相关论文: Full Asymptotic Expansion of Monodromy Data for th…
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…
We consider three special cases of the initial value problem of the first Painlev\'e equation (PI). Our approach is based on the method of uniform asymptotics introduced by Bassom, Clarkson, Law and McLeod. A rigorous proof of a property of…
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…
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…
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 the second discrete Painlev\'{e} equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region…
We consider a connection problem of the first Painlev\'{e} equation ($\mathrm{P_I}$), trying to connect the local behavior (Laurent series) near poles and the asymptotic behavior as the variable $t$ tends to negative infinity for real…
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 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…
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…
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…
Leading terms of asymptotic expansions for the general complex solutions of the fifth Painlev\'e equation as $t\to\imath\infty$ are found. These asymptotics are parameterized by monodromy data of the associated linear ODE. $$…
For more than a century, the Painlev\'e I equation has played an important role in both physics and mathematics. Its two-parameter family of solutions was studied in many different ways, yet still leads to new surprises and discoveries. Two…
A one-parameter family of trans-series asymptotics of solutions to the Degenerate Painlev\'{e} III Equation (DP3E) are parametrised in terms of the monodromy data of an associated two-by-two linear auxiliary problem via the isomonodromy…
In previous work, Bender and Komijani (2015 \textit{J. Phys. A: Math. Theor.} 48, 475202) studied the first Painlev\'e (PI) equation and showed that the sequence of initial conditions giving rise to separatrix solutions could be…
We analyze the one parameter family of tronqu\'ee solutions of the Painlev\'e equation \P1 in the pole-free sectors together with the region of the first array of poles. We find a convergent expansion for these solutions, containing one…
We introduce a new rigorous method, based on Borel summability and asymptotic constants of motion generalizing \cite{invent} and \cite{ode1}, to analyze singular behavior of nonlinear ODEs in a neighborhood of infinity and provide global…
In this paper, we revisit large variable asymptotic expansions of tronqu\'ee solutions of the Painlev\'e I equation, obtained via the Riemann-Hilbert approach and the method of steepest descent. The explicit construction of an extra local…
We apply the uniform asymptotics method proposed by Bassom, Clarkson, Law and McLeod to a special Painlev\'{e} V equation, and we provide a simpler and more rigorous proof of the connection formulas for a special solution of the equation,…