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

Author

Listed:
  • Craig, Walter
  • Henderson, Diane M.
  • Oscamou, Maribeth
  • Segur, Harvey

Abstract

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.

Suggested Citation

  • Craig, Walter & Henderson, Diane M. & Oscamou, Maribeth & Segur, Harvey, 2007. "Stable three-dimensional waves of nearly permanent form on deep water," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 74(2), pages 135-144.
    • Handle: RePEc:eee:matcom:v:74:y:2007:i:2:p:135-144
    DOI: 10.1016/j.matcom.2006.10.032
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    Cited by:

    1. Carter, John D., 2012. "Plane-wave solutions of a dissipative generalization of the vector nonlinear Schrödinger equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(6), pages 1038-1046.

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