相关论文: Application of Uniform Asymptotics to the Second P…

The self-consistent expansion (SCE) is a powerful technique for obtaining perturbative solutions to problems in statistical physics but it suffers from a subtle problem - too much freedom! The SCE can be used to generate an enormous number…

We investigate the long-time asymptotic behavior of a class of solutions to the defocusing Manakov system under nonzero boundary conditions. These solutions are characterized by a $3 \times 3$ matrix Riemann Hilbert problem. We find that…

The asymptotics of the generic second Painleve transcendent in the complex domain is found and justified via the direct asymptotic analysis of the associated Riemann-Hilbert problem based on the Deift-Zhou nonlinear steepest descent method.…

We study the asymptotic behavior of a special smooth solution y(x,t) to the second member of the Painleve I hierarchy. This solution arises in random matrix theory and in the study of Hamiltonian perturbations of hyperbolic equations. The…

This paper is a continuation of our analysis, begun in arXiv:1310.2276, of the rational solutions of the inhomogeneous Painleve-II equation and associated rational solutions of the homogeneous coupled Painleve-II system in the limit of…

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…

This paper proposes a new approach to the asymptotic analysis of Painlev\'e equations. The approach is based on representing solutions of the Painlev\'e equations using formal series in two variables, $\sum_{k=0}^{\infty}y^kA_k(x)$, with…

The integrability (solvability via an associated single-valued linear problem) of a differential equation is closely related to the singularity structure of its solutions. In particular, there is strong evidence that all integrable…

We study the asymptotic behavior of the Ablowitz-Segur solutions for the second Painlev\'e equation using the Riemann-Hilbert approach and methods based on asymptotic expansions of classical special functions. Recent results show that the…

The generalized Hastings-McLeod solutions to the inhomogeneous Painlev\'{e}-II equation arise in multi-critical unitary random matrix ensembles, the chiral two-matrix model for rectangular matrices, non-intersecting squared Bessel paths,…

We give a classification for the small-$\tau$ asymptotic behaviours of solutions to the degenerate third Painlev\'e equation, $u^{''}(\tau) = \frac{(u^{\prime}(\tau))^{2}}{u(\tau)} - \frac{u^{\prime}(\tau)}{\tau} + \frac{1}{\tau}\left(-8…

The degenerate third Painlev\'{e} equation, $u^{\prime \prime} = \frac{(u^{\prime})^{2}}{u} - \frac{u^{\prime}}{\tau} + \frac{1}{\tau}(-8 \epsilon u^{2} + 2ab) + \frac{b^{2}}{u}$, where $\epsilon,b \in \mathbb{R}$, and $a \in \mathbb{C}$,…

By application of the theory for second-order linear differential equations with two turning points developed in \cite{Olver1975}, uniform asymptotic approximations are obtained for the Lam\'{e} and Mathieu functions with a large real…

Extrapolation is a generic problem in physics and mathematics: how to use asymptotic data in one parametric regime to learn about the behavior of a function in another parametric regime. For example: extending weak coupling expansions to…

Unstable separatrix solutions for the first and second Painlev\'e transcendents are studied both numerically and analytically. For a fixed initial condition, say $y(0)=0$, there is a discrete set of initial slopes $y'(0)=b_n$ that give rise…

Painleve transcendents are usually considered as complex functions of a complex variable, but in applications it is often the real cases that are of interest. Under a reasonable assumption (concerning the behavior of a dynamical system…

The main subject of the paper is the so-called Discrete Painlev\'e-1 Equation (DP1). Solutions of DP1 are classified under criterion of their behavior while argument tends to infinity. The Isomonodromic Deformations Method yields asymptotic…

We obtain asymptotics of polynomials satisfying the orthogonality relations $$ \int_{\mathbb{R}} z^k P_n(z; t , N) \mathrm{e}^{-N \left(\frac{1}{4}z^4 + \frac{t}{2}z^2 \right)} \mathrm{d} z = 0 \quad \text{ for } \quad k = 0, 1, ..., n-1,…

We discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of Lyapunov functions. The approach is new in this field and allows for simpler proofs of general…

A multidomain spectral approach for Painlev\'e transcendents on unbounded domains is presented. This method is designed to study solutions determined uniquely by a, possibly divergent, asymptotic series valid near infinity in a sector and…