Related papers: Airy and Painlev\'e asymptotics for the mKdV equat…
We further study the orthogonal polynomials with respect to the generalized Airy weight based on the work of Clarkson and Jordaan [{\em J. Phys. A: Math. Theor.} {\bf 54} ({2021}) {185202}]. We prove the ladder operator equations and…
Asymptotic solutions are derived for inhomogeneous differential equations having a large real or complex parameter and a simple turning point. They involve Scorer functions and three slowly varying analytic coefficient functions. The…
Usually when solving differential or difference equations via series solutions one encounters divergent series in which the coefficients grow like a factorial. Surprisingly, in the $q$-world the $n$th coefficient is often of the size…
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…
Asymptotic formula is derived for the behavior of the fundamental solution of the second-order elliptic self-adjoint operator with a piecewise-smooth coefficient in front of the senior derivatives near the discontinuity surface of the…
In this paper we discuss Airy solutions of the second Painlev\'e equation (\mbox{\rm P$_{\rm II}$}) and two related equations, the Painlev\'e XXXIV equation ($\mbox{\rm P}_{34}$) and the Jimbo-Miwa-Okamoto $\sigma$ form of \mbox{\rm P$_{\rm…
We study the asymptotics of solutions to a particular class of systems of linear wave equations, namely, of silent equations. We obtain asymptotic estimates of all orders for the solutions, and show that solutions are uniquely determined by…
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…
We consider time-inhomogeneous ODEs whose parameters are governed by an underlying ergodic Markov process. When this underlying process is accelerated by a factor $\varepsilon^{-1}$, an averaging phenomenon occurs and the solution of the…
A novel extension of the canonical solitonic mKdV equation is introduced which admits hybrid Ermakov-Painlev\'e II symmetry reduction. Application of the latter is made to obtain exact solution of Airy-type to a class of moving boundary…
In this paper we obtain large $z$ asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the Painlev\'e II differential equation. Using the fact that these tau functions can be written…
In this paper, We study the asymptotics of the leading coefficients and the recurrence coefficients for the orthogonal polynomials with repect to the Laguerre weight with singularity of root type and jump type at the soft edge via the…
It is shown that a generalization of the Painlev\'e-II equation (P-II) to a system of coupled equations with symmetry breaking terms is integrable. A Lax pair for this system is used to relate the asymptotic behavior of the solutions at…
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…
The main subject of the paper is the so-called Discrete Painlev\'e-1 Equation (DP1). Solutions of the DP1 are classified under criterion of their behavior while argument tends to infinity. The appropriate theorems of existence are proved.
In this paper, we investigate the large-order asymptotics of multi-rational solitons of the focusing complex modified Korteweg-de Vries (c-mKdV) equation with nonzero background via the Riemann-Hilbert problems. First, based on the Lax…
In this article, we derive the asymptotic expansion, up to an arbitrary order in theory, for the solution of a two-dimensional elliptic equation with strongly anisotropic diffusion coefficients along different directions, subject to the…
This paper concerns the use of asymptotic expansions for the efficient solving of forward and inverse problems involving a nonlinear singularly perturbed time-dependent reaction--diffusion--advection equation. By using an asymptotic…
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,…
Asymptotic formulas are derived for the Stieltjes-Wigert polynomials $S_n(z;q)$ in the complex plane as the degree $n$ grows to infinity. One formula holds in any disc centered at the origin, and the other holds outside any smaller disc…