Help Privacy Policy Disclaimer
  Advanced SearchBrowse




Journal Article

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


Igonin,  Sergei Aleksandrovich
Max Planck Institute for Mathematics, Max Planck Society;

External Resource
Fulltext (restricted access)
There are currently no full texts shared for your IP range.
Fulltext (public)

(Preprint), 503KB

Supplementary Material (public)
There is no public supplementary material available

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
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.