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The Painlevé transcendents with solvable monodromy
, 2007
"... Abstract: We will study special solutions of the fourth, fifth and sixth Painlevé equations with generic values of parameters whose linear monodromy can be calculated explicitly. We will show the relation between Umemura’s classical solutions and our solutions. 1 ..."
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Abstract: We will study special solutions of the fourth, fifth and sixth Painlevé equations with generic values of parameters whose linear monodromy can be calculated explicitly. We will show the relation between Umemura’s classical solutions and our solutions. 1
ON THE ALGEBRAIC INDEPENDENCE OF GENERIC PAINLEVÉ TRANSCENDENTS.
"... Abstract. We prove that if y ′ ′ = f(y, y ′ , t, α, β,...) is a generic Painlevé equation from among the classes II to V, and if y1,..., yn are distinct solutions, then tr.deg (C(t)(y1, y ′ 1,..., yn, y ′ n)/C(t)) = 2n. (This was proved by Nishioka for the single equation PI.) For generic Painlevé ..."
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Cited by 2 (1 self)
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Abstract. We prove that if y ′ ′ = f(y, y ′ , t, α, β,...) is a generic Painlevé equation from among the classes II to V, and if y1,..., yn are distinct solutions, then tr.deg (C(t)(y1, y ′ 1,..., yn, y ′ n)/C(t)) = 2n. (This was proved by Nishioka for the single equation PI.) For generic
R–MATRIX CONSTRUCTION OF ELECTROMAGNETIC MODELS FOR THE PAINLEVÉ TRANSCENDENTS
, 1994
"... The Painlevé transcendents PI–PV and their representations as isomonodromic deformation equations are derived as nonautonomous Hamiltonian systems from the classical R–matrix Poisson bracket structure on the dual space ˜ sl ∗ R (2) of the loop algebra ˜ slR(2). The Hamiltonians are obtained by compo ..."
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Cited by 5 (1 self)
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The Painlevé transcendents PI–PV and their representations as isomonodromic deformation equations are derived as nonautonomous Hamiltonian systems from the classical R–matrix Poisson bracket structure on the dual space ˜ sl ∗ R (2) of the loop algebra ˜ slR(2). The Hamiltonians are obtained
Quasilinear Stokes phenomenon for the second Painlevé transcendent
, 2001
"... Using the RiemannHilbert approach, we study the quasilinear Stokes phenomenon for the second Painlevé equation yxx = 2y 3 + xy − α. The precise description of the exponentially small jump in the dominant solution approaching α/x as x  → ∞ is given. For the asymptotic power expansion of the dom ..."
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Cited by 12 (5 self)
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Using the RiemannHilbert approach, we study the quasilinear Stokes phenomenon for the second Painlevé equation yxx = 2y 3 + xy − α. The precise description of the exponentially small jump in the dominant solution approaching α/x as x  → ∞ is given. For the asymptotic power expansion
Growth models, random matrices and Painlevé transcendents
 Nonlinearity
"... The Hammersley process relates to the statistical properties of the maximum length of all up/right paths connecting random points of a given density in the unit square from (0,0) to (1,1). This process can also be interpreted in terms of the height of the polynuclear growth model, or the length of t ..."
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Cited by 8 (2 self)
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the calculation of the scaled cumulative distribution, in which a particular Painlevé II transcendent plays a prominent role. 1
A τFUNCTION SOLUTION TO THE SIXTH PAINLEVE TRANSCENDENT
"... Abstract. We represent and analyze the general solution of the sixth Painleve ́ transcendent in the Picard–Hitchin–Okamoto class in the Painleve ́ form as the logarithmic derivative of the ratio of certain τ–functions. These functions are expressible explicitly in terms of the elliptic Legendre int ..."
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Cited by 4 (0 self)
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Abstract. We represent and analyze the general solution of the sixth Painleve ́ transcendent in the Picard–Hitchin–Okamoto class in the Painleve ́ form as the logarithmic derivative of the ratio of certain τ–functions. These functions are expressible explicitly in terms of the elliptic Legendre
Painlevé Transcendents in the neighbourhood of fixed sjngular points
 Funkcial. Ekvac
, 1982
"... Consider the sixth Painleve equation (VI) $y^{¥prime¥prime}=¥frac{1}{2}(¥frac{1}{y}+¥frac{1}{y1}+¥frac{1}{yx})(y^{¥prime})^{2}(¥frac{1}{x}+¥frac{1}{x1}+¥frac{1}{yx})y^{¥prime}$ $+¥frac{y(y1)(yx)}{x^{2}(x1)^{2}}(¥alpha+¥beta¥frac{x}{y^{2}}+¥gamma¥frac{x1}{(y1)^{2}}+¥delta¥frac{x(x1)}{(yx) ..."
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Cited by 2 (0 self)
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Consider the sixth Painleve equation (VI) $y^{¥prime¥prime}=¥frac{1}{2}(¥frac{1}{y}+¥frac{1}{y1}+¥frac{1}{yx})(y^{¥prime})^{2}(¥frac{1}{x}+¥frac{1}{x1}+¥frac{1}{yx})y^{¥prime}$ $+¥frac{y(y1)(yx)}{x^{2}(x1)^{2}}(¥alpha+¥beta¥frac{x}{y^{2}}+¥gamma¥frac{x1}{(y1)^{2}}+¥delta¥frac{x(x1)}{(y
© Hindawi Publishing Corp. APPLICATION OF UNIFORM ASYMPTOTICS TO THE FIFTH PAINLEVÉ TRANSCENDANT
, 2001
"... We apply the uniform asymptotics method to the fifth Painlevé transcendants, find its asymptotics of the form y =−1+t−1/2A(t) as t→ ∞ along the positive taxis, and obtain the corresponding monodromy data. 2000 Mathematics Subject Classification: 34E05. 1. Introduction. We ..."
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We apply the uniform asymptotics method to the fifth Painlevé transcendants, find its asymptotics of the form y =−1+t−1/2A(t) as t→ ∞ along the positive taxis, and obtain the corresponding monodromy data. 2000 Mathematics Subject Classification: 34E05. 1. Introduction. We
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