The potential form of the fourth-order evolution equation for deep-water gravity–capillary waves

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Abstract
The fourth‐order evolution equation for the wave envelope of a train of deep‐water gravity–capillary waves is derived in terms of the complex velocity potential Ψ(x,t). Despite a difference between Ψ(x,t) and the complex amplitude a(x,t) to this order, the evolution equations are curiously almost identical.

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