Routes to stability for spatially periodic breather solutions of a damped NLS equation.

 The spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger (NLS) equation, i.e. the heteroclinic orbits of unstable Stokes waves, are typically unstable. In this talk  we examine  the effects of dissipation on the  one- mode SPBs  U^(j)(x,t) as well as multi-mode SPBs U^(j,k)(x,t) using a damped  NLS equation which incorporates both uniform linear damping and nonlinear damping  of the mean flow,
for a range of parameters typically encountered in experiments. The damped wave dynamics is viewed as near integrable, allowing one to use the spectral theory of the NLS equation to interpret the perturbed flow. A broad categorization of how the route to stability for the SPBs  depends on the mode structure of the SPB and whether the damping is linear or nonlinear is obtained 
as well as the distinguishing features of the stabilized state.  Time permitting, a reduced, finite dimensional dynamical system that goverms the linearly damped SPBs will be presented