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

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