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1,102
Rational surfaces associated with affine root systems and geometry of the Painlevé equations
, 1999
"... We present a geometric approach to the theory of Painlev'e equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anticanonical divisor D of canonical type. We classify all such surfaces X. To each X, there corresponds a root subsyste ..."
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Cited by 160 (6 self)
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subsystems of E (1) 8 inside the Picard lattice of X. We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces. We show that the translation part of the affine Weyl group gives rise to discrete Painlev'e equations, and that the above action
On the location of poles for the Ablowitz Segur family of solutions to the second Painleve equation
 Nonlinearity
"... ] Centre de recherches mathématiques ..."
The YablonskiiVorob’ev polynomials for the second Painlevé hierarchy
, 2006
"... Special polynomials associated with rational solutions of the second Painlevé equation and other equations of its hierarchy are studied. A new method, which allows one to construct each family of polynomials is presented. The structure of the polynomials is established. Formulaes for their coefficie ..."
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Cited by 2 (2 self)
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Special polynomials associated with rational solutions of the second Painlevé equation and other equations of its hierarchy are studied. A new method, which allows one to construct each family of polynomials is presented. The structure of the polynomials is established. Formulaes
On the geometry of the first and second Painlevé equations
, 2009
"... In this paper we explicitly compute the transformation that maps the generic second order differential equation y ′ ′ = f(x, y, y ′ ) to the Painlevé first equation y ′ ′ = 6y 2 + x (resp. the Painlevé second equation y ′ ′ = 2y 3 + yx + α). This change of coordinates, which is function of f and ..."
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Cited by 2 (0 self)
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In this paper we explicitly compute the transformation that maps the generic second order differential equation y ′ ′ = f(x, y, y ′ ) to the Painlevé first equation y ′ ′ = 6y 2 + x (resp. the Painlevé second equation y ′ ′ = 2y 3 + yx + α). This change of coordinates, which is function of f
On the Linearization of the First and Second Painlevé Equations
, 806
"... We found Fuchs–Garnier pairs in 3×3 matrices for the first and second Painlevé equations which are linear in the spectral parameter. As an application of our pairs for the second Painlevé equation we use the generalized Laplace transform to derive an invertible integral transformation relating two i ..."
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We found Fuchs–Garnier pairs in 3×3 matrices for the first and second Painlevé equations which are linear in the spectral parameter. As an application of our pairs for the second Painlevé equation we use the generalized Laplace transform to derive an invertible integral transformation relating two
THE COALESCENCE LIMIT OF THE SECOND PAINLEVÉ EQUATION
, 1996
"... In this paper, we study a well known asymptotic limit in which the second Painlevé equation (PII) becomes the first Painlevé equation (PI). The limit preserves the Painlevé property (i.e. that all movable singularities of all solutions are poles). Indeed it has been commonly accepted that the movabl ..."
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Cited by 1 (1 self)
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In this paper, we study a well known asymptotic limit in which the second Painlevé equation (PII) becomes the first Painlevé equation (PI). The limit preserves the Painlevé property (i.e. that all movable singularities of all solutions are poles). Indeed it has been commonly accepted
wavelet solutions of the Second Painleve equations
 Iranian Journal of Science and Technology
"... Dynamically adaptive numerical methods have been developed to find solutions for differential equations. The subject of wavelet has attracted the interest of many researchers, especially, in finding efficient solutions for differential equations. Wavelets have the ability to show functions at differ ..."
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Cited by 1 (0 self)
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at different levels of resolution. In this paper, a numerical method is proposed for solving the second Painleve equation based on the Legendre wavelet. The solutions of this method are compared with the analytic continuation and Adomian Decomposition methods and the ability of the Legendre wavelet method
On the Extension of the Painlevé Property to Difference Equations
, 1999
"... It is well known that the integrability of a differential equation is related to the singularity structure of its solutions in the complex domain. A number of ways of extending this philosophy to discrete equations are explored. First, following the classical work of Julia, Birkhoff and others, a na ..."
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Cited by 49 (10 self)
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functions can produce spurious branching in solutions which must be factored out of the analysis. Second, examples and theorems from the theory of difference equations are presented which show that, modulo these periodic functions, solutions of a large class of difference equations are meromorphic
Physics © by SpringerVerlag 1980 Monodromy and SpectrumPreserving Deformations I
"... Abstract. A method for solving certain nonlinear ordinary and partial differential equations is developed. The central idea is to study monodromy preserving deformations of linear ordinary differential equations with regular and irregular singular points. The connections with isospectral deformation ..."
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equations. Several classes of solutions are discussed, and in particular the general expression for rational solutions for the second Painleve equation family is shown to be — — ln(A+/A__ \ where Δ+ and Δ _ are determinants. We also demonstrate that each of these equations is an exactly integrable
An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlevé Equation (and Generalizations) ⋆
"... Abstract. We construct a family of secondorder linear difference equations parametrized by the hypergeometric solution of the elliptic Painlevé equation (or higherorder analogues), and admitting a large family of monodromypreserving deformations. The solutions are certain semiclassical biorthogon ..."
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Cited by 2 (1 self)
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Abstract. We construct a family of secondorder linear difference equations parametrized by the hypergeometric solution of the elliptic Painlevé equation (or higherorder analogues), and admitting a large family of monodromypreserving deformations. The solutions are certain semiclassical
Results 1  10
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