Non-linear propagation characteristics of Bleustein-Gulyaev waves

A pair of semi-linear partial differential equations governing the slow variation in the fundamental and the third harmonic amplitudes of a quasi-monochromatic finite-amplitude Bleustein-Gulyaev (BG) wave on a crystal belonging to either the 6 mm or the 4 mm symmetry class is derived by using an extension of the method of multiple scales. The analysis of the exact solution of these equations in terms of Jacobian elliptic functions reveals the existence of growth-decay cycles in the amplitude variation of the various harmonics, as in the case of non-linear Rayleigh waves.