We propose a new approach for modeling weakly nonlinear waves, based on enhancing truncated amplitude equations with exact linear dispersion. Our example is based on the nonlinear Schrödinger ͑NLS͒ equation for deep-water waves. The enhanced NLS equation reproduces exactly the conditions for nonlinear four-wave resonance ͑the ''figure 8'' of Phillips͒ even for bandwidths greater than unity. Sideband instability for uniform Stokes waves is limited to finite bandwidths only, and agrees well with exact results of McLean; therefore, sideband instability cannot produce energy leakage to high-wave-number modes for the enhanced equation, as reported previously for the NLS equation. The new equation is extractable from the Zakharov integral equation, and can be regarded as an intermediate between the latter and the NLS equation. Being solvable numerically at no additional cost in comparison with the NLS equation, the new model is physically and numerically attractive for investigation of wave evolution.

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Unstable thin liquid films on solid substrates dewet by hole nucleation on defects or by a linear surface instability (spinodal dewetting). A system with destabilizing short-range and stabilizing long-range molecular interactions is investigated. We show that, for a subrange within the linearly unstable film thickness range, nucleation determines the final structure, whereas spinodal dewetting is of negligible influence. The results are also applicable to the spinodal decomposition of binary mixtures.

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