Stability of plane waves on deep water with dissipation

Nathan E. Canney and John D. Carter

Mathematics and Computers in Simulation (MATCOM), 2007, vol. 74, issue 2, 159-167

Abstract: The Benjamin–Feir modulational instability effects the evolution of perturbed plane-wave solutions of the cubic nonlinear Schrödinger equation (NLS), the modified NLS, and the band-modified NLS. Recent work demonstrates that the Benjamin–Feir 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.

Keywords: Benjamin–Feir; Dissipation; Plane waves; NLS; Stability (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:74:y:2007:i:2:p:159-167

DOI: 10.1016/j.matcom.2006.10.010

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