Painlevé sixth equation as isomonodromic
                    deformations equation of an irregular system

Mazzocco, Marta

doi:10.1090/crmp/032/12

Cited by 29 publications

(59 citation statements)

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“…Clarkson, however, in these studies, the Laplace transform was not applied and the relation with the Jimbo-Miwa Fuchs-Garnier pair (1.7), (1.8b) was not yet realized. In this paper, we report the Fuchs-Garnier pairs of the type (1.9) for P 3 − P 6 : the pairs for P 3 , P 4 , and P 5 are new, the pair for P 6 coincides with the known one [18,29]. We note that our Fuchs-Garnier pair for P 4 has a different singularity structure comparing to the one by Noumi-Yamada [33].…”

Section: Introductionmentioning

confidence: 69%

“…We note that system (3.9a) coincides exactly with the system given in [29] if we make the gauge transformation Φ → JΦ where J is the constant matrix…”

Section: Reduction Of the 3 × 3 Fuchs-garnier Pair To The 2 × 2 Pair mentioning

confidence: 78%

“…Following [29] we assume that 0 is one of eigenvalues of the matrix B 6 0 (t). Note that this condition is a normalization of system (3.9) rather than a restriction.…”

Section: Reduction Of the 3 × 3 Fuchs-garnier Pair To The 2 × 2 Pair mentioning

confidence: 99%

“…Instead, following [29], we briefly outline how this pair can be mapped to the 2 × 2 Fuchs-Garnier pair for P 6 given by Jimbo and Miwa, which is defined by equation (1.8d). As a result we obtain a parametrization of the 3WRI system in terms of the solutions of P 6 .…”

Section: Reduction Of the 3 × 3 Fuchs-garnier Pair To The 2 × 2 Pair mentioning

confidence: 99%

“…More details of the calculation explained in the previous section can be found in [29]. Here we present the final result, the parametrization of the functions w j ,w j in terms of P 6 together with the corresponding reduction to get solutions of the 3WRI system.…”

Section: Similarity Solution Of 3wri System In Terms Of the Sixth Paimentioning

confidence: 99%

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On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

Joshi

Kitaev

Treharne

2007

Journal of Mathematical Physics

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We extend similarity reductions of the coupled (2+1)-dimensional three-wave resonant interaction system to its Lax pair. Thus we obtain new 3 × 3 matrix Fuchs-Garnier pairs for the third, fourth, and fifth Painlevé equations, together with the previously known Fuchs-Garnier pair for the sixth Painlevé equation. These Fuchs-Garnier pairs have an important feature: they are linear with respect to the spectral parameter. Therefore we can apply the Laplace transform to study these pairs. In this way we found reductions of all pairs to the standard 2 × 2 matrix Fuchs-Garnier pairs obtained by M. Jimbo and T. Miwa. As an application of the 3 × 3 matrix pairs, we found an integral auto-transformation for the standard Fuchs-Garnier pair for the fifth Painlevé equation. It generates an Okamoto-like Bäcklund transformation for the fifth Painlevé equation. Another application is an integral transformation relating two different 2 × 2 matrix Fuchs-Garnier pairs for the third Painlevé equation.

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Section: Introductionmentioning

confidence: 69%

“…We note that system (3.9a) coincides exactly with the system given in [29] if we make the gauge transformation Φ → JΦ where J is the constant matrix…”

Section: Reduction Of the 3 × 3 Fuchs-garnier Pair To The 2 × 2 Pair mentioning

confidence: 78%

“…Following [29] we assume that 0 is one of eigenvalues of the matrix B 6 0 (t). Note that this condition is a normalization of system (3.9) rather than a restriction.…”

Section: Reduction Of the 3 × 3 Fuchs-garnier Pair To The 2 × 2 Pair mentioning

confidence: 99%

Section: Reduction Of the 3 × 3 Fuchs-garnier Pair To The 2 × 2 Pair mentioning

confidence: 99%

Section: Similarity Solution Of 3wri System In Terms Of the Sixth Paimentioning

confidence: 99%

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On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

Joshi

Kitaev

Treharne

2007

Journal of Mathematical Physics

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The Sixth Painlevé Equation as Similarity Reduction of $${\widehat{\mathfrak{gl}}_3}$$ Generalized Drinfel’d–Sokolov Hierarchy

Kakei

Kikuchi

2007

Lett Math Phys

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Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

Guzzetti

2021

Lett Math Phys

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We consider a Pfaffian system expressing isomonodromy of an irregular system of Okubo type, depending on complex deformation parameters $$u=(u_1,\ldots ,u_n)$$ u = ( u 1 , … , u n ) , which are eigenvalues of the leading matrix at the irregular singularity. At the same time, we consider a Pfaffian system of non-normalized Schlesinger-type expressing isomonodromy of a Fuchsian system, whose poles are the deformation parameters $$u_1,\ldots ,u_n$$ u 1 , … , u n . The parameters vary in a polydisc containing a coalescence locus for the eigenvalues of the leading matrix of the irregular system, corresponding to confluence of the Fuchsian singularities. We construct isomonodromic selected and singular vector solutions of the Fuchsian Pfaffian system together with their isomonodromic connection coefficients, so extending a result of Balser et al. (I SIAM J Math Anal 12(5): 691–721, 1981) and Guzzetti (Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case, including confluence of singularities. Then, we introduce an isomonodromic Laplace transform of the selected and singular vector solutions, allowing to obtain isomonodromic fundamental solutions for the irregular system, and their Stokes matrices expressed in terms of connection coefficients. These facts, in addition to extending (Balser et al. in I SIAM J Math Anal 12(5): 691–721, 1981; Guzzetti in Funkcial Ekvac 59(3): 383–433, 2016) to the isomonodromic case (with coalescences/confluences), allow to prove by means of Laplace transform the main result of Cotti et al. (Duke Math J arXiv:1706.04808, 2017), namely the analytic theory of non-generic isomonodromic deformations of the irregular system with coalescing eigenvalues.

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Painlevé sixth equation as isomonodromic deformations equation of an irregular system

Cited by 29 publications

References 0 publications

On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

On the linearization of the Painlevé III–VI equations and reductions of the three-wave resonant system

The Sixth Painlevé Equation as Similarity Reduction of $${\widehat{\mathfrak{gl}}_3}$$ Generalized Drinfel’d–Sokolov Hierarchy

Isomonodromic Laplace transform with coalescing eigenvalues and confluence of Fuchsian singularities

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