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 $\mu<\mu_{1}\approx0.274$. 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.
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