Collective modes in uniaxial incommensurate-commensurate systems with   the real order parameter
  The basic Landau model for uniaxial systems of the II class is nonintegrable, and allows for various stable and metastable periodic configurations, beside that representing the uniform (or dimerized) ordering. In the present paper we complete the analysis of this model by performing the second order variational procedure, and formulating the combined Floquet-Bloch approach to the ensuing nonstandard linear eigenvalue problem. This approach enables an analytic derivation of some general conclusions on the stability of particular states, and on the nature of accompanied collective excitations. Furthermore, we calculate numerically the spectra of collective modes for all states participating in the phase diagram, and analyze critical properties of Goldstone modes at all second order and first order transitions between disordered, uniform and periodic states. In particular it is shown that the Goldstone mode softens as the underlying soliton lattice becomes more and more dilute. 
V. Dananic, A. Bjelis and M. Latkovic