Abstract
From Zakharov’s integral equation a nonlinear evolution equation for broader bandwidth gravity waves in deep water is obtained, which is one order higher than the corresponding equation derived by Trulsen and Dysthe [“A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water,” Wave Motion 24, 281 (1996)]. The instability regions in the perturbed wave-number space for a uniform Stokes wave obtained from this equation are in surprisingly good agreement with the exact results obtained by McLean et al. [“Three dimensional instability of finite amplitude water waves,” Phys. Rev. Lett. 46, 817 (1981)].
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