On the persistence of breathers at deep water

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 < μ 1 ≈ 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 μ 1 . 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 ≪ μ 1 , in agreement with recent experimental and numerical observations of breathers.