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

TL;DR

This paper investigates Lie algebras associated with zero-curvature representations of (1+1)-dimensional scalar evolution PDEs, deriving necessary conditions for integrability and analyzing algebra structures for well-known integrable equations.

Contribution

It introduces a framework using Lie algebras F(E) to determine integrability conditions and explores their structure for classical integrable equations like KdV and mKdV.

Findings

Derived necessary conditions for PDE integrability using F(E)

Proved non-integrability of certain fifth-order scalar PDEs

Analyzed algebra structures for classical integrable equations

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.