Basic expansions of solutions to the sixth Painlevé equation in the generic case

We consider the sixth Painlevé equation for generic values of its four complex parameters. By methods of power geometry, we obtain those asymptotic expansions of solutions to the equation near the singular point x = 0 for which the order of the first term is less than unity. We refer to these expansions as basic expansions. They form 10 families and include expansions of four types, namely, power, power-logarithmic, complicated, and exotic. All other asymptotic expansions of solutions to the equation near the three singular points x = 0, x = 1, and x = ∞ can be computed from the basic expansions with the use of symmetries of the equation. Most of these expansions are new.     

On connection formulas for asymptotic expansions for some special solutions of the fifth Painlevé equation

Asymptotic expansions for some special solutions of the fifth Painlevé equation for x→0 and for x→+∞ and the corresponding connection formulas are obtained. An example of application of the obtained formulas to the third Painlevé equation is given. Bibliography: 6 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 19–29. Translated by F. V. Andreev.     

Hierarchies of exact solutions for the discrete third Painlevé equation

In this paper, we construct hierarchies of rational solutions of the discrete third Painlevé equation (d-PIII) by applying Schlesinger transformations to simple initial solutions. We show how these solutions reduce in the continuous limit to the hierarchies of rational solutions of the third Painlevé equation (PIII). We also study the solutions of d-PIII which are expressed in terms of discrete Bessel functions and show that these solutions reduce in the continuous limit the hierarchies of special function solutions of PIII.     

Power expansions of solutions to the modified third Painlevé equation in a neighborhood of zero

The modified third Painlevé equation

Total integrals of Ablowitz-Segur solutions for the inhomogeneous Painlevé II equation

In this paper, we establish a formula determining the value of the Cauchy principal value integrals of the real and purely imaginary Ablowitz-Segur solutions for the inhomogeneous second Painlevé equation. Our approach relies on the analysis of the corresponding Riemann-Hilbert problem and the construction of an appropriate parametrix in a neighborhood of the origin. Obtained integral formulas are consistent with already known analogous results for the Ablowitz-Segur solutions of the homogeneous Painlevé II equation.     

On some special solutions of the fifth Painlevé equation

Four kinds of special solutions of the fifth Painlevé equation are described. Their asymptotic expansions for t→+∞ are given. The corresponding monodromy data are calculated. This gives the possibility of obtaining connection formulas. Bibliography: 7 titles. Dedicated to the memory of A. P. Oskolkov Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 243, 1997, pp. 10–18. Translated by F. V. Andreev.     

Weak nonlinear asymptotic solutions for the fourth order analogue of the second Painlevé equation

The fourth-order analogue of the second Painlevé equation is considered. The monodromy manifold for a Lax pair associated with the P ₂ ² equation is constructed. The direct monodromy problem for the Lax pair is solved. Asymptotic solutions expressed via trigonometric functions in the Boutroux variables along the rays ? = \(\frac{2}{5}\)π(2n + 1) on the complex plane have been found by the isomonodromy deformations technique.     

On a relationship between solutions of the third Painlevé equation

The direct and inverse Bäcklund transformations for the third Painlevé equation in the case O is used to obtain a nonlinear functional relationship connecting the solutions of this equation for different values of the parameters that occur in it.Belarus State University of Information Technology and Electronics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 364–366, March, 1995.     

Solution of the equivalence problem for the Painlevé IV equation

We solve the equivalence problem for the Painlevé IV equation, formulating the necessary and sufficient conditions in terms of the invariants of point transformations for an arbitrary second-order differential equation to be equivalent to the Painlevé IV equation. We separately consider three pairwise nonequivalent cases: both equation parameters are zero, a = b = 0; only one parameter is zero, b = 0; and the parameter b ?? 0. In all cases, we give an explicit point substitution transforming an equation satisfying the described test into the Painlevé IV equation and also give expressions for the equation parameters in terms of invariants.     

Asymptotic behavior of the solutions of the p-Laplacian equation

The asymptotic behavior of the solutions for p-Laplacian equations as p→∞ is studied.     

Picard and Chazy solutions to the Painlevé VI equation

We study the solutions of a particular family of Painlevé VI equations with parameters and , for . We show that in the case of half-integer , all solutions can be written in terms of known functions and they are of two types: a two-parameter family of solutions found by Picard and a new one-parameter family of classical solutions which we call Chazy solutions. We give explicit formulae for them and completely determine their asymptotic behaviour near the singular points and their nonlinear monodromy. We study the structure of analytic continuation of the solutions to the PVI equation for any such that . As an application, we classify all the algebraic solutions. For half-integer, we show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For integer, we show that all algebraic solutions belong to a one-parameter family of rational solutions. Received: 23 February 1999 / Accepted: 10 January 2001 / Published online: 18 June 2001     

Uniqueness theorems for solutions of Painlevé transcendents

The paper deals with the uniqueness problems when two meromorphic functions f and g share three distinct values CM and f satisfies the first, second or fourth Painlevé transcendents.     

Numerical solution of the Painlevé V equation

A numerical method for solving the Cauchy problem for the fifth Painlevé equation is proposed. The difficulty of the problem is that the unknown function can have movable singular points of the pole type; moreover, the equation has singularities at the points where the solution vanishes or takes the value 1. The positions of all of these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Numerical results illustrating the potentials of this method are presented.     

Expansions of solutions to the fifth Painlevé equation near its nonsingular point

Doklady Mathematics -     

Numerical solution of the Painlevé IV equation

A numerical method for solving the Cauchy problem for the fourth Painlevé equation is proposed. The difficulty of the problem is that the unknown function can have movable singular points of the pole type; moreover, the equation may have singularities at the points where the solution vanishes. The positions of poles and zeros of the solution are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities in the corresponding point and its neighborhood. Numerical results confirming the efficiency of this method are presented.     

Numerical solution of the Painlevé VI equation

A numerical method for solving the Cauchy problem for the sixth Painlevé equation is proposed. The difficulty of this problem, as well as the other Painlevé equations, is that the unknown function can have movable singular points of the pole type; moreover, the equation may have singularities at the points where the solution takes the values 0 or 1 or is equal to the independent variable. The positions of all of these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. The main results of this paper are the derivation of the auxiliary equations and the formulation of transition criteria. Numerical results illustrating the potentials of this method are presented.     

Euler integral symmetries for the confluent Heun equation and symmetries of the Painlevé equation PV

Euler integral symmetries relate solutions of ordinary linear differential equations and generate integral representations of the solutions in several cases or relations between solutions of constrained equations. These relations lead to the corresponding symmetries of the monodromy matrices for the differential equations. We discuss Euler symmetries in the case of the deformed confluent Heun equation, which is in turn related to the Painlevé equation PV. The existence of symmetries of the linear equations leads to the corresponding symmetries of the Painlevé equation of the Okamoto type. The choice of the system of linear equations that reduces to the deformed confluent Heun equation is the starting point for the constructions. The basic technical problem is to choose the bijective relation between the system parameters and the parameters of the deformed confluent Heun equation. The solution of this problem is quite large, and we use the algebraic computing system Maple for this.