Painlevé Differential Equations in the Complex Plane

Громак, Валерий Иванович; Laine, Ilpo; 下村, 俊

doi:10.1515/9783110198096

Cited by 321 publications

(414 citation statements)

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“…In fact, the Nevanlinna test seems more natural when compared to the well-known complex analytic Painlevé test as an integrability test for second order ordinary differential equations; see [2, p. 362]. We mention that the prime integrable difference equations are the discrete Painlevé equations which can be obtained from the classical Painlevé differential equations [12] via suitable discretizations.…”

Section: Discussionmentioning

confidence: 99%

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

2008

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Abstract. We investigate the growth of the Nevanlinna Characteristic of f (z + η) for a fixed η ∈ C in this paper. In particular, we obtain a precise asymptotic relation between T`r, f (z + η)´and T (r, f ), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f (z + η)/f (z) which is a discrete version of the classical logarithmic derivative estimates of f (z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker [40] concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst [1] concerning integrable difference equations.

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Section: Discussionmentioning

confidence: 99%

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

2008

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“…The series (3.2.8) and (3.4.22) converge for sufficiently small |x|. The C ∞ 0 for integer values of r and the family B 10 were known earlier [54].…”

Section: Introductionmentioning

confidence: 79%

“…The families B 1 -B 6 , B τ 0 , B τ 1 , B τ 2 , B τ 6 with τ = ±1 exhaust all expansions corresponding to the left vertical edge of the polygon in Figure 1. For families B 1 , B 2 , only subfamilies with constant coefficients and integer exponents were known before [54]. The families B τ 0 , B τ 1 , B τ 2 , B τ 6 , B 3 -B 6 are new.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2010

Trans. Moscow Math. Soc.

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Abstract. We obtain all asymptotic expansions of solutions of the sixth Painlevé equation near all three singular points x = 0, x = 1, and x = ∞ for all values of four complex parameters of this equation. The expansions are obtained for solutions of five types: power, power-logarithmic, complicated, semiexotic, and exotic. They form 117 families. These expansions may contain complex powers of the independent variable x. First we use methods of two-dimensional power algebraic geometry to obtain those asymptotic expansions of all five types near the singular point x = 0 for which the order of the leading term is less than 1. These expansions are called basic expansions. They form 21 families. All other asymptotic equations near three singular points are obtained from basic ones using symmetries of the equation. The majority of these expansions are new. Also, we present examples and compare our results with previously known ones.

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“…Equation (18) is believed to define a new transcendent. Now we will apply the algorithm introduced in the introduction to (17). We find that ψ j in (11) are given by…”

Section: The Cosgrove's Fif-i Equationmentioning

confidence: 99%

Bäcklund Transformations for Fifth-order Painlevé Equations

Sakka

2005

Zeitschrift Für Naturforschung A

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In this article we study Bäcklund transformations of the fifth-order Painlevé equations Fif-I, Fif-II, Fif-III and Fif-IV. We derive Bäcklund transformations between these equations and new fifth-order Painlevé equations. The method of derivation is based on the idea of seeking transformations that preserve the Painlevé property. Moreover we give first integrals for the new equations. -MSC2000 classification scheme numbers: 34M55, 35Q51, 37K35

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Painlevé Differential Equations in the Complex Plane

Cited by 321 publications

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On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane

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Bäcklund Transformations for Fifth-order Painlevé Equations

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