u\sp 3-4/u,$ the third Painlevé equation. The method then leads to an asymptotic representation of certain solutions of PII, PIII-equation respectively and to an asymptotic representation of the pole-distribution of those solutions. After chapters 0 to 4, mainly concerning information on the general theory, there are seven chapters with the essential results and four chapters with applications and four chapters appendix. In this appendix there are some generalizations of the theory to other Painlevé equations and a synopsis of formulas. The headers of the essential chapters are: V: Asymptotic solution to a direct problem of the monodromy theory for the system (2). VI: Asymptotic solution to a direct problem of the monodromy theory for the system (3). VII: The manifold of solutions of Painlevé II equation decreasing as $x\to -\infty$. Parametrization of their asymptotics through the monodromy data. Ablowitz-Segur connection formulae for real valued solutions decreasing exponentially as $x\to +\infty.$ VIII: The manifold of solutions of Painlevé III equation. The connection formulae for the asymptotics of real valued solutions of the Cauchy problem. IX: The manifold of solutions to Painlevé II equation increasing as $x\to \infty$. The expression of their asymptotics through the monodromy data. The connection formulae for pure imaginary solutions. X: The movable poles of real value solutions to Painlevé II equation and the eigenfunctions of the anharmonic oscillator. XI: The movable poles of the solution to Painlevé III equation and their connection with Mathieu functions. Aside from some misprints the text is very well written and there is a list of 71 references and a subject index at the end of the paper.
Reviewer: G.Jank

##### MSC:
 34-02 Research monographs (ordinary differential equations) 34E05 Asymptotic expansions (ODE) 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE) 35Q99 PDE of mathematical physics and other areas
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