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

##### External Resource

https://doi.org/10.1016/j.geomphys.2020.103596

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##### Fulltext (public)

arXiv:1810.09280.pdf

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##### Citation

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.

Cite as: https://hdl.handle.net/21.11116/0000-0007-CD28-6

##### 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.

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.