An integrable deformation of the sine(sinh)-Gordon model -- Malcev algebra

Abstract

In this paper we present an integrable model in two dimensions. It is a deformation of the sine(sinh)-Gordon model. We give its Lax connection. We also obtain its (classical) $r$-matrix. It satisfies the classical Yang-Baxter equation. Thus, the model is a classical integrable field theory in two dimensions. The underlying algebra is a ${\cal Z}$-graded non-Lie Malcev algebra. It is a direct sum of $sl(2, {\rm I\!R})$ Lie algebras with a shared common Cartan. A Malcev algebra is the tangent space at the identity of an analytic Moufang loop as a Lie algebra is the tangent space at the identity of a Lie group. We expect the model to be integrable at the quantum level. We also give a family of classically integrable models which are related to the Poisson-Boltzmann equation in two dimensions.