Lie algebras responsible for zero-curvature representations of scalar evolution 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. For any (1+1)-dimensional scalar evolution equation $E$, we define 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. To achieve this, we find a normal form for ZCRs with respect to the action of the group of local gauge transformations. As we show in other publications, using these algebras, one obtains some necessary conditions for integrability of the considered PDEs (where integrability is understood in the sense of soliton theory) and necessary conditions for existence of a Bäcklund transformation between two given equations. Examples of proving non-integrability and applications to obtaining non-existence results for Bäcklund transformations are presented in other publications as well. In our 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. In this paper we describe general properties of $F(E)$ and present generators and relations for these algebras. In other publications we study the structure of $F(E)$ for equations of KdV, Krichever-Novikov, Kaup-Kupershmidt, Sawada-Kotera types. Among the obtained algebras, one finds infinite-dimensional Lie algebras of certain matrix-valued functions on rational and elliptic algebraic curves.

23 pages; v4: some results have been moved to other preprints

References in corpus (5)

• Integrable evolution equations with constant separant
• Cyclic bases of zero-curvature representations: five illustrations to one concept
• Infinite-dimensional prolongation Lie algebras and multicomponent Landau-Lifshitz systems associated with higher genus curves
• On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs
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Cited by in corpus (6)

• On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs
• Higher jet prolongation Lie algebras and Backlund transformations for (1+1)-dimensional PDEs
• Particle-like, dyx-coaxial and trix-coaxial Lie algebra structures for a multi-dimensional continuous Toda type system
• Simplifications of Lax pairs for differential-difference equations by gauge transformations and (doubly) modified integrable equations
• On Lie algebras responsible for zero-curvature representations and Backlund transformations of (1+1)-dimensional scalar evolution PDEs
• On Lie algebras responsible for zero-curvature representations of multicomponent (1+1)-dimensional evolution PDEs