Theory of real supersolids

We review the main properties of a supersolid. We describe first the macroscopic equation that satisfies a supersolid based on general arguments and symmetries and show that such solids might exhibit simultaneously or independently both elastic behavior and superfluidity. We then explain why a supersolid state should exist for solids at very low temperature but with a very small superfluid fraction. Finally, we propose a mean-field model, based on the Gross-Pitaevski\u{\i} equation, which presents the general properties expected for a supersolid and should therefore provide a consistent framework to study its dynamical properties.