A STRUCTURAL VIEWPOINT ON THE SINE-GORDON EQUATION IN INCOMMENSURATE PHASES

The spatially inhomogeneous order parameter considered in the Landau description of incommensurate phases is analysed from a structural viewpoint using the superspace approach. All modulation functions, including that of the order parameter, are defined along the internal space. Taking a displacive case for concreteness and within the constant-amplitude approximation, a general differential equation for the order parameter phase, theta, which generalizes the well known sine-Gordon equation, is derived. No reference is made to any particular expression for the free-energy expansion; only consistency arguments between the Landau description and the structural properties of an incommensurate phase are used. It is shown that, under certain quite common conditions, this general equation can be approximately reduced to the sine-Gordon equation. The sine-Gordon equation can, therefore, be considered of rather general validity when modelling the structure of incommensurate phases irrespective of their particular thermodynamic potential. The introduction of the theta modulation along the internal space further simplifies the equation; its parameters become system independent, being a function of only the soliton density.