On the critical behavior, the connection problem and the elliptic representation of a Painlevé VI equation
The sixth Painlevé equation with particular coefficients \[ \begin{multlined} \frac{d^2y}{dx^2}=\frac 12 \Bigl[\frac 1y+\frac 1{y-1} +\frac 1{y-x}\Bigr]\Bigl(\frac {dy}{dx}\Bigr)^2 -\Bigl[\frac 1x +\frac 1{x-1}+\frac 1{y-x}\Bigr] \frac{dy}{dx} \\ +\frac {y(y-1)(y-x)}{2x^2(x-1)^2}\Bigl[(2\mu-1)^2+\frac {x(x-1)}{(y-x)^2}\Bigr], \qquad \mu\in\mathbb{C},\end{multlined} \tag{PVI\(_{\mu}\)} \] is equivalent to the WDVV equation of associativity in two-dimensional topological field theory. The solutions to \(\text{PVI}_{\mu}\) are parametrized by a triple of complex numbers \((x_0,x_1,x_{\infty})\) such that \(x_0^2+x_1^2+x_{\infty}^2-x_0x_1x_{\infty}=4\sin ^2 (\pi\mu).\) By starting from (PVI\(_{\mu}\)) with \(\mu=-1,\) \(x_0=x_1=x_{\infty}=3,\) it is possible to compute Gromov-Witten invariants for the quantum cohomology of the two-dimensional projective space. Here, under the condition \(x_i\not=\pm 2\), \(i=0,1,\infty,\) the author gives an asymptotic expression for solutions near the critical point \(x=0;\) from which behaviors near \(x=1\) and \(x=\infty\) can be seen by the use of a symmetric property of the critical points. He also solves the connection problem, namely he finds a relation among the parameters of each solution near these different critical points. In the proof of these results, the isomonodromic deformation for the Fuchsian system \[ \frac {dY}{dz}=\Bigl[\frac {A_0(x)}{z}+\frac{A_x(x)} {z-x}+\frac{A_1(x)}{z-1}\Bigr]Y, \] \(A_i(x)\) being certain \(2\times 2\)-matrices, plays an essential role. Furthermore, another expression of solutions of the form \[ y(x)=\wp (u(x)/2; \omega_1(x),\omega_2(x))+\frac 13 (1+x) \] is given near \(x=0.\) Here, \(u(x)\) solves a nonlinear second-order differential equation, and \(\omega_j(x)\), \(j=1,2\), are expressible in terms of hypergeometric series. As a special case, this elliptic representation describes oscillatory solutions.
scientific article; zbMATH DE number 1742922