ABSTRACT:

We consider (some generalizations of) the Frenkel-Kontorova and the Ginzburg-Landau-Allen-Cahn models in a (generalized) periodic setting.
We discuss the existence of minimal laminations and of non-minimal solutions which possess a (generalized) rotation vector.
The symmetry and rigidity properties of entire monotone solutions of phase transition PDEs will be also considered, with extensions to fractional operators.