Related papers: On Airy Solutions of the Second Painlev\'e Equatio…
We show that the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlev\'e equation, constructed from the Airy functions, as well as associated solutions of P$_\mathrm{XXXIV}$ and S$_\mathrm{II}$, can be expressed…
We describe the pole-free regions of the one-parameter family of special solutions of P$_\mathrm{II}$, the second Painlev\'e equation, constructed from the Airy functions. This is achieved by exploiting the connection between these…
We show that the alternative discrete Painlev\'e I equation (alt-dP$_{\rm I}$) has a unique solution which remains positive for all $n \geq 0$. Furthermore, we identify this positive solution in terms of a special solution of the second…
The increasing tritronqu\'ee solutions of the Painlev\'e-II equation with parameter $\alpha$ exhibit square-root asymptotics in the maximally-large sector $|\arg(x)|<\tfrac{2}{3}\pi$ and have recently appeared in applications where it is…
By analogy to the continuous Painlev\'e II equation, we present particular solutions of the discrete Painlev\'e II (d-P$\rm_{II}$) equation. These solutions are of rational and special function (Airy) type. Our analysis is based on the…
We study the solutions of the second Painlev\'e equation in the space of initial conditions first constructed by Okamoto, in the limit as the independent variable, x, goes to infinity. Simultaneously, we study solutions of the related…
Using a simple operator-norm estimate we show that the solution to the second Painlev\'e equation within the Ablowitz-Segur family is pole-free in a well defined region of the complex plane of the independent variable. The result is…
It is well-known that the first and second Painlev\'e equations admit solutions characterised by divergent asymptotic expansions near infinity in specified sectors of the complex plane. Such solutions are pole-free in these sectors and…
In this paper, we are concerned with the following noncommutative Painlev\'{e} II equation \begin{equation*} \mathbf{D}^2 \beta_1 = 4\mathbf{s} \beta_1 +4 \beta_1 \mathbf{s} +8 \beta_1^3, \end{equation*} where $\beta_1=\beta_1(\vec{s})$ is…
In a recent paper, Aimo Hinkkanen and Ilpo Laine proved that the transcendental solutions to Painleve's second differential equation w"=a+zw+w^3 have either order of growth 3 or else 3/2. We complete this result by proving that there exist…
In the current paper we study auto-B\"acklund transformations of the non-stationary second Painlev\'e hierarchy $\text{P}_\text{II}^{(n)}$ depending on $n$ parameters: a parameter $\alpha_n$ and times $t_1, \dots, t_{n-1}$. Using generators…
For a pair of coupled Painlev\'e equations obtained as a similarity reduction of the Hirota-Satsuma systems we describe special parameter-families of solutions given in terms of mixtures of rational and Airy functions, and in terms of a…
We consider the higher order asymptotics for the mKdV equation in the Painlev\'e sector. We first show that the solution admits a uniform expansion to all orders in powers of $t^{-1/3}$ with coefficients that are smooth functions of…
We review non-autonomous Hamiltonian systems, polynomial in two dependent variables, with the property that all of their solutions are meromorphic functions in the complex plane. These are related to known Hamiltonian systems with the…
The Painlev\'e equations possess transcendental solutions $y(t)$ with special initial values that are symmetric under rotation or reflection in the complex $t$-plane. They correspond to monodromy problems that are explicitly solvable in…
A perturbation series solution is constructed in terms of Airy functions for a nonlinear two-point boundary-value problem arising in an established model of steady electrodiffusion in one dimension, for two ionic species carrying equal and…
The paper discusses P$_{III-V}$ equation for special values of its parameters for which this equation reduces to P$_{III}$, I$_{12}$, as well as, to some special cases of I$_{38}$ and I$_{49}$ equations from the Ince's list of $50$ second…
Bilinear structure for the discrete Painlev\'e I equation is investigated. The solution on semi-infinite lattice is given in terms of the Casorati determinant of discrete Airy function. Based on this fact, the discrete Painlev\'e I equation…
We consider the second Painlev\'e equation $$ u"(x)=2u^3(x)+xu(x)-\alpha, $$ where $\alpha $ is a nonzero constant. Using the Deift-Zhou nonlinear steepest descent method for Riemann-Hilbert problems, we rigorously prove the asymptotics as…
The first five classical Painlev\'e equations are known to have solutions described by divergent asymptotic power series near infinity. Here we prove that such solutions also exist for the infinite hierarchy of equations associated with the…