Abstract
The long-time behavior of a perturbation to a uniform wavetrain of the compact Zakharov equation is studied near the modulational instability threshold. A multiple-scale analysis reveals that the perturbation evolves in accord with a focusing nonlinear Schrodinger equation for values of wave steepness μ < μ1 ≈ 0.274. The long-time dynamics is characterized by interacting breathers, homoclinic orbits to an unstable wavetrain. The associated Benjamin-Feir index is a decreasing function of μ, and it vanishes at μ1. Above this threshold, the perturbation dynamics is of defocusing type and breathers are suppressed. Thus, homoclinic orbits persist only for small values of wave steepness μ ≪ μ1, in agreement with recent experimental and numerical observations of breathers.
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Fedele, F. On the persistence of breathers at deep water. Jetp Lett. 98, 523–527 (2014). https://doi.org/10.1134/S0021364013220050
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DOI: https://doi.org/10.1134/S0021364013220050