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

The spatially inhomogeneous order parameter considered in the Landau d escription of incommensurate phases is analysed from a structural view point using the superspace approach. All modulation functions, includi ng that of the order parameter, are defined along the internal space. Taking a displacive case for concreteness and within the constant-ampl itude approximation, a general differential equation for the order par ameter phase, theta, which generalizes the well known sine-Gordon equa tion, is derived. No reference is made to any particular expression fo r the free-energy expansion; only consistency arguments between the La ndau description and the structural properties of an incommensurate ph ase 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 r ather general validity when modelling the structure of incommensurate phases irrespective of their particular thermodynamic potential. The i ntroduction of the theta modulation along the internal space further s implifies the equation; its parameters become system independent, bein g a function of only the soliton density.