Examinando por Autor "Wattis, Jonathan A D"

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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.
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.