The Generalised mKdV Equations for Level -3 of $\hat{sl}_2$

Abstract

A certain generalisation of the hierarchy of mKdV equations (modified KdV), which forms an integrable system, is studied here. This generalisation is based on a Lax operator associated to the equations, with principal components of degrees between -3 and 0. The results are the following ones: 1) an isomorphism between the space of jets of the system and a quotient of ${Sl}_2({\CC}((t)))$; 2) the fact that the monodromy matrixes of the Lax operators have, morover, Poisson brackets given by the trigonometric r-matrix; 3) a definition of the action of screening operators on the densities; 4) an identification of the intersection of the kernel with the integrals of motion.