Affine Weyl group symmetries in Painlevé type equations (Q2744100)

This is an overview on the unified theory of the symmetries of Painlevé-type equations originated by the authors.NEWLINENEWLINE They begin with the typical differential system NEWLINE\[NEWLINEf_i'= f_i(f_{i+1}- f_{1+2})+ \alpha_i,\quad i\in\mathbb{Z}/3\mathbb{Z},NEWLINE\]NEWLINE possessing the symmetry of the affine Weyl group of type \(A^{(1)}_2\). The system is equivalent to the fourth Painlevé equation \(P_{\text{IV}}\) under some normalization and is called the symmetric form of \(P_{\text{IV}}\).NEWLINENEWLINE First, the symmetries of the system are discussed in detail, as a prototype of the theory. The Bäcklund transformations are described by means of the Cartan matrix and the orientation matrix of type \(A^{(1)}_2\). As for the special solutions, there is a 1-parameter family of classical solutions along each reflection line and there is a rational solution at the barycenter. The other solutions than these special ones are nonclassical in the sense of Umemura. Notice that these two kinds of solutions are obtained from certain seed solutions by the application of Bäcklund transformations. The system can also be written as a Hamiltonian system with some polynomial Hamiltonian. Moreover, one can introduce \(\tau\)-functions to which the Bäcklund transformations are lifted. The \(\tau\)-functions satisfy certain Hirota bilinear equations. From this fact, the fourth Painlevé equation \(P_{\text{IV}}\) can be regarded as a similarity reduction of the 3-reduced modified KP hierarchy. In addition to it, the system with affine Weyl group of type \(A^{(1)}_3\) is introduced, which is equivalent to the fifth Painlevé equation \(P_{\text{V}}\).NEWLINENEWLINE The structure of Bäcklund transformations can be formulated in the general framework of root systems. For an aribitrary generalized Cartan matrix \(A\) and the associated orientation matrix, the Weyl group \(W= W(A)\) is realized as a group of automorphisms of the field of rational functions of constants \(\alpha_i\)'s, \(f\)-variables and \(\tau\)-variables. Then, one can introduce a series of differential systems in the variables \(f_i\), \(i=0,1,\dots,\ell\), with the affine Weyl group symmetry of type \(A^{(1)}_\ell\). The forms of the systems are different according to the parity of \(\ell\). Bäcklund transformations, Poisson structures and Hamiltonian systems, \(\tau\)-functions, Hirota bilinear equations and Lax formalism are discussed for these differential systems.NEWLINENEWLINEFor the entire collection see [Zbl 0969.00055].