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Global Meromorphy of Solutions of the Painlevé Equations and Their Hierarchies
AbstractWe show that all local holomorphic solutions of all equations constituting the hierarchies of the first and second Painlevé equations can be...
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Riccati Equations Revisited: Linearization and Analytic Interpretation of Instanton-Type Solutions
As a prototype of Painlevé equations, we discuss instanton-type solutions of the Riccati equation with a large parameter in this paper. Based on the...
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Rational Solutions of Equations Associated with the Second Painlevé Equation
Nonlinear differential equations associated with the second Painlevé equation are considered. Transformations for solutions of the singular manifold...
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Explicit Meromorphic Solutions of a Second Order Briot–Bouquet Differential Equation
In this paper, a special second-order Briot–Bouquet differential equation is studied. We construct explicit meromorphic solutions by the...
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Dynamics of the Fifth Painlevé Foliation
The leaves of the Painlevé foliations appear as the isomonodromic deformations of a rank 2 linear connection on a moduli space of connections....
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Systems of Vector Fields for the Integration of Ordinary Differential Equations
In this work, we investigate different classes of vector fields that can be used to find exact solutions of ordinary differential equations. The...
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Explicit Meromorphic Solutions of a Certain Briot-Bouquet Differential Equations
This paper deals with the Briot-Bouquet differential equations with degree three. The previous result shows that all the meromorphic solutions belong...
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Lax Pairs and Special Polynomials Associated with Self-similar Reductions of Sawada — Kotera and Kupershmidt Equations
Self-similar reductions of the Sawada-Kotera and Kupershmidt equations are studied. Results of Painlevé’s test for these equations are given. Lax...
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Polygons of Petrović and Fine, algebraic ODEs, and contemporary mathematics
In this paper, we study the genesis and evolution of geometric ideas and techniques in investigations of movable singularities of algebraic ordinary...
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Necessary Conditions for Point Equivalence of Second-Order Odes to the Sixth Painlevé Equation
We consider the equivalence problem for projective-type scalar second-order ordinary differential equations with respect to invertible point changes...
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On the Integrability of the Abel and of the Extended Liénard Equations
We present some exact integrability cases of the extended Liénard equation y ″ + f ( y )( y ′) n + k ( y )( y ′) m + g ( y ) y ′ + h ( y ) = 0, with n > 0 and m > 0...
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On Various Approaches to Asymptotics of Solutions to the Third Painlevé Equation in a Neighborhood of Infinity
We examine asymptotic expansions of the third Painlevé transcendents for αδ ≠ = 0 and γ = 0 in the neighborhood of infinity in a sector of aperture < 2
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Rational and Special Solutions for Some Painlevé Hierarchies
A self-similar reduction of the Korteweg–de Vries hierarchy is considered. A linear system of equations associated with this hierarchy is presented....
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Value Distribution and Growth of Solutions of Certain Painlevé Equations
As a result of classification of second order ordinary differential equations without movable branch points, a number of the so-called Painlevé...
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On the integrability conditions for a family of Liénard-type equations
We study a family of Liénard-type equations. Such equations are used for the description of various processes in physics, mechanics and biology and...
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The Calculation and Use of Generalized Symmetries for Second-Order Ordinary Differential Equations
New relationships between generalized symmetries, commuting symmetries, and generalized...
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Generalities on the Painlevé Equations
The Painlevé equations are six families P I , P II , P...
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Higher-Order Algebraic Differential Equations
In this chapter we will extend the investigations of the previous chapter to second-order algebraic differential equations and two-dimensional...