  A group-theoretical approach for studying localized periodic and
quasiperiodic vibrations in 2D and 3D lattice dynamical models is developed.
This approach is demonstrated for the scalar models on the plane square
lattice. The symmetry-determined invariant manifolds admitting existence of
localized vibrations are found and some types of discrete breathers are
constructed on these manifolds. A general method using the apparatus of matrix
representations of symmetry groups to simplify the standard linear stability
analysis is discussed. This method allows one to decompose the corresponding
system of linear differential equations with time-dependent coefficients into a
number of independent subsystems whose dimensions are less than the full
dimension of the considered system.

Pattern Formation and Solitons