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 $e^{i\omega x}$ in $[-1,1]$, where $\omega\geq 0$. Since this weight function is complex-valued and, for large $\omega$, 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 $\omega$, 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 $\omega$, an asymptotic expansion of the polynomials in terms of rescaled Laguerre polynomials near $\pm 1$ 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

Characteristic polynomials of complex random matrices and Painlevé transcendents

We study expectations of powers and correlation functions for characteristic polynomials of N × N non-Hermitian random matrices. For the 1-point and 2-point correlation function, we obtain several characterizations in terms of Painlev´e transcendents, both at finite-N and asymptotically as N → ∞. In the asymptotic analysis, two regimes of interest are distinguished: boundary asymptotics where parameters of the correlation function can touch the boundary of the limiting eigenvalue support and bulk asymptotics where they are strictly inside the support. For the complex Ginibre ensemble this involves Painlev´e IV at the boundary as N → ∞. Our approach, together with the results in [49] suggests that this should arise in a much broader class of planar models. For the bulk asymptotics, one of our results can be interpreted as the merging of two ‘planar Fisher-Hartwig singularities’ where Painlev´e V arises in the asymptotics. We also discuss the correspondence of our results with a normal matrix model with d-fold rotational symmetries known as the lemniscate ensemble, recently studied in [14,18]. Our approach is flexible enough to apply to non-Gaussian models such as the truncated unitary ensemble or induced Ginibre ensemble; we show that in the former case Painlev´e VI arises at finite-N. Scaling near the boundary leads to Painlev´e V, in contrast to the Ginibre ensemble