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**Proof of the Dubrovin conjecture and analysis of the tritronquée solutions of PI.**
*(English)*
Zbl 1305.34151

Based on a number of numeric evidences, B. Dubrovin et al. [J. Nonlinear Sci. 19, No. 1, 57–94 (2009; Zbl 1220.37048)] conjectured that the tritronquée solutions of the first Painlevé equation PI, \(y''=6y^2+z\), are analytic in a sector of width \(8\pi/5\) including the origin and in some neighborhood of the origin itself as well. The authors of the present paper give a complete and rigorous proof of this conjecture.

The idea of the proof is very classical and involves a construction of an approximate solution to PI and an analysis of a small difference between the approximate solution \(y_0(z)\) and the genuine tritronquée solution \(y_t(z)\) using the nonlinear integral equation technique and contractive mapping arguments. The construction of the aproximate solution depends on the domain of the complex plane \(z\): in the sectors separated from the boundaries of the sector of analyticity and lying apart from the origin, \(y_0(z)\) is simply two initial terms of the asymptotic series of \(y_t(z)\) at infinity. The approximate solution near the boundary of the sector of analyticity involves also several exponential terms of the trans-series for \(y_t(z)\) that reflect the effects of the nearby poles. In a neighborhood of the origin, the authors approximate and extend one of the above quasi-solutions \(y_0(z)\) by projecting its Taylor series to a sum of the Chebyshev polynomials.

The idea of the proof is very classical and involves a construction of an approximate solution to PI and an analysis of a small difference between the approximate solution \(y_0(z)\) and the genuine tritronquée solution \(y_t(z)\) using the nonlinear integral equation technique and contractive mapping arguments. The construction of the aproximate solution depends on the domain of the complex plane \(z\): in the sectors separated from the boundaries of the sector of analyticity and lying apart from the origin, \(y_0(z)\) is simply two initial terms of the asymptotic series of \(y_t(z)\) at infinity. The approximate solution near the boundary of the sector of analyticity involves also several exponential terms of the trans-series for \(y_t(z)\) that reflect the effects of the nearby poles. In a neighborhood of the origin, the authors approximate and extend one of the above quasi-solutions \(y_0(z)\) by projecting its Taylor series to a sum of the Chebyshev polynomials.

Reviewer: Andrei A. Kapaev (St. Petersburg)

### MSC:

34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |

33E17 | Painlevé-type functions |

### Citations:

Zbl 1220.37048
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\textit{O. Costin} et al., Duke Math. J. 163, No. 4, 665--704 (2014; Zbl 1305.34151)

### References:

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