Negative even grade mKdV hierarchy and its soliton solutions

In this paper, we provide an algebraic construction for the negative even mKdV hierarchy which gives rise to time evolutions associated with even graded Lie algebraic structures. We propose a modification of the dressing method, in order to incorporate a non-trivial vacuum configuration and construct a deformed vertex operator for sl-hat(2) that enable us to obtain explicit and systematic solutions for the whole negative even grade equations.