We demonstrate a variety of nonlinear phenomena at extremely low powers in cryogenic acoustic cavities fabricated from quartz material, which have not undergone any electrodiffusion processes. Nonlinear phenomena observed include lineshape discontinuities, power response discontinuities, quadrature oscillations, and self-induced transparency. These phenomena are attributed to nonlinear dissipation through a large number of randomly distributed heavy trapped ions, which would normally be removed by electrodiffusion. A simple mean-field model predicts most of the observed phenomena. In contrast to Duffing-like systems, this system shows an unusual mechanism of nonlinearity, which is not related to crystal anharmonicity. © 2013 AIP Publishing LLC.