Related papers: Stokes Phenomena in Discrete Painlev\'e II
In this study, we consider the asymptotic behaviour of the first discrete Painlev\{e} equation in the limit as the independent variable becomes large. Using an asymptotic series expansion, we identify two types of solutions which are…
We consider the asymptotic behaviour of solutions of the first $q$-difference Painlev\'{e} equation in the limits $|q|\rightarrow 1$ and $n\rightarrow\infty$. Using asymptotic power series, we describe four families of solutions that…
We use exponential asymptotic analysis to identify the relevance of Stokes' phenomenon to integrability in discrete systems. We study Stokes' phenomenon in two discrete problems with the same (leading-order) continuous limit, a…
The higher-order Stokes phenomenon can emerge in the asymptotic analysis of many problems governed by singular perturbations. Indeed, over the last two decades, the phenomena has appeared in many physical applications, from acoustic and…
Using the Riemann-Hilbert approach, we explicitly construct the asymptotic $\Psi$-function corresponding to the solution $y\sim\pm\sqrt{-x/2}$ as $|x|\to\infty$ to the second Painlev\'e equation $y_{xx}=2y^3+xy-\alpha$. We precisely…
Using the Riemann-Hilbert approach, we study the quasi-linear Stokes phenomenon for the second Painlev\'e equation $y_{xx}=2y^3+xy-\alpha$. The precise description of the exponentially small jump in the dominant solution approaching…
In this paper, we study the isomonodromy deformation equations for the $n\times n$ system of first order meromorphic linear ordinary differential equations with two second order poles. We analyze the asymptotic behaviour of the solutions at…
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 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…
In this paper, we study the asymptotic behavior and connection problem of Painlev\'e I (PI) equation through a detailed analysis of the Stokes multipliers associated with its solutions. Focusing on the regime where the derivative at the…
Singularly-perturbed ordinary differential equations often exhibit Stokes' phenomenon, which describes the appearance and disappearance of oscillating exponentially small terms across curves in the complex plane known as Stokes curves.…
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 consider two special cases of the connection problem for the second Painlev\'e equation (PII) using the method of uniform asymptotics proposed by Bassom et al.. We give a classification of the real solutions of PII on the negative…
In this work we propose a new method for investigating connection problems for the class of nonlinear second-order differential equations known as the Painlev{\'e} equations. Such problems can be characterized by the question as to how the…
We study the asymptotic behaviour of two multiplicative- ($q$-) discrete Painlev\'e equations as their respective independent variable goes to infinity. It is shown that the generic asymptotic behaviours are given by elliptic functions. We…
When studying fluid-body interactions in the low-Froude limit, traditional asymptotic theory predicts a waveless free-surface at every order. This is due to the fact that the waves are in fact exponentially small---that is, beyond all…
In this paper, we find all possible asymptotic behaviours of the solutions of the second Painlev\'e equation $y''=2y^3+xy +\alpha$ as the parameter $\alpha\to\infty$ in the local region $x\ll\alpha^{2/3}$. We prove that these are asymptotic…
For equation P$_I^2$, the second member in the P$_I$ hierarchy, we prove existence of various degenerate solutions depending on the complex parameter $t$ and evaluate the asymptotics in the complex $x$ plane for $|x|\to\infty$ and…
We study the asymptotic behaviour of the solutions of the generic ($D_6^{(1)}$-type) third Painlev\'e equation in the space of initial values as the independent variable approaches infinity (or zero) and show that the limit set of each…