  In this paper we address the stability of resonantly forced density waves in
dense planetary rings.
  Already by Goldreich & Tremaine (1978) it has been argued that density waves
might be unstable, depending on the relationship between the ring's viscosity
and the surface mass density.
  In the recent paper Schmidt et al. (2016) we have pointed out that when -
within a fluid description of the ring dynamics - the criterion for viscous
overstability is satisfied, forced spiral density waves become unstable as
well.
  In this case, linear theory fails to describe the damping, but nonlinearity
of the underlying equations guarantees a finite amplitude and eventually a
damping of the wave.
  We apply the multiple scale formalism to derive a weakly nonlinear damping
relation from a hydrodynamical model.
  This relation describes the resonant excitation and nonlinear viscous damping
of spiral density waves in a vertically integrated fluid disk with density
dependent transport coefficients.
  The model consistently predicts density waves to be (linearly) unstable in a
ring region where the conditions for viscous overstability are met.
  Sufficiently far away from the Lindblad resonance, the surface mass density
perturbation is predicted to saturate to a constant value due to nonlinear
viscous damping.
  The wave's damping lengths of the model depend on certain input parameters,
such as the distance to the threshold for viscous overstability in parameter
space and the ground state surface mass density.
