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The unified solution of the incommensurate phase in ferroelectrics

We derive a unified multi-harmonic modulation solution of the incommensurate phase with a simple model by a method like that used in solving pendulum problems in mechanics. This solution reduces to single-harmonic modulation at the upper end of temperatures, and transforms into kink solution at the lower end of temperatures, The basic wave-vector decreases monotonically with temperature, as was observed in experimentation.