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

TL;DR

This paper studies how viscosity influences the stability and rogue wave formation in a nonlinear Schrödinger model for deep water waves, revealing mechanisms for spectral downshifting and instability transitions.

Contribution

It provides new analysis of viscosity effects on wave stability, rogue wave enhancement, and spectral downshifting in a higher order nonlinear Schrödinger equation.

Findings

Viscosity affects the linear stability of Stokes waves.

Enhanced rogue wave activity due to viscosity.

New criteria for spectral downshifting and its relation to wave momentum.

Abstract

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.