B\"acklund transformations for fourth Painlev\'e hierarchies

B\"acklund transformations (BTs) for ordinary differential equations (ODEs), and in particular for hierarchies of ODEs, are a topic of great current interest. Here we give an improved method of constructing BTs for hierarchies of ODEs. This approach is then applied to fourth Painlev\'e ($P_{IV}$) hierarchies recently found by the same authors [{\em Publ. Res. Inst. Math. Sci. (Kyoto)} {\bf 37} 327--347 (2001)]. We show how the known pattern of BTs for $P_{IV}$ can be extended to our $P_{IV}$ hierarchies. Remarkably, the BTs required to do this are precisely the Miura maps of the dispersive water wave hierarchy. We also obtain the important result that the fourth Painlev\'e equation has only one nontrivial fundamental BT, and not two such as is frequently stated.Comment: 23 pages, 2 figures, to appear Journal of Differential Equation

Behaviour of the extended Toda lattice

We consider the first member of an extended Toda lattice hierarchy. This system of equations is differential with respect to one independent variable and differential-delay with respect to a second independent variable. We use asymptotic analysis to consider the long wavelength limits of the system. By considering various magnitudes for the parameters involved, we derive reduced equations related to the Korteweg-de Vries and potential Boussinesq equations

The second Painlevé equation, a related nonautonomous semidiscrete equation, and a limit to the first Painlevé equation: scalar and matrix cases

In this paper we consider the matrix nonautonomous semidiscrete (or lattice) equation D dtUn = (2n − 1)(Un+1 − Un−1)−1, as well as the scalar case thereof. This equation was recently derived in the context of auto-Bäcklund transformations for a matrix partial differential equation. We use asymptotic techniques to reveal a connection between this equation and the matrix (or, as appropriate, scalar) first Painlevé equation. In the matrix case, we also discuss our asymptotic analysis more generally, as well as considering a component-wise approach. In addition, Hamiltonian formulations of the matrix first and second Painlevé equations are given, as well as a discussion of classes of solutions of the matrix second Painlevé equation

Integrability and asymptotic behaviour of a differential-difference matrix equation

In this paper we consider the matrix lattice equation U_{n,t} (U_{n+1} − U_{n−1} ) = g(n)I, in both its autonomous (g(n) = 2) and nonautonomous (g(n) = 2n − 1) forms. We show that each of these two matrix lattice equations are integrable. In addition, we explore the construction of Miura maps which relate these two lattice equations, via intermediate equations, to matrix analogs of autonomous and nonautonomous Volterra equations, but in two matrix dependent variables. For these last systems, we consider cases where the dependent variables belong to certain special classes of matrices, and obtain integrable coupled systems of autonomous and nonautonomous lattice equations and corresponding Miura maps. Moreover, in the nonautonomous case we present a new integrable nonautonomous matrix Volterra equation, along with its Lax pair. Asymptotic reductions to the matrix potential Korteweg-de Vries and matrix Korteweg-de Vries equations are also given

Mappings preserving locations of movable poles: a new extension of the truncation method to ordinary differential equations

The truncation method is a collective name for techniques that arise from truncating a Laurent series expansion (with leading term) of generic solutions of nonlinear partial differential equations (PDEs). Despite its utility in finding Backlund transformations and other remarkable properties of integrable PDEs, it has not been generally extended to ordinary differential equations (ODEs). Here we give a new general method that provides such an extension and show how to apply it to the classical nonlinear ODEs called the Painleve equations. Our main new idea is to consider mappings that preserve the locations of a natural subset of the movable poles admitted by the equation. In this way we are able to recover all known fundamental Backlund transformations for the equations considered. We are also able to derive Backlund transformations onto other ODEs in the Painleve classification.Comment: To appear in Nonlinearity (22 pages

Hyperoxemia and excess oxygen use in early acute respiratory distress syndrome : Insights from the LUNG SAFE study

Publisher Copyright: © 2020 The Author(s). Copyright: Copyright 2020 Elsevier B.V., All rights reserved.Background: Concerns exist regarding the prevalence and impact of unnecessary oxygen use in patients with acute respiratory distress syndrome (ARDS). We examined this issue in patients with ARDS enrolled in the Large observational study to UNderstand the Global impact of Severe Acute respiratory FailurE (LUNG SAFE) study. Methods: In this secondary analysis of the LUNG SAFE study, we wished to determine the prevalence and the outcomes associated with hyperoxemia on day 1, sustained hyperoxemia, and excessive oxygen use in patients with early ARDS. Patients who fulfilled criteria of ARDS on day 1 and day 2 of acute hypoxemic respiratory failure were categorized based on the presence of hyperoxemia (PaO2 > 100 mmHg) on day 1, sustained (i.e., present on day 1 and day 2) hyperoxemia, or excessive oxygen use (FIO2 ≥ 0.60 during hyperoxemia). Results: Of 2005 patients that met the inclusion criteria, 131 (6.5%) were hypoxemic (PaO2 < 55 mmHg), 607 (30%) had hyperoxemia on day 1, and 250 (12%) had sustained hyperoxemia. Excess FIO2 use occurred in 400 (66%) out of 607 patients with hyperoxemia. Excess FIO2 use decreased from day 1 to day 2 of ARDS, with most hyperoxemic patients on day 2 receiving relatively low FIO2. Multivariate analyses found no independent relationship between day 1 hyperoxemia, sustained hyperoxemia, or excess FIO2 use and adverse clinical outcomes. Mortality was 42% in patients with excess FIO2 use, compared to 39% in a propensity-matched sample of normoxemic (PaO2 55-100 mmHg) patients (P = 0.47). Conclusions: Hyperoxemia and excess oxygen use are both prevalent in early ARDS but are most often non-sustained. No relationship was found between hyperoxemia or excessive oxygen use and patient outcome in this cohort. Trial registration: LUNG-SAFE is registered with ClinicalTrials.gov, NCT02010073publishersversionPeer reviewe

Behaviour of the extended Volterra lattice

We investigate the behaviour of solutions of the recently proposed extended Volterra lattice. A variety of methods are used to determine the effects of the new terms on small amplitude equations, and, following approximation of the partial differential delay equations by PDEs we also determine similarity reductions

Multicomponent equations associated to non-isospectral scattering problems

We consider a non-isospectral scattering problem having as its spatial part an energy-dependent Schrodinger operator. This gives rise to new completely integrable multicomponent systems of equations in (2 + 1) dimensions. Their reductions to systems in (1 + 1) dimensions have isospectral scattering problems and include multicomponent extensions of the AKNS equation and also a generalization of the Dym equation. An extension of the Fuchssteiner-Fokas-Camassa-Holm equation to (2 + 1) dimensions is also presented