Stability of plane waves on deep water with dissipation

Abstract

The Benjamin-Feir modulational instability effects the evolution of perturbed planewave solutions of the cubic nonlinear Schrödinger equation (NLS), the modified NLS, and the band-modified NLS. Recent work demonstrates that the BenjaminFeir instability in NLS is "stabilized" when a linear term representing dissipation is added. In this paper, we add a linear term representing dissipation to the modified NLS and band-modified NLS equations and establish that the plane-wave solutions of these equations are linearly stable. Although the plane-wave solutions are stable, some perturbations grow for a finite period of time. We analytically bound this growth and present approximate time-dependent regions of wave-number space that correspond to perturbations that have increasing amplitudes.

On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations

“…This paper also contained the results of some experiments backing up these findings. Their results were extended by Canney & Carter [19] who generalised (1.8) by including higher order terms up to and including the fifth derivative (ie ones like XXXYY A ). They performed a stability analysis and their conclusions agreed with the earlier ones that dissipation stabilizes the waves.…”

Packet of gravity surface waves at high Reynolds numbers

“…To allow for viscosity it is conventional-purely phenomenologically, that is, from general considerations,-to add a linear-in-amplitude term in these evolutionary equations [3,4]. The conditions, at which the introduction of an additional term into these equations may be correct, are not discussed.…”