We investigated theoretically the nonlinear phenomena (a self-focusing, a self-trapping, stimulated light scattering and four-wave mixing) in smectic A liquid crystals (SA) caused by a new physical mechanism of cubic nonlinearity determined by the propagating mode of a layer normal displacement, or the so-called second sound. A nonlinear part of a refractive index caused by a layer normal deformation which is responsible for a self-focusing effect, and a gain coefficient in the case of a stimulated light scattering (SLS) on the second sound are one or two orders of magnitude greater than the similar quantities for isotropic organic liquids. The light-induced dynamic grating of layer deformations generates a high-frequency longitudinal electric field due to the flexoelectric effect. The light-induced dynamic grating of layer normal displacement ultimately excites a rotational hydrodynamic how with a considerable velocity. Using the slow varying amplitude approximation we calculated the light wave and second sound wave amplitudes explicitly for some important cases.