On the Stokes geometry of a unified family of \(P_{\mathrm{J}}\)-hierarchies (J=I, II, IV, 34)
The exact WKB analysis has been developed for linear and nonlinear equations with a large parameter including higher-order Painlevé equations, in which the Stokes geometry or the topological graph consisting of Stokes curves, turning points and singular points plays a crucial role [\textit{T. Kawai} and \textit{Y. Takei}, Algebraic analysis of singular perturbation theory. Transl. from the Japanese by Goro Kato. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1100.34004)]. In the \(P_{\mathrm{J}}\)-hierarchy of (higher-order) Painlevé equations of \(P_{\mathrm{J}}\)-type, where \(\mathrm{J}=\text{I, II, IV, 34}\), let \((P_{\mathrm{J}})_m\) \((m=1,2,3,\ldots)\) be each member with unknown functions \(u_j=u_j(t),\) \(v_j=v_j(t)\) \((1\le j\le m)\). Between the Stokes geometry of \((P_{\mathrm{J}})_m\) and that of the associated Lax pair, an intrinsic relationship is known [\textit{T. Kawai} et al., Astrisque 297, 117--166 (2004; Zbl 1086.34072)]. These hierarchies are written in a unified generating function in \(\theta\) of the form \[ \begin{aligned} \eta^{-1} \frac{d}{dt} \begin{pmatrix} \mathcal{U} \theta \\ \mathcal{V} \theta \end{pmatrix} \equiv \begin{pmatrix} f_1 \\ f_2 \end{pmatrix} (1-\mathcal{U}) +\begin{pmatrix} 0 -1 \\ 1 0 \end{pmatrix} \begin{pmatrix} \frac{\partial H} {\partial{\mathcal{U}}} \\ \frac{\partial H}{\partial {\mathcal{V}}} \end{pmatrix} + \begin{pmatrix} 0 \\ \frac{H(\mathcal{U}, \mathcal{V})}{1-\mathcal{U}} \end{pmatrix} \pmod{\theta^{m+2}}\end{aligned}\tag{1} \] [the author, ``General formal solutions for a unified family of PJ-hierarchies (J =I, II, IV, 34)'', J. Math. Soc. Japan 71, 979--1003 (2019)]. Here, (i) \(\eta\) is a large parameter; (ii) \(\mathcal{U}(\theta) = \sum_{k=1}^{m+1} u_k\theta^k,\) \(\mathcal{V}(\theta) = \sum_{k=1}^{m+1} v_k\theta^k\); (iii) \(f_1\), \(f_2\) are certain polynomials in \(\theta,\) \(u_1,\) \(v_1\) and in holomorphic functions \(y_1(t),\) \(y_2(t),\) \(z_1(t),\) \(z_2(t)\); (iv) \(H(\mathcal{U}, \mathcal{V})\) is a quadratic polynomial in \(\mathcal{U},\) \(\mathcal{V}.\) This is called a unified family of \(P_{\mathrm{J}}\)-hierarchies \((\mathrm{J}=\text{I, II, IV, 34})\). In this paper, under certain conditions, a Lax pair of the form \[ (\gamma \theta^{m+l} \partial/\partial\theta -\eta A)\psi(\theta, t)=0, \quad ( \partial/\partial t -\eta B)\psi(\theta, t)=0 \quad (l=2\text{ or }3)\tag{2} \] associated with (1) is given, and, for the Stokes geometry of (1) and (2) along a zero-parameter solution, the same relationship as of the individual members of \(P_{\mathrm{J}}\)-hierarchy is shown: for example, at a turning point \(t=\tau^{\mathrm{I}}\) for (1) of the first kind, a double turning point \(\theta=b_j(t)\) merges with a simple one \(\theta=a_k(t)\) in the Stokes geometry of (2).
scientific article; zbMATH DE number 7036258
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