## On WKB analysis of higher order Painlevé equations with a large parameter.(English)Zbl 1067.34088

The authors announce a generalization of their reduction theorem [Adv. Math. 118, 1–33 (1996; Zbl 0848.34005)] for 0-parameter solutions of the traditional Painlevé equations with a large parameter to those of some higher-order Painlevé equations [“On the Stokes geometry of higher order Painlevé equations”, RIMS Preprint Series No. 1443 (2004)].

### MSC:

 34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies 34M40 Stokes phenomena and connection problems (linear and nonlinear) for ordinary differential equations in the complex domain 34M60 Singular perturbation problems for ordinary differential equations in the complex domain (complex WKB, turning points, steepest descent)

### Keywords:

WKB analysis; Painlevé transcendent

Zbl 0848.34005
Full Text:

### References:

 [1] Aoki, T., Kawai, T., Koike, T., and Takei, Y.: On the exact WKB analysis of microdifferential operators of WKB type. RIMS Preprint Series No. 1429 (2003). · Zbl 1079.34070 [2] Aoki, T., Kawai, T., and Takei, Y.: WKB analysis of Painlevé transcendents with a large parameter. II. Structure of Solutions of Differential Equations, World Sci. Publishing, Singapore, pp. 1-49 (1996). · Zbl 0848.34005 [3] Gordoa, P. R., Joshi, N., and Pickering, A.: On a generalized $$2+1$$ dispersive water wave hierarchy. Publ. Res. Inst. Math. Sci., 37 , 327-347 (2001). · Zbl 0997.35094 [4] Gordoa, P. R., and Pickering, A.: Nonisospectral scattering problems: A key to integrable hierarchies. J. Math. Phys., 40 , 5749-5786 (1999). · Zbl 1063.37544 [5] Kawai, T., Koike, T., Nishikawa, Y., and Takei, Y.: On the Stokes geometry of higher order Painlevé equations. RIMS Preprint Series No. 1443 (2004). · Zbl 1086.34072 [6] Kawai, T., and Takei, Y.: On the structure of Painlevé transcendents with a large parameter. Proc. Japan Acad., 69A , 224-229 (1993). · Zbl 0794.34049 [7] Kawai, T., and Takei, Y.: WKB analysis of Painlevé transcendents with a large parameter. I. Adv. Math., 118 , 1-33 (1996). · Zbl 0848.34005 [8] Nishikawa, Y., and Takei, Y.: On the structure of the Riemann surface in the Painlevé hierarchies. (In preparation). [9] Shimomura, S.: Painlevé property of a degenerate Garnier system of $$(9/2)$$-type and of a certain fourth order non-linear ordinary differential equation. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 29 , 1-17 (2000). · Zbl 0952.35139
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.