TY - JOUR

T1 - Stable three-dimensional waves of nearly permanent form on deep water

AU - Craig, Walter

AU - Henderson, Diane M.

AU - Oscamou, Maribeth

AU - Segur, Harvey

N1 - Funding Information:
We gratefully acknowledge financial support from the National Science Foundation (DMS-FRG 0139842, DMS-FRG 0139847 and DMS 9810751).

PY - 2007/3/7

Y1 - 2007/3/7

N2 - Consider a uniform train of surface waves with a two-dimensional, bi-periodic surface pattern, propagating on deep water. One approximate model of the evolution of these waves is a pair of coupled nonlinear Schrödinger equations, which neglects any dissipation of the waves. We show that in this model, such a wave train is linearly unstable to small perturbations in the initial data, because of a Benjamin-Feir-type instability. We also show that when the model of coupled equations is generalized to include appropriate wave damping, the corresponding wave train is linearly stable to perturbations in the initial data. Therefore, according to the damped model, the two-dimensional surface wave patterns studied by Hammack et al. [J.L. Hammack, D.M. Henderson, H. Segur, Progressive waves with persistent, two-dimensional surface patterns in deep water, J. Fluid Mech. 532 (2005) 1-51] are linearly stable in the presence of wave damping.

AB - Consider a uniform train of surface waves with a two-dimensional, bi-periodic surface pattern, propagating on deep water. One approximate model of the evolution of these waves is a pair of coupled nonlinear Schrödinger equations, which neglects any dissipation of the waves. We show that in this model, such a wave train is linearly unstable to small perturbations in the initial data, because of a Benjamin-Feir-type instability. We also show that when the model of coupled equations is generalized to include appropriate wave damping, the corresponding wave train is linearly stable to perturbations in the initial data. Therefore, according to the damped model, the two-dimensional surface wave patterns studied by Hammack et al. [J.L. Hammack, D.M. Henderson, H. Segur, Progressive waves with persistent, two-dimensional surface patterns in deep water, J. Fluid Mech. 532 (2005) 1-51] are linearly stable in the presence of wave damping.

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U2 - 10.1016/j.matcom.2006.10.032

DO - 10.1016/j.matcom.2006.10.032

M3 - Article

AN - SCOPUS:33846914485

VL - 74

SP - 135

EP - 144

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

SN - 0378-4754

IS - 2-3

ER -