Miura transformations and the various guises of integrable lattice equations

Abstract

We present various Miura-type transformations that exist between integrable lattice equations, which lead to some new and quite unexpected relations between these lattice equations. In particular, we show that in the discrete case, contrary to the continuous one, the sine-Gordon and mKdV equations are essentially the same. We also examine two new equations recently proposed by Hydon and Viallet and show that they can be transformed to the discrete mKdV and/or sine-Gordon equations.