We develop an analytic theory to describe spiral density waves propagating in a shearing disc in the weakly non-linear regime. Such waves are generically found to be excited in simulations of turbulent accretion discs, in particular if the said turbulence arises from the magneto-rotational instability (MRI). We derive a modified Burgers equation governing their dynamics, which includes the effects of non-linear steepening, dispersion and a bulk viscosity to support shocks. We solve this equation approximately to obtain non-linear sawtooth solutions that are asymptotically valid at late times. In this limit, the presence of shocks is found to cause the wave amplitude to decrease with time as t−2. The validity of the analytic description is confirmed by direct numerical solution of the full non-linear equations of motion. The asymptotic forms of the wave profiles of the state variables are also found to occur in MRI simulations, indicating that dissipation due to shocks plays a significant role apart from any effects arising from direct coupling to the turbulence.