Incommensurate Phase Transitions under the Existence of the Lifshitz Invariant

The numerical solutions of the phase transitions under the existence of the Lifshitz invariant are presented. The temperature dependences of thermodynamic potential and dielectric susceptibility are shown for some values of coefficients appearing in a model thermodynamic potential. The feature of space modulation of order parameters is also shown. The results will be discussed in the relationship with some ferroelectric materials in which the incommensurate phase is actually realized in some temperature range.