On weakly nonlinear modulation of waves on deep water

We propose a new approach for modeling weakly nonlinear waves, based on enh ancing truncated amplitude equations with exact linear dispersion. Our exam ple is based on the nonlinear Schrodinger (NLS) equation for deep-water wav es. The enhanced NLS equation reproduces exactly the conditions for nonline ar four-wave resonance (the "figure 8" of Phillips) even for bandwidths gre ater than unity. Sideband instability for uniform Stokes waves is limited t o finite bandwidths only, and agrees well with exact results of McLean; the refore, sideband instability cannot produce energy leakage to high-wave-num ber modes for the enhanced equation, as reported previously for the NLS equ ation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equa tion. Being solvable numerically at no additional cost in comparison with t he NLS equation, the new model is physically and numerically attractive for investigation of wave evolution. (C) 2000 American Institute of Physics. [ S1070-6631(00)50010-5].