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 $\mu<\mu_{1}\approx0.274$. 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 $\mu$, and it vanishes at $\mu_{1}$. 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 $\mu\ll\mu_{1}$, in agreement with recent experimental and numerical observations of breathers.