Asymptotic behaviour around a boundary point of the<i>q</i>-Painlevé VI equation and its connection problem

Mano, Toshiyuki

doi:10.1088/0951-7715/23/7/004

Cited by 18 publications

(44 citation statements)

References 12 publications

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“…The formula for y above can be regarded as an extension of Mano's asymptotic expansion to all orders for the solution of q-P VI [18]. Theorem 2.1 was obtained by constructing the fundamental solution of the Lax-pair for q-P VI in [16], in terms of q-conformal blocks in [2].…”

Section: )mentioning

confidence: 99%

Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations

Matsuhira¹,

Nagoya

2019

SIGMA

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We derive series representations for the tau functions of the q-Painlevé V, III 1 , III 2 , and III 3 equations, as degenerations of the tau functions of the q-Painlevé VI equation in [Jimbo M., Nagoya H., Sakai H., J. Integrable Syst. 2 (2017), xyx009, 27 pages]. Our tau functions are expressed in terms of q-Nekrasov functions. Thus, our series representations for the tau functions have explicit combinatorial structures. We show that general solutions to the q-Painlevé V, III 1 , III 2 , and III 3 equations are written by our tau functions. We also prove that our tau functions for the q-Painlevé III 1 , III 2 , and III 3 equations satisfy the three-term bilinear equations for them.

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Section: )mentioning

confidence: 99%

Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations

Matsuhira¹,

Nagoya

2019

SIGMA

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“…In fact, the Stokes structure in the k th -adjacent domains are identical to those in D0 since W0 is 2πi-periodic. Each adjacent domain contributes an exponential contribution to the asymptotic solution described by (41). From figure 3.4a, we see that the presence of the exponential contribution from D−1 does not affect the dominance of (41) in D0 and hence its associated Stokes multiplier may be freely specified.…”

Section: Stokes Structurementioning

confidence: 98%

“…Using the naming convention Figure 3.2a illustrates the value S3, which is defined by (40) and (39), as Stokes curves are crossed. Figure 3.2b illustrates the regions of validity of the asymptotic solution described by (41). The dark and light grey shaded regions denote regions in which the exponential contribution associated with χ3 is large and small respectively.…”

Section: Stokes Structurementioning

confidence: 99%

“…From figure 3.4a, we see that the presence of the exponential contribution from D−1 does not affect the dominance of (41) in D0 and hence its associated Stokes multiplier may be freely specified. However, the presence of the exponential contribution from D1 dominates the asymptotic solution, (41). In order for (41) to remain valid in D0, we require the value of the Stokes multiplier associated with exponential contribution in D1 must be chosen to be equal to zero.…”

Section: Stokes Structurementioning

confidence: 99%

“…However, there have been very few asymptotic studies on q-Painlevé equations. The asymptotic study of variants of the sixth q-Painlevé equation have been investigated by Mano [41] and Joshi and Roffelson [33] using the q-analogue of the isomonodromic deformation approach. The first q-Painlevé equation has also been investigated in [28,30].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Nonlinear q-Stokes phenomena for q-Painlevé I

Joshi¹,

Lustri²,

Luu³

2019

J. Phys. A: Math. Theor.

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We consider the asymptotic behaviour of solutions of the first q-difference Painlevé equation in the limits |q| → 1 and n → ∞. Using asymptotic power series, we describe four families of solutions that contain free parameters hidden beyond-all-orders. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. In order to investigate such phenomena we apply exponential asymptotic techniques to obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we also determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the Stokes curves are described by curves known as q-spirals.

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Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

Gavrylenko

Lisovyy

2018

Commun. Math. Phys.

131

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We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with n regular singular points on the Riemann sphere and generic monodromy in GL (N , C). The corresponding operator acts in the direct sum of N (n − 3) copies of L 2 S 1 . Its kernel has a block integrable form and is expressed in terms of fundamental solutions of n−2 elementary 3-point Fuchsian systems whose monodromy is determined by monodromy of the relevant n-point system via a decomposition of the punctured sphere into pairs of pants. For N = 2 these building blocks have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to n = 4 gives a series representation of the general solution to Painlevé VI equation.

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Asymptotic behaviour around a boundary point of theq-Painlevé VI equation and its connection problem

Cited by 18 publications

References 12 publications

Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations

Combinatorial Expressions for the Tau Functions of q-Painlevé V and III Equations

Nonlinear q-Stokes phenomena for q-Painlevé I

Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

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