Robert Buckingham

Asymptotics of Tracy-Widom Distributions and the Total Integral of a Painlevé II Function

The Tracy-Widom distribution functions involve integrals of a Painlevé II function starting from positive infinity. In this paper, we express the Tracy-Widom distribution functions in terms of integrals starting from minus infinity. There are two consequences of these new representations. The first is the evaluation of the total integral of the Hastings-McLeod solution of the Painlevé II equation. The second is the evaluation of the constant term of the asymptotic expansions of the Tracy-Widom distribution functions as the distribution parameter approaches minus infinity. For the GUE Tracy-Widom distribution function, this gives an alternative proof of the recent work of Deift, Its, and Krasovsky. The constant terms for the GOE and GSE Tracy-Widom distribution functions are new.

Large-degree asymptotics of rational Painlevé-II functions: critical behaviour

This paper is a continuation of our analysis, begun in Buckingham and Miller (2014 Nonlinearity 27 2489–577), 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 large degree. In this paper we establish asymptotic formulae valid near a certain curvilinear triangle in the complex plane that was previously shown to separate two distinct types of asymptotic behaviour. Our results display both a trigonometric degeneration of the rational Painleve-II functions and also a degeneration to the tritronquee solution of the Painleve-I equation. Our rigorous analysis is based on the steepest descent method applied to a Riemann–Hilbert representation of the rational Painleve-II functions, and supplies leading-order formulae as well as error estimates.

Semiclassical Soliton Ensembles for the Three-Wave Resonant Interaction Equations

The three-wave resonant interaction equations are a non-dispersive system of partial differential equations with quadratic coupling describing the time evolution of the complex amplitudes of three resonant wave modes. Collisions of wave packets induce energy transfer between different modes via pumping and decay. We analyze the collision of two or three packets in the semiclassical limit by applying the inverse-scattering transform. Using WKB analysis, we construct an associated semiclassical soliton ensemble, a family of reflectionless solutions defined through their scattering data, intended to accurately approximate the initial data in the semiclassical limit. The map from the initial packets to the soliton ensemble is explicit and amenable to asymptotic and numerical analysis. Plots of the soliton ensembles indicate the space–time plane is partitioned into regions containing either quiescent, slowly varying, or rapidly oscillatory waves. This behavior resembles the well-known generation of dispersive shock waves in equations such as the Korteweg–de Vries and nonlinear Schrödinger equations, although the physical mechanism must be different in the absence of dispersion.

Original article: Semiclassical spectral confinement for the sine-Gordon equation

The inverse scattering method for solving the sine-Gordon equation in laboratory coordinates requires the analysis of the Faddeev-Takhtajan eigenvalue problem. This problem is not self-adjoint and the eigenvalues may lie anywhere in the complex plane, so it is of interest to determine conditions on the initial data that restrict where the eigenvalues can be. We establish bounds on the eigenvalues for a broad class of zero-charge initial data that are applicable in the semiclassical or zero-dispersion limit. It is shown that no point off the coordinate axes or turning point curve can be an eigenvalue if the dispersion parameter is sufficiently small.

Understanding the Barriers to Hospice Care in Saudi Arabia.

Dear Editor: To access the students’ knowledge of hospice philosophy, practice and principles, a survey of Saudi Arabian students studying at a Midwestern public university was taken. Hospice care was established in Saudi Arabia in 1992 at the King Faisal Specialist Hospital. The key barriers to the full implementation of the analgesic ladder are religious, acceptance, knowledge, and availability of the medications necessary for palliative care. Access to palliative care services is limited by a lack of awareness about palliative care, knowledgeable providers, and availability of community resources. The Western concept of good death does not have the same meaning in Muslim communities, as death is anchored in the Qur’an, Allah, and faith. For palliative and hospice care, the role of religion in Saudi Arabia presents unique limitations in the use of opioids. We surveyed all 477 Saudi Arabian students studying at a Midwestern public university and recorded a response rate of 86 (19.0%) students. Questions addressed in the survey included demographics, enrollment status, and if they knew what hospice care is. The survey showed that 74 (86.0%) of 86 Saudi Arabian students attending a Midwestern public university did not know what hospice care is, and only 12 (14.0%) of 86 Saudi Arabian students had some knowledge about hospice care. Interestingly, the gender distribution for Saudi Arabian students attending a Midwestern public university accounted for males at 59.0% and females at 41.0%, both very close to Saudi Arabia’s national demographics of 58.0% males and 42.0% females. We interpreted the gender distribution at the Midwestern university to be representative of Saudi Arabia. Caring for dying family members in Muslim society is seen as the duty of relatives, which makes having patients enter hospice care a ‘‘shirking’’ of that responsibility. The barriers currently faced by hospice care in Saudi Arabia in part differ from the Western experience, but not from what is considered a universal philosophy of hospice care. In sum, the education of the general population and physicians about the philosophy and practice of hospice and palliative care may indeed be worthwhile for consideration to aid the terminally ill in Saudi Arabia.