64 research outputs found
On the Riemann-Hilbert approach to asymptotics of tronqu\'ee solutions of Painlev\'e I
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 parametrix around the recessive stationary point of the phase
function, in terms of complementary error functions, makes it possible to give
detailed information about non-perturbative contributions beyond standard
Poincar\'e expansions for tronqu\'ee and tritronqu\'ee solutions.Comment: 28 pages, 6 figures. Second revision, some (more) typos correcte
On systems of differential equations with extrinsic oscillation
We present a numerical scheme for an efficient discretization of nonlinear systems of differential equations subjected to highly oscillatory perturbations. This method is superior
to standard ODE numerical solvers in the presence of high frequency forcing terms,and is based on asymptotic expansions of the solution in inverse powers of the oscillatory
parameter w, featuring modulated Fourier series in the expansion coefficients. Analysis of numerical stability and numerical examples are included
Simulation of MEMRISTORS in the presence of a high-frequency forcing function
This reported work is concerned with the simulation of MEMRISTORS when they are subject to high-frequency forcing functions. A novel asymptotic-numeric simulation method is applied. For systems involving high-frequency signals or forcing functions, the superiority of the proposed method in terms of accuracy and efficiency when compared to standard simulation techniques shall be illustrated. Relevant dynamical properties in relation to the method shall also be considered
Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions
We consider polynomials P_n orthogonal with respect to the weight J_? on [0,?), where J_? is the Bessel function of order ?. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros are complex and accumulate as n?? near the vertical line Rez=??2. We prove this fact for the case 0???1/2 from strong asymptotic formulas that we derive for the polynomials Pn in the complex plane. Our main tool is the Riemann-Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift-Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for ??1/2
Asymptotic solvers for second-order differential equation systems with multiple frequencies
In this paper, an asymptotic expansion is constructed to solve
second-order dierential equation systems with highly oscillatory forcing terms involving multiple frequencies. An asymptotic expansion is derived in inverse of powers of the oscillatory parameter and its truncation results in a very eective method of dicretizing the dierential equation system in question. Numerical experiments illustrate the eectiveness of the asymptotic method in contrast to the standard Runge-Kutta method
Efficient computation of delay differential equations with highly oscillatory terms.
This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation
The kissing polynomials and their Hankel determinants
We study a family of polynomials that are orthogonal with respect to the
weight function in , where . Since this
weight function is complex-valued and, for large , highly oscillatory,
many results in the classical theory of orthogonal polynomials do not apply. In
particular, the polynomials need not exist for all values of the parameter
, and, once they do, their roots lie in the complex plane. Our results
are based on analysing the Hankel determinants of these polynomials,
reformulated in terms of high-dimensional oscillatory integrals which are
amenable to asymptotic analysis. This analysis yields existence of the
even-degree polynomials for large values of , an asymptotic expansion
of the polynomials in terms of rescaled Laguerre polynomials near and a
description of the intricate structure of the roots of the Hankel determinants
in the complex plane. This work is motivated by the design of efficient
quadrature schemes for highly oscillatory integrals.Comment: 31 pages, 11 figures. Revised version, Section 8 edite
Large z Asymptotics for Special Function Solutions of Painlevé II in the Complex Plane
In this paper we obtain large z asymptotic expansions in the complex plane for the tau function corresponding to special function solutions of the PainlevĂ© II differential equation. Using the fact that these tau functions can be written as nĂn Wronskian determinants involving classical Airy functions, we use Heine's formula to rewrite them as n-fold integrals, which can be asymptotically approximated using the classical method of steepest descent in the complex plane
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