Related papers: Explicit error term of the elliptic asymptotics fo…
For the fifth Painlev\'e equation it is known that a general solution is represented asymptotically by an elliptic function in cheese-like strips near the point at infinity. We present an explicit asymptotic formula for the error term of…
For the fourth Painlev\'e transcendents we derive elliptic asymptotic representations, which were announced by late Professor Kapaev without proofs. Then we newly obtain related results including the correction function.
For the fifth Painlev\'e transcendents an asymptotic representation by the Jacobi $\mathrm{sn}$-function is presented in cheese-like strips along generic directions near the point at infinity. Its elliptic main part may be understood to…
In this paper, we revisit the asymptotic formulas of real Painlev\'e I transcendents as the independent variable tends to negative infinity, which were initially derived by Kapaev with the complex WKB method. Using the Riemann-Hilbert…
For a general solution of the third Painlev\'e equation of complete type we show the Boutroux ansatz near the point at infinity. It admits an asymptotic representation in terms of the Jacobi sn-function in cheese-like strips along generic…
For a general solution of the degenerate third Painlev\'e equation we show the Boutroux ansatz near the point at infinity. It admits an asymptotic representation in terms of the Weierstrass pe-function in cheese-like strips along generic…
The discrete Painlev\'e equations have mathematical properties closely related to those of the differential Painlev\'e equations. We investigate the appearance of elliptic functions as limiting behaviours of $q$-Painlev\'e transcendents,…
In this survey, we review asymptotic results of some orthogonal polynomials which are relate to the Painlev\'e transcendents. They are obtained by applying Deift-Zhou's nonlinear steepest descent method for Riemann-Hilbert problems. In the…
We study the asymptotic behaviour of solutions of the fourth Pain\-lev\'e equation as the independent variable goes to infinity in its space of (complex) initial values, which is a generalisation of phase space described by Okamoto. We show…
The classical Painlev\'e equations are so well known that it may come as a surprise to learn that the asymptotic description of its solutions remains incomplete. The problem lies mainly with the description of families of solutions in the…
The asymptotic solution for the Painleve-2 equation with small parameter is considered. The solution has algebraic behavior before point $t_*$ and fast oscillating behavior after the point $t_*$. In the transition layer the behavior of the…
We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painleve equation. For the general solution, explicit recursive formulae for the Taylor expansion of the…
We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the…
The paper concerns asymptotic studies for the sixth Painlev\'e transcendent as independent variable tends to infinity. The primary tool is averaging and the Whitham method. Elliptic ansatz, appropriate modulation equation and asymptotics…
We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show that the extension provides a method of finding singularities of solutions of nonlinear differential equations, using…
We present expressions for the coefficients which arise in asymptotic expansions of multiple integrals of Laplace type (the first term of which is known as Laplace's approximation) in terms of asymptotic series of the functions in the…
The solutions of the perturbed first Painlev\'e equation $y"=6y^2-x^\mu$, $\mu>-4$, are uniquely determined by the free constant $C$ multiplying the exponentially small terms in the complete large $x$ asymptotic expansions. Full details are…
The Lie point symmetries of ordinary differential equations (ODEs) that are candidates for having the Painlev\'e property are explored for ODEs of order $n =2, \dots ,5$. Among the 6 ODEs identifying the Painlev\'e transcendents only…
In this paper, we compute the small and large $x$ asymptotics of the special function solutions of Painlev\'e-III equation in the complex plane. We use the representation in terms of Toeplitz determinants of Bessel functions obtained in…
We study the full asymptotic expansion of the monodromy data ({\it i.e.}, Stokes multipliers) for the first Painlev\'{e} transcendent (PI) with large initial data or large pole parameters. Our primary approach involves refining the complex…