Higher-order modulation effects on solitary wave envelopes in deep water

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Abstract
The envelope equation of Dysthe (1979), which provides an extension of the nonlinear Schrödinger equation (NLS) to fourth order in wave steepness, is used to discuss higher-order modulation effects on the long-time evolution of solitary wave envelopes in deep water. The Dysthe equation admits solitary-wave solutions, similar to those of the NLS. Using perturbation methods, it is shown that an initial disturbance in the form of a solitary wave group of the NLS evolves to a solitary wave of the Dysthe equation having lower peak amplitude and moving with higher speed than the original wave; the increase in wave speed is caused by a downshift in wavefrequency. Asymptotic expressions are derived for this amplitude decrease and frequency downshift, which are consistent with numerical and experimental results.
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