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Painlevé equations through symmetry. Translated from the 2000 Japanese original by the author. (English) Zbl 1077.34003
Translations of Mathematical Monographs 223. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3221-2/hbk). x, 156 p. \$ 69.00 (2004).

The Painlevé equations have attracted a lot of attention not only in mathematics but also in application to physics and engineering. This book is based on the recent extensive study of the author developed in collaboration with Y. Yamada concerning Painlevé equations and related discrete integrable systems. The main theme is the symmetry of the Painlevé equations, in particular, the second and the fourth Painlevé equations

$\begin{array}{cc}& {y}^{\text{'}\text{'}}=2{y}^{3}+ty+\alpha ,\hfill \\ & {y}^{\text{'}\text{'}}=\frac{{\left({y}^{\text{'}}\right)}^{2}}{2y}+\frac{3}{2}{y}^{3}+4t{y}^{2}+2\left({t}^{2}-\alpha \right)y+\frac{\beta }{y},\hfill \end{array}$

where $\alpha$ and $\beta$ are complex parameters. Using the symmetric property, the author describes the structure of groups of Bäcklund transformations in a natural and transparent manner, and systematically investigates their $\tau$ functions and rational (or classical special) solutions.

The first chapter is devoted to introductory explanations of Bäcklund transformations and related topics for P${}_{\text{II}}$ written as the Hamiltonian system

${q}^{\text{'}}=\frac{\partial H}{\partial p},\phantom{\rule{2.em}{0ex}}{p}^{\text{'}}=-\frac{\partial H}{\partial q},\phantom{\rule{2.em}{0ex}}H=\frac{1}{2}p\left(p-2{q}^{2}-t\right)-bq·\phantom{\rule{2.em}{0ex}}\left(1\right)$

As an application, classical special solutions are discussed.

Chapter 2 begins with the symmetric form of P${}_{\text{IV}}$

$\begin{array}{cc}& \left\{\begin{array}{c}{f}_{0}^{\text{'}}={f}_{0}\left({f}_{1}-{f}_{2}\right)+{\alpha }_{0},\hfill \\ {f}_{1}^{\text{'}}={f}_{1}\left({f}_{2}-{f}_{0}\right)+{\alpha }_{1},\hfill \\ {f}_{2}^{\text{'}}={f}_{2}\left({f}_{0}-{f}_{1}\right)+{\alpha }_{2},\hfill \end{array}\right\\hfill \\ & {\alpha }_{0}+{\alpha }_{1}+{\alpha }_{2}=1,\phantom{\rule{1.em}{0ex}}{f}_{0}+{f}_{1}+{f}_{2}=t·\hfill \end{array}$

It is shown that the extended affine Weyl group of type ${A}_{2}^{\left(1\right)}$ acts on the unknown variables ${f}_{0},{f}_{1},{f}_{2}$ and the parameters ${\alpha }_{0},{\alpha }_{1},{\alpha }_{2}$ of (2) as a group of Bäcklund transformations. The corresponding Cartan matrix and Dynkin diagram are also discussed. A similar treatment is possible for P${}_{\text{II}}$; it is shown that the extended affine Weyl group of type ${A}_{1}^{\left(1\right)}$ acts on a symmetric form of P${}_{\text{II}}·$

Chapter 3 deals with $\tau$-functions for P${}_{\text{II}}$ and P${}_{\text{IV}}$. For the symmetric forms of P${}_{\text{II}}$ and P${}_{\text{IV}}$, their suitable $\tau$-functions are defined. The $\tau$-functions satisfy bilinear differential equations of Hirota type. The extended affine Weyl groups stated above act also on these $\tau$-functions, yielding Bäcklund transformations for them.

Chapter 4 is devoted to the formulation of $\tau$-functions on a lattice. For a $\tau$-function of P${}_{\text{II}}$, the iteration of a Bäcklund transformation generates a sequence of the form $\left\{{\tau }_{n}={T}^{n}\left({\tau }_{0}\right)\right\}$, $n\in ℤ·$ This sequence satisfies the Toda equation; and it defines the $\phi$-factors, which are polynomials in unknown variables and parameters of the symmetric system of P${}_{\text{II}}·$ Substitution of a seed polynomial solution into $\phi$-factors yields Yablonski-Vorobiev polynomials appearing in the expressions of rational solutions of P${}_{\text{II}}$. For $\tau$-functions of P${}_{\text{IV}}$, the iteration of Bäcklund transformations generates a family of $\tau$-functions on the triangular lattice. They satisfy the Toda equations in three directions, and define the $\phi$-factors. In this case, specialisation of $\phi$-factors yields Okamoto polynomials associated with rational solutions of P${}_{\text{IV}}·$ A pair of iterative sequences for ${f}_{0}$ and ${f}_{1}$ (of (2)) satisfies a discrete Painlevé equation $d{\text{P}}_{\text{II}}$. Furthermore, the action of the extended affine Weyl group of type ${A}_{n-1}^{\left(1\right)}$ is formulated, which yields a discrete system of type ${A}_{n-1}^{\left(1\right)}$ corresponding to Bäcklund transformations for a certain system of nonlinear differential equations. In this framework, the associated family of $\tau$-functions is also formulated on a lattice of type ${A}_{n-1}^{\left(1\right)}$.

The $\phi$-factors defined by the $\tau$-functions above are polynomials in the variables ${f}_{j}$ and ${\alpha }_{j}$ of the discrete system of type ${A}_{n-1}^{\left(1\right)}$. Chapter 5 gives a determinant expression of the $\phi$-factors called a formula of Jacobi-Trudi type. It is shown that suitably specialised $\phi$-factors, containing Yablonski-Vorobiev polynomials and Okamoto polynomials, are represented in terms of Schur functions.

Chapters 6 and 7 are devoted to the proof of the Jacobi-Trudi formula above. By the use of Gauss decomposition of matrices, the author constructs birational transformations for the case of the finite root system of type ${A}_{n-1}$, which act on $\tau$-functions yielding their $\phi$-factors with the Jacobi-Trudi formula of type ${A}_{n-1}·$ These transformations, $\tau$-functions and $\phi$-factors can be extended to those of type ${A}_{\infty }$ with the indexing set $ℤ·$ Folding it to $ℤ/nℤ$ yields $\phi$-factors of type ${A}_{n-1}^{\left(1\right)}$ admitting the Jacobi-Trudi formula of this type.

In the final chapter, the relation between these birational transformations and Lax formalism is discussed. It is shown that they are interpreted as the compatibility condition for a Lax representation. For P${}_{\text{II}},$ P${}_{\text{IV}}$ and a certain system of nonlinear partial differential equations, the corresponding Lax representations are illustrated.

In the appendix, several related topics are collected, including the history of Painlevé equations.

Bäcklund transformations and $\tau$-functions are basic and important in the study of the structure of Painlevé equations and integrable systems. This book provides a new perspective on these materials, and is recommended to those who are interested in this field.

##### MSC:
 34-02 Research monographs (ordinary differential equations) 34M55 Painlevé and other special equations; classification, hierarchies 37K35 Lie-Bäcklund and other transformations 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies 14E05 Rational and birational maps 20F55 Reflection groups; Coxeter groups 34C14 Symmetries, invariants (ODE)
##### Keywords:
Painlevé equations