On some algebraic properties of the vector mKdV and vector sG equation

George Papamikos, University of Leeds

We present the construction of the vector modified KdV (vmKdV) hierarchy using formal Darboux transformations. This construction, due to Drinfeld and Sokolov, is completely algebraic and algorithmic. We use the algebraic properties of the Lax representation of the hierarchy in order to construct an exact recursion operator for the hierarchy. Moreover, we prove a loop group factorisation of the formal Darboux transformation which leads to the construction of a generating function for the conserved densities of the vmKdV. We also sketch the construction of Darboux transformations in closed form and their use in obtaining soliton solutions for the whole hierarchy.

Next we will consider the vector sine-Gordon equation (vsG) as an inverse flow of the vmKdV hierarchy. We present its Darboux-Baclund transformation and two related integrable differential difference equations which are related to each other via a discrete Miura transformation. Finally, considering the refactorisation problem of the product of two Darboux matrices we construct a related Yang-Baxter map and a discrete vector sine-Gordon equation.