Related papers: Asymptotic Series for Some Painlev\'E VI Solutions
The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear…
We consider the initial-value problem for the Sasa-Satsuma equation on the line with decaying initial data. Using a Riemann-Hilbert formulation and steepest descent arguments, we compute the long-time asymptotics of the solution in the…
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…
By exploiting an old idea first used by Pizzetti for the classical Laplacian, we introduce a notion of {\it asymptotic average solutions} making pointwise solvable every Poisson equation $\mathcal{L} u(x)=-f(x)$ with continuous data $f$,…
As a new application of the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod, we provide a simpler and more rigorous proof of the connection formulas of some special solutions of the fifth Painlev\'e equation,…
In this paper, we will develop an iterative procedure to determine the detailed asymptotic behaviour of solutions of a certain class of nonlinear vector differential equations which approach a nonlinear sink as time tends to infinity. This…
We propose an iterative estimating equations procedure for analysis of longitudinal data. We show that, under very mild conditions, the probability that the procedure converges at an exponential rate tends to one as the sample size…
These lectures introduce the method of nonlinear steepest descent for Riemann-Hilbert problems. This method finds use in studying asymptotics associated to a variety of special functions such as the Painlev\'{e} equations and orthogonal…
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…
We use a method, inspired by Pohozeav's work, to study asymptotic behaviors of non-variational elliptic systems in dimension n greater than two. The results apply to changing sign solutions.
In this lectures various methods which give a possibility to extend an area of applicability of perturbation series and hence to omit their local character are analysed. While applying asymptotic methods as a rule the following situation…
The automation of the traditional Painleve test in Mathematica is discussed. The package PainleveTest.m allows for the testing of polynomial systems of ordinary and partial differential equations which may be parameterized by arbitrary…
Purely numerical methods do not always provide an accurate way to find all the global solutions to nonlinear ODE on infinite intervals. For example, finite-difference methods fail to capture the asymptotic behavior of solutions, which might…
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…
Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered, for the case of polynomial forcing terms. The expansions…
We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollazek Polynomials as an example. The…
The solutions of the (nonlinear) Painleve VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto's affine F4 Weyl group action and many properties of the solutions will be…
A determinant formula for a class of algebraic solutions to Painlev\'e VI equation (P$_{\rm VI}$) is presented. This expression is regarded as a special case of the universal characters. The entries of the determinant are given by the…
It is proved that the Painlev\'{e} VI equation $(PVI_{\al,\be,\ga,\de})$ for the special values of constants $(\al=\frac{\nu^2}{4},\be=-\frac{\nu^2}{4}, \ga=\frac{\nu^2}{4},\de=\f1{2}-\frac{\nu^2}{4})$ is a reduced hamiltonian system. Its…
We present the hyperasymptotic expansions for a certain group of solutions of the heat equation. We extend this result to a more general case of linear PDEs with constant coefficients. The generalisation is based on the method of Borel…