The wave equation for elastic waves in piezoelectric crystals is found, including the third-order terms in the expansion of the thermodynamic potential in the strain and electric field. The effective nonlinear parameters for a general crystal are obtained. In the general case these parameters depend on the second-order and third-order coefficients: nonlinear piezoelectric, electrostrictive, and linear electro-optic. These coefficients are considered for piezoelectric and nonpiezoelectric crystals. Some estimates are given of the values of the nonlinear coefficients.
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V. E. Ljamov (1972) studied this question.