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Keywords = Miura maps

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9 pages, 254 KiB  
Article
Novel Bäcklund Transformations for Integrable Equations
by Pilar Ruiz Gordoa and Andrew Pickering
Mathematics 2022, 10(19), 3565; https://doi.org/10.3390/math10193565 - 29 Sep 2022
Cited by 1 | Viewed by 1242
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Completely Integrable Equations: Algebraic Aspects and Applications)
8 pages, 221 KiB  
Article
Dispersionless BKP Equation, the Manakov–Santini System and Einstein–Weyl Structures
by Leonid V. Bogdanov
Symmetry 2021, 13(9), 1699; https://doi.org/10.3390/sym13091699 - 15 Sep 2021
Cited by 1 | Viewed by 2026
Abstract
We construct a map from solutions of the dispersionless BKP (dBKP) equation to solutions of the Manakov–Santini (MS) system. This map defines an Einstein–Weyl structure corresponding to the dBKP equation through the general Lorentzian Einstein–Weyl structure corresponding to the MS system. We give [...] Read more.
We construct a map from solutions of the dispersionless BKP (dBKP) equation to solutions of the Manakov–Santini (MS) system. This map defines an Einstein–Weyl structure corresponding to the dBKP equation through the general Lorentzian Einstein–Weyl structure corresponding to the MS system. We give a spectral characterisation of reduction in the MS system, which singles out the image of the dBKP equation solution, and also consider more general reductions of this class. We define the BMS system and extend the map defined above to the map (Miura transformation) of solutions of the BMS system to solutions of the MS system, thus obtaining an Einstein–Weyl structure for the BMS system. Full article
(This article belongs to the Special Issue Symmetry of Hamiltonian Systems: Classical and Quantum Aspects)
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