Symmetries: a path from discrete to continuous integrable systems

Abstract: Symmetries provide useful tools to study and classify differential and discrete equations, as well as to construct solutions for these equations. A new and interesting application of symmetries is that they provide a link between discrete and differential equations. Specifically, one can employ the symmetries of the former in order to derive systems of differential equations. In particular, the discrete potential KdV equation and its symmetries will be used as an illustrative example to present this derivation. It will be shown that integrability aspects, like multidimensional consistency and Bäcklund transformation, are inherited to the resulting system of differential equations by its discrete counterpart. Finally, this analysis will be extended to the class of equations to which the discrete potential KdV belongs.