A family of solutions of a nonlinear ordinary differential equation and its application to Painlevé equations (III), (V) and (VI)

Shimomura, Shun

doi:10.2969/jmsj/03940649

Cited by 25 publications

(39 citation statements)

References 3 publications

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“…A technique similar to that used in the proof of Theorem 1.1 was first introduced by S. Shimomura in [15,31,32,33,34] for a class of functional equations.…”

Section: )mentioning

confidence: 99%

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The elliptic representation of the general Painlevé VI equation

Guzzetti

2002

Comm Pure Appl Math

136

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“…A technique similar to that used in the proof of Theorem 1.1 was first introduced by S. Shimomura in [15,31,32,33,34] for a class of functional equations.…”

Section: )mentioning

confidence: 99%

“…The nongeneric case β = γ = 1 − 2δ = 0 is studied in [8,13,22] for its applications to topological field theory. Studies on the critical behavior can be also found in [15,31,32,33,34].…”

Section: Introductionmentioning

confidence: 99%

The elliptic representation of the general Painlevé VI equation

Guzzetti

2002

Comm Pure Appl Math

136

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“…In this approach, the Painlevé equation is written as an integrability condition of a linear system, and by analyzing this Lax pair various properties of the Painlevé equation can be derived including exact solutions, special integrals, connection formulae, and Bäcklund transformations; studies of P III using the isomonodromy deformation method include [38,[42][43][44][45][46][47][48][49][50][51][52]. Other studies of P III include [10,23,[53][54][55][56][57].…”

Section: Introductionmentioning

confidence: 99%

Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation

1997

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In this article our concern is with the third Painlevé equation d2y/dx2= (1/y)(dy/dx)2− (1/x)(dy/dx) + (αy2+β)/x+γy3+δ/y where α, β, γ, and δ are arbitrary constants. It is well known that this equation admits a variety of types of solution and here we classify and characterize many of these. Depending on the values of the parameters the third Painlevé equation can admit solutions that may be either expressed as the ratio of two polynomials in either x or x1/3 or related to certain Bessel functions. It is thought that all exact solutions of (1) can be categorized into one or other of these hierarchies. We show how, given a few initial solutions, it is possible to use the underlying structures of these hierarchies to obtain many other solutions. In addition, we show how this knowledge concerning the continuous third Painlevé equation (1) can be adapted and used to derive exact solutions of a suitable discretized counterpart of (1). Both the continuous and discrete solutions we find are of potential importance as it is known that the third Painlevé equation has a large number of physically significant applications.

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“…If θ 0 − θ x = θ ∞ = 0, then by [22,Theorem 5.6], equation (V) admits a family of solutions given by tanh 2 (…”

Section: Proof Of Theorem 22mentioning

confidence: 99%

“…For more general integration constants, a family of solutions near x = 0 expanded into convergent series in spiral domains or sectors was given by the present author [22]. Kaneko and Ohyama [16] presented certain Taylor series solutions around x = 0 such that each corresponding linear system (1.1) is solvable in terms of hypergeometric functions and that the monodromy may be calculated explicitly.…”

Section: Introductionmentioning

confidence: 99%

Critical Behaviours of the Fifth Painlevé Transcendents and the Monodromy Data

Shimomura

2017

Kyushu J. Math.

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Abstract. For the fifth Painlevé equation, we present families of convergent series solutions near the origin and the corresponding monodromy data for the associated isomonodromy linear system. These solutions are of complex power type, of inverse logarithmic type and of Taylor series type. For generic parameters the total set of these critical behaviours is almost complete. For the complex power type of solutions in the generic case, we clarify the structure of the analytic continuation on the universal covering around the origin, and examine the distribution of zeros, poles and 1-points. It is shown that two kinds of spiral domains including a sector as a special case are alternately arrayed; the domains of one kind contain sequences both of zeros and of poles, and those of the other kind sequences of 1-points.

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A family of solutions of a nonlinear ordinary differential equation and its application to Painlevé equations (III), (V) and (VI)

Cited by 25 publications

References 3 publications

The elliptic representation of the general Painlevé VI equation

The elliptic representation of the general Painlevé VI equation

Bäcklund Transformations and Solution Hierarchies for the Third Painlevé Equation

Critical Behaviours of the Fifth Painlevé Transcendents and the Monodromy Data

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