The authors analyze the statistical properties of a liquid-liquid interface in a Couette flow for a wide range of shear rates. This study reveals strong similarities with Burgers turbulence. However, their relaxation equation is derived for overdamped fluctuations in the limit of vanishing Reynolds number, whereas turbulence and the Burgers equation are studied in the limit of high Reynolds number. The statistics are Gaussian at low shear. Then, above a critical shear rate a transition to a nonlinear phenomenology is observed by the authors. An energy criterion allows them to predict the onset of non-Gaussian statistics. It also provides an explanation for the development of shock singularities through the exchange of kinetic energy from regions with positive to negative gradients in the shear direction.