## Geometric properties of the Riemann surfaces associated with the Noumi-Yamada systems with a large parameter.(English)Zbl 1242.34154

The Noumi-Yamada systems $$(NY)_l$$, $$l=2,3,4,\dots$$, are systems of $$l+1$$ first order ODEs for $$l+1$$ unknowns which enjoy the affine Weyl group symmetry of type $$A_l^{(1)}$$. These systems can be understood as the generalizations of the classical Painlevé equations PII, PIV and PV. Using appropriate scaling, it is possible to introduce the large parameter $$\eta$$ and to study particular $$0$$-parameter solutions to $$(NY)_l$$ using the so-called exact WKB analysis.
The above mentioned $$0$$-parameter formal series in $$\eta^{-1}$$ is completely specified by the leading order term satisfying a system of algebraic equations. Earlier, the authors proved the existence of a $$0$$-parameter solution in the case of even $$l$$. In the present paper, they treat the cases of even and odd $$l$$ in a unified manner proving that the relevant algebraic system determines a variety which is isomorphic to the variety determined by a hyperbolic system with an appropriate normalization. In this way, the authors find the number of the possible leading order terms (i.e. the number of the branches of a multi-valued holomorphic function satisfying the algebraic system) outside the set of the turning points of the first kind (zeros of the Jacobian of the algebraic system) as well as the number of such turning points.

### MSC:

 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent) 34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations 34M25 Formal solutions and transform techniques for ordinary differential equations in the complex domain

### Keywords:

Noumi-Yamada systems; Painlevé hierarchy; formal solutions
Full Text:

### References:

 [1] T. Aoki and N. Honda, Regular sequence associated with the Noumi-Yamada equations with a large parameter, Algebraic Analysis of Differential Equations, Springer, 2008, pp.,45-53. · Zbl 1162.34072 [2] T. Aoki, N. Honda and Y. Umeta, On the number of the turning points of the second kind of the Noumi-Yamada system, RIMS Kôkyûroku Bessatsu, · Zbl 1315.34100 [3] T. Suwa, Dual class of a subvariety, Tokyo J. Math., 23 (2000), 51-68. · Zbl 0971.32005 [4] Y. Takei, Toward the exact WKB analysis for higher-order Painlevé equations - The case of Noumi-Yamada Systems -, Publ. Res. Inst. Math. Sci., 40 (2004), 709-730. · Zbl 1076.34099 [5] M. Noumi and Y. Yamada, Higher order Painlevé equations of type $$A_l^{(1)}$$, Funkcial Ekvac., 41 (1998), 483-503. · Zbl 1140.34303 [6] M. Noumi and Y. Yamada, Symmetry in Painlevé equations, In: Toward the Exact WKB Analysis of Differential Equations, (eds. C. J. Howls, T. Kawai and Y. Takei), Linear or Non-Linear, Kyoto Univ. Press, 2000, pp.,245-260.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.