Solutions of the first and second Painlevé equations are meromorphic

Hinkkanen, A.; Laine, Ilpo

doi:10.1007/bf02788247

Cited by 66 publications

(66 citation statements)

References 4 publications

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“…Note 4. One gets higher orders in the asymptotic expansions of (x n , s n ) by formal Picard iterations, using (21) and (22) in (17) and (16); these lead, after inversion of (17) and (16), to an asymptotic expansion of h(x) and h ′ (x). This is quite straightforward and fairly short, but a formal calculation will introduce uncontrolled errors, and a good part of the technical sections of the paper deals with rigorizing the analysis.…”

Section: Direct Methods To Find Stokes Multipliers In Closed Formmentioning

confidence: 99%

A direct method to find Stokes multipliers in closed form for P$_1$ and more general integrable systems

Costin

Huang

2015

Trans. Amer. Math. Soc.

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Abstract. We calculate the Stokes multipliers in closed form for P1 using a direct approach. For this purpose, we introduce a new rigorous method, based on Borel summability and asymptotic constants of motion generalizing previous results, to analyze singular behavior of nonlinear ODEs in a neighborhood of infinity and provide global information about their solutions in C. In equations with the Painlevé-Kowalevski (P-K) property (stating that movable singularities are not branched) the method allows for solving connection problems. The analysis is carried in detail for P1, y ′′ = 6y 2 + z, for which we find the Stokes multipliers in closed form and global asymptotics in C for solutions having power-like behavior in some direction, in particular for the tritronquées.Calculating the Stokes multipliers solely relies on the P-K property and does not use linearization techniques such as Riemann-Hilbert or isomonodromic reformulations.We develop methods for finding asymptotic expansions in sectors where solutions have infinitely many singularities. These techniques do not rely on integrability and apply to more general second order ODEs which, after normalization, are asymptotically close to autonomous Hamiltonian systems.

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Section: Direct Methods To Find Stokes Multipliers In Closed Formmentioning

confidence: 99%

A direct method to find Stokes multipliers in closed form for P$_1$ and more general integrable systems

Costin

Huang

2015

Trans. Amer. Math. Soc.

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“…Their solutions are meromorphic functions in the sense that every local solution has a continuation to a function meromorphic in C. For recent proofs see Hinkkanen and Laine in [10] for P 2 or Steinmetz in [29] for both equations. The solutions are also known to be of finite order [26,27,30].…”

Section: Painlevé Equations and Value Distribution Theorymentioning

confidence: 99%

Meromorphic solutions of P_4,34 and their value distribution

Ciechanowicz

Filipuk²

2016

Ann. Acad. Sci. Fenn. Math.

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Abstract. The unified equation P 4,34 is closely related to the well-known Painlevé equations P 2 and P 4 . We discuss various properties of solutions of P 4,34 , including one-parameter families of solutions, Bäcklund transformations, regular systems for expansions around zeros and poles and value distribution. In particular, we give estimates of defects and multiplicity indices of transcendental meromorphic solutions of this equation. Moreover, we study solutions of P 4,34 from the perspective of Petrenko's theory, which is also new for P 2 , P 4 and P 34 . We give estimates of deviations and analyse the sets of exceptional values in the sense of Petrenko for equations P 2 , P 4 , P 34 and the unified equation P 4,34 .

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“…A proof of freedom from movable essential singularities for the first Painlevé equation, using the approach originally due to Painlevé, appears in chapter 14 of Ince's book [21], but these arguments have only been made completely rigorous quite recently [17]. Here we shall not attempt to prove the non-existence of movable essential singularities for the ODE (2.2), since this is a very delicate issue.…”

Section: Connection Problemsmentioning

confidence: 99%

Solutions of Abreu's Equation with Rotation Invariance

Hone¹

2004

Methods and Applications of Analysis

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Abstract. We consider a fourth order nonlinear partial differential equation in n-dimensional space introduced by Abreu in the context of Kähler metrics on toric varieties. Rotation invariant similarity solutions, depending only on the radial coordinate in R n , are determined from the solutions of a second order ordinary differential equation (ODE), with a non-autonomous Lagrangian formulation. A local asymptotic analysis of solutions of the ODE in the neighbourhood of singular points is carried out, and the existence of a class of solutions on an interval of the positive real semi-axis is proved using a nonlinear integral equation. The integrability (or otherwise) of Abreu's equation is discussed.

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Solutions of the first and second Painlevé equations are meromorphic

Cited by 66 publications

References 4 publications

A direct method to find Stokes multipliers in closed form for P$_1$ and more general integrable systems

A direct method to find Stokes multipliers in closed form for P$_1$ and more general integrable systems

Meromorphic solutions of P_4,34 and their value distribution

Solutions of Abreu's Equation with Rotation Invariance

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