Expansions on special solutions of the first $q$-Painlevé equation around the infinity

Ohyama, Yousuke

doi:10.3792/pjaa.86.91

Cited by 5 publications

(9 citation statements)

References 5 publications

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“…where P n is a solution of the linear homogeneous equation (17) and α 0 and n 0 are constants. If we use the convention that the summation is zero when the upper bound is less than the lower bound, then α 0 = v n 0 /P n 0 .…”

Section: Lemma 5 Let P N Be a Solution Of The Linear Homogeneous Difmentioning

confidence: 99%

“…Another version is

f_{n + 1} f_{n - 1} = - \frac{s}{f_{n}} + \frac{s}{f_{n}^{2}},

where s is an exponential function of n and

f_{n} = f (s)

. This equation is due to and studied in and . It can be transformed to Equation , by taking a new dependent variable:

f_{n} = v_{n} / ρ_{n}

, which leads to

v_{n + 1} v_{n - 1} = ρ_{n + 1} ρ_{n} ρ_{n - 1} (- \frac{s}{v_{n}} + \frac{s ρ_{n}}{v_{n}^{2}}) .

This becomes Equation , when we take

ρ_{n} = - β (α x)

and

s = α^{4} x^{4} / β^{3}

, with

v_{n} = v (s) = g (x) = g_{n}

.…”

Section: Introductionmentioning

confidence: 99%

“…The Newton polygon of(17) is an equilateral triangle with its base on the interval [0, 2] and remaining sides of slope 3 and −3, respectively. See the definition of Newton polygons and associated theory of asymptotic behavior of solutions in Section 2.2 of[21].…”

mentioning

confidence: 99%

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Quicksilver Solutions of a q‐Difference First Painlevé Equation

Joshi

2014

Stud Appl Math

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In this paper, we present new, unstable solutions, which we call quicksilver solutions, of a q‐difference Painlevé equation in the limit as the independent variable approaches infinity. The specific equation we consider in this paper is a discrete version of the first Painlevé equation (qPI), whose phase space (space of initial values) is a rational surface of type A7(1). We describe four families of almost stationary behaviors, but focus on the most complicated case, which is the vanishing solution. We derive this solution's formal power series expansion, describe the growth of its coefficients, and show that, while the series is divergent, there exist true analytic solutions asymptotic to such a series in a certain q‐domain. The method, while demonstrated for qPI, is also applicable to other q‐difference Painlevé equations.

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Section: Lemma 5 Let P N Be a Solution Of The Linear Homogeneous Difmentioning

confidence: 99%

“…Another version is

f_{n + 1} f_{n - 1} = - \frac{s}{f_{n}} + \frac{s}{f_{n}^{2}},

where s is an exponential function of n and

f_{n} = f (s)

. This equation is due to and studied in and . It can be transformed to Equation , by taking a new dependent variable:

f_{n} = v_{n} / ρ_{n}

, which leads to

v_{n + 1} v_{n - 1} = ρ_{n + 1} ρ_{n} ρ_{n - 1} (- \frac{s}{v_{n}} + \frac{s ρ_{n}}{v_{n}^{2}}) .

This becomes Equation , when we take

ρ_{n} = - β (α x)

and

s = α^{4} x^{4} / β^{3}

, with

v_{n} = v (s) = g (x) = g_{n}

.…”

Section: Introductionmentioning

confidence: 99%

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Quicksilver Solutions of a q‐Difference First Painlevé Equation

Joshi

2014

Stud Appl Math

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“…Fixed points are important for studying limiting solutions. The cases where the solutions of Equation (1.1) approach a steady state as n → +∞ have been considered in [12,16]. Joshi [12] showed that there exists a true solution satisfying |w| → 0 as n → +∞, which is asymptotic to a divergent asymptotic series expansion in powers of 1/ξ, but is not a singularity of the limiting invariant K. This vanishing, unstable solution was called the quicksilver solution in [12] and its trajectory lies in a neighbourhood of the origin in S, which is punctured at two base points (called b 1 , b 2 below) that approach the origin.…”

Section: Fixed Pointsmentioning

confidence: 99%

Singular dynamics of aq-difference Painlevé equation in its initial-value space

Joshi

Lobb

2015

J. Phys. A: Math. Theor.

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We construct the initial-value space of a q-discrete first Painlevé equation explicitly and describe the behaviours of its solutions w(n) in this space as n → ∞, with particular attention paid to neighbourhoods of exceptional lines and irreducible components of the anti-canonical divisor. These results show that trajectories starting in domains bounded away from the origin in initial value space are repelled away from such singular lines. However, the dynamical behaviours in neighbourhoods containing the origin are complicated by the merger of two simple base points at the origin in the limit. We show that these lead to a saddle-point-type behaviour in a punctured neighbourhood of the origin.Dedicated to Rodney J. Baxter on the occasion of his 75th birthday.

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“…; a n Þ A N n , jaj ¼ a 1 þ Á Á Á þ a n , and studied the isomonodromy deformation problem for the equation (1.2). Nishioka [12] and Ohyama [13] and Menous [11] obtained that tyðqtÞ ¼ y þ dðy; tÞ ð1:5Þ was changed…”

Section: Introductionmentioning

confidence: 99%

Holomorphic and Singular Solutions of <i>q</i>-Difference-Differential Equations of Briot-Bouquet Type

Yamazawa

2016

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Abstract. In 1990, Gérard-Tahara [4] introduced the Briot-Bouquet type partial di¤erential equation tq t u ¼ F ðt; x; u; q x uÞ, and they determined the structure of holomorphic and singular solutions provided that the characteristic exponent rðxÞ satisfies rð0Þ B f1; 2; . . .g. In this paper the author shows existences of holomorphic and singular solutions of the following type of di¤erence-di¤erential equations tD q u ¼ F ðt; x; u; q x uÞ.

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Expansions on special solutions of the first $q$-Painlevé equation around the infinity

Cited by 5 publications

References 5 publications

Quicksilver Solutions of a q‐Difference First Painlevé Equation

Quicksilver Solutions of a q‐Difference First Painlevé Equation

Singular dynamics of aq-difference Painlevé equation in its initial-value space

Holomorphic and Singular Solutions of <i>q</i>-Difference-Differential Equations of Briot-Bouquet Type

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