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  On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs

Igonin, S. A., & Manno, G. (2020). On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs. Journal of Geometry and Physics, 150: 103596. doi:10.1016/j.geomphys.2020.103596.

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Igonin, Sergei Aleksandrovich1, Author              
Manno, Gianni, Author
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1Max Planck Institute for Mathematics, Max Planck Society, ou_3029201              

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Free keywords: Nonlinear Sciences, Exactly Solvable and Integrable Systems, Mathematical Physics, Mathematics, Rings and Algebras
 Abstract: Zero-curvature representations (ZCRs) are one of the main tools in the theory of integrable PDEs. In particular, Lax pairs for (1+1)-dimensional PDEs can be interpreted as ZCRs. In [arXiv:1303.3575], for any (1+1)-dimensional scalar evolution equation $E$, we defined a family of Lie algebras $F(E)$ which are responsible for all ZCRs of $E$ in the following sense. Representations of the algebras $F(E)$ classify all ZCRs of the equation $E$ up to local gauge transformations. In [arXiv:1804.04652] we showed that, using these algebras, one obtains necessary conditions for existence of a B\"acklund transformation between two given equations. The algebras $F(E)$ are defined in terms of generators and relations. In this paper we show that, using the algebras $F(E)$, one obtains some necessary conditions for integrability of (1+1)-dimensional scalar evolution PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution PDEs of order $5$. Also, we prove a result announced in [arXiv:1303.3575] on the structure of the algebras $F(E)$ for certain classes of equations of orders $3$, $5$, $7$, which include KdV, mKdV, Kaup-Kupershmidt, Sawada-Kotera type equations. Among the obtained algebras for equations considered in this paper and in [arXiv:1804.04652], one finds infinite-dimensional Lie algebras of certain polynomial matrix-valued functions on affine algebraic curves of genus $1$ and $0$. In this approach, ZCRs may depend on partial derivatives of arbitrary order, which may be higher than the order of the equation $E$. The algebras $F(E)$ generalize Wahlquist-Estabrook prolongation algebras, which are responsible for a much smaller class of ZCRs.

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Language(s): eng - English
 Dates: 2020
 Publication Status: Published in print
 Pages: 24
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 Table of Contents: -
 Rev. Type: Peer
 Identifiers: arXiv: 1810.09280
DOI: 10.1016/j.geomphys.2020.103596
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Title: Journal of Geometry and Physics
  Abbreviation : J. Geom. Phys.
Source Genre: Journal
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Publ. Info: Elsevier
Pages: - Volume / Issue: 150 Sequence Number: 103596 Start / End Page: - Identifier: -