Solution classes of the matrix second Painlevé hierarchy

Gordoa, Pilar R.; Pickering, Andrew; Wattis, Jonathan A. D.

doi:10.1016/j.physd.2022.133295

Cited by 5 publications

(1 citation statement)

References 57 publications

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“…We will see in the next section that the system (34)-(37) admits a reduction to a matrix fourth Painlevé system given in [1] (a system of matrix ordinary differential equations), which would then provide one way of obtaining solutions of (34)-(37). However, the question of solutions of this matrix fourth Painlevé system has yet to be explored, and is beyond the scope of the present paper (as an example of the consideration of solution classes of matrix ordinary differential equations, we refer to [19]). In the case of the scalar reduction, it can be shown that our auto-Bäcklund transformation (38)-( 43) can allow a new solution to be obtained from a given initial solution.…”

Section: Auto-bäcklund Transformationsmentioning

confidence: 99%

Novel Bäcklund Transformations for Integrable Equations

Gordoa

Pickering

2022

Mathematics

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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.

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Section: Auto-bäcklund Transformationsmentioning

confidence: 99%

Novel Bäcklund Transformations for Integrable Equations

Gordoa

Pickering

2022

Mathematics

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The extended second Painlevé hierarchy: Auto-Bäcklund transformations and special integrals

Gordoa,

Pickering

2025

Journal of Differential Equations

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Higher-order nonlinear special functions: Painlevé hierarchies, a survey

Gordoa,

Pickering

2024

Contemporary Mathematics

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The six Painlevé transcendents are widely accepted as nonlinear special functions. Over the last quarter of a century or so, there has been a surge of interest in higher-order analogues of the Painlevé equations, most often defined as members of hierarchies of equations of increasing order, i.e., of so-called Painlevé hierarchies. We give here a survey of such Painlevé hierarchies, including of their derivation and the derivation of their properties. Amongst other aspects, we discuss the relationships between the properties of completely integrable hierarchies, e.g., Hamiltonian structures and Miura maps, nonisospectral scattering problems, and those of Painlevé hierarchies, e.g., Lax pairs, Bäcklund and auto-Bäcklund transformations, and sequences of special solutions. Given the large number of papers published on Painlevé hierarchies, we hope this review will serve as a useful future reference.

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Solution classes of the matrix second Painlevé hierarchy

Cited by 5 publications

References 57 publications

Novel Bäcklund Transformations for Integrable Equations

Novel Bäcklund Transformations for Integrable Equations

The extended second Painlevé hierarchy: Auto-Bäcklund transformations and special integrals

Higher-order nonlinear special functions: Painlevé hierarchies, a survey

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