Examinando por Autor "Pickering, Andrew"

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Bäcklund transformations for a new extended Painlevé hierarchy
(Elsevier, 2019-04) Gordoa, Pilar R; Pickering, Andrew
In a recent paper we introduced an extended second Painlevé hierarchy and studied its properties. The approach developed in order to derive these results is widely applicable. Here we use it to obtain a second example of an extended Painlevé hierarchy. We also give results on Bäcklund transformations, auto-Bäcklund transformations and other properties of this and related hierarchies of ordinary differential equations, as well as on the nesting of equations whereby we obtain relations between systems of different orders but of the same form.
Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations
(Elsevier, 2020) Hernández-Bermejo, Benito; Iagar, Razvan G; Gordoa, Pilar R; Pickering, Andrew; Sánchez, Ariel
In this work, we construct a transformation between the solutions to the following reaction-convection-diffusion equation u_t = (u^m)_{xx} + a(x)(u^m)_x + b(x)u^m, posed for x ∈ R, t ≥ 0 and m > 1, where a, b are two continuous real functions, and the solutions to the nonhomogeneous diffusion equation of porous medium type f(y)θ_τ = (θ^m)_{yy}, posed in the half-line y ∈ [0, ∞) with τ ≥ 0, m > 1 and suitable density functions f(y). We apply this correspondence to the case of constant coefficients a(x) = 1 and b(x) = K > 0. For this case, we prove that compactly supported solutions to the first equation blow up in finite time, together with their interfaces, as x → −∞. We then establish the large time behavior of solutions to a homogeneous Dirichlet problem associated to the first equation on a bounded interval. We also prove a finite time blow-up of the interfaces for compactly supported solutions to the second equation when f(y) = y^{−γ} with γ > 2.
Integrability and asymptotic behaviour of a differential-difference matrix equation
(Elsevier, 2021) Gordoa, Pilar R; Pickering, Andrew; Wattis, Jonathan A D
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.
Novel Bäcklund Transformations for Integrable Equations
(MDPI, 2022) Gordoa, Pilar R; Pickering, Andrew
In this paper, we construct a new matrix partial differential equation having a structure and properties which mirror those of a matrix fourth Painlevé equation recently derived by the current authors. In particular, we show that this matrix equation admits an auto-Bäcklund transformation analogous to that of this matrix fourth Painlevé equation. Such auto-Bäcklund transformations, in appearance similar to those for Painlevé equations, are quite novel, having been little studied in the case of partial differential equations. Our work here shows the importance of the underlying structure of differential equations, whether ordinary or partial, in the derivation of such results. The starting point for the results in this paper is the construction of a new completely integrable equation, namely, an inverse matrix dispersive water wave equation.
On matrix fourth Painlevé hierarchies
(Elsevier, 2021) Gordoa, Pilar R; Pickering, Andrew
We define new matrix fourth Painlevé hierarchies and a new matrix second Painlevé hierarchy. For our matrix fourth Painlevé hierarchies, properties such as auto-Bäcklund transformations, Bäcklund transfor mations and special integrals are derived, and known relations between fourth Painlevé transformations are extended to the matrix ODE hierarchies. The matrix ODE hierarchies presented here are derived as reductions of a new nonisospectral matrix dispersive water wave hierarchy and modifications thereof.
Solution classes of the matrix second Painlevé hierarchy
(Elsevier, 2022) Gordoa, Pilar R; Pickering, Andrew; Wattis, Jonathan A D
We explore the generation of classes of solutions of the matrix second Painlevé hierarchy. This involves the consideration of the application of compositions of auto-Bäcklund transformations to different initial solutions, with the number of distinct solutions obtained for each value of the parameter appearing in the hierarchy depending on the symmetry properties of the chosen initial solution. This paper not only extends our previous results for the matrix second Painlevé equation itself, given in a recent paper, to the matrix second Painlevé hierarchy, but also provides a more detailed account of the underlying process.
The second Painlevé equation, a related nonautonomous semidiscrete equation, and a limit to the first Painlevé equation: Scalar and matrix cases
(Elsevier, 2019) Pickering, Andrew; Gordoa, Pilar R; Wattis, Jonathan A D
In this paper we consider the matrix nonautonomous semidiscrete (or lattice) equation U_{n,t} = (2n − 1)(U_{n+1} − U_{n−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.
Ultrasonic Waves in Bubbly Liquids: An Analytic Approach
(MDPI, 2021) Gordoa, Pilar R; Pickering, Andrew
We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instan taneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.