  Supersolids are theoretically predicted quantum states that break the
continuous rotational and translational symmetries of liquids while preserving
superfluid transport properties. Over the last decade, much progress has been
made in understanding and characterizing supersolid phases through numerical
simulations for specific interaction potentials. The formulation of an
analytically tractable framework for generic interactions still poses
theoretical challenges. By going beyond the usually considered quadratic
truncations, we derive a systematic higher-order generalization of the
Gross-Pitaevskii mean field model in conceptual similarity with the
Swift-Hohenberg theory of pattern formation. We demonstrate the tractability of
this broadly applicable approach by determining the ground state phase diagram
and the dispersion relations for the supersolid lattice vibrations in terms of
the potential parameters. Our analytical predictions agree well with numerical
results from direct hydrodynamic simulations and earlier quantum Monte-Carlo
studies. The underlying framework is universal and can be extended to
anisotropic pair potentials with complex Fourier-space structure.
