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

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 Bäcklund transformation between two given equations. The algebras F(E) are defined in [arXiv:1303.3575] in terms of generators and relations.In this preprint we show that, using the algebras F(E), one obtains some necessary conditions for integrability of the considered PDEs, where integrability is understood in the sense of soliton theory. Using these conditions, we prove non-integrability for some scalar evolution equations 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.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.

Lie algebras responsible for zero-curvature representations of scalar evolution equations

“…Zero-curvature representations and Bäcklund transformations belong to the main tools in the theory of integrable PDEs (see, e.g., [5,22,30]). This paper along with [10,11,12] is part of a research program on investigating the structure of zero-curvature representations (ZCRs) for partial differential equations (PDEs) of various types. The study of ZCRs performed in this paper leads to some results on Bäcklund transformations and integrability, which are described in [11,12].…”

“…The number d ≥ 1 in (1) is such that the function F may depend only on x, t, u k for k ≤ d. The symbol Z ≥0 denotes the set of nonnegative integers. Methods of this paper can also be applied to (1+1)-dimensional multicomponent evolution PDEs, see [10].…”

“…As we show in the preprints [12,11], using F p (E, a), one obtains some necessary conditions for integrability of equations (1) and necessary conditions for existence of a Bäcklund transformation between two given equations. To get such results, one needs to study certain properties of ZCRs (3), (4) with arbitrary p, and we do this by means of the algebras F p (E, a).…”

“…Applications of F p (E, a) to obtaining necessary conditions for integrability of equations (1) are presented in [12]. Examples of the use of these conditions in proving non-integrability for some equations of order 5 are presented in [12] as well. Applications to obtaining non-existence results for Bäcklund transformations between two given equations are described in [11].…”

“…In this paper and in [12,11] we present also a number of results on the structure of the algebras F p (E, a) for some classes of scalar evolution equations of orders 3, 5, 7 and concrete examples. In particular, the KdV equation is considered in Theorem 7 in this paper.…”