The Effects of Viscosity on the Linear Stability of Damped Stokes Waves, Downshifting, and Rogue Wave Generation

We investigate a higher order nonlinear Schr\"odinger equation with linear damping and weak viscosity, recently proposed as a model for deep water waves exhibiting frequency downshifting. Through analysis and numerical simulations, we discuss how the viscosity affects the linear stability of the Stokes wave solution, enhances rogue wave formation, and leads to permanent downshift in the spectral peak. The novel results in this work include the analysis of the transition from the initial Benjamin-Feir instability to a predominantly oscillatory behavior, which takes place in a time interval when most rogue wave activity occurs. In addition, we propose new criteria for downshifting in the spectral peak and determine the relation between the time of permanent downshift and the location of the global minimum of the momentum and the magnitude of its second derivative.