Exact periodic solutions are found for three-dimensional lattices with fourth-order on-site electron-ion potentials. Numerical results are presented for ferroelectric SnTe. The coupling of these nonlinear waves with phonons is investigated for phase transitions with finite critical wave vectors. For K₂{SeO}₄$ both the paraelectric-incommensurate and the incommensurate-commensurate ferroelectric transitions are analyzed. The cubic power of the nonlinear force leads to a lock-in transition at the wave vector 2π3.
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Bilz et al. (1982) studied this question.
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