  We theoretically and experimentally examine the effect of forcing and damping
on systems that can be described by the nonlinear Schr\"odinger equation
(NLSE), by making use of the phase-space predictions of the three-wave
truncation of the spectrum. In the latter, only the fundamental frequency and
the upper and lower sidebands are retained. Plane wave solutions to the NLSE
exhibit modulation instability (MI) within a frequency band determined by a
linear stability analysis. For modulation frequencies inside the MI-band, we
experimentally demonstrate that forcing and damping cause a separatrix crossing
during the evolution. Our experiments are performed on deep water waves, which
are better described by the higher-order NLSE, the Dysthe equation. We
therefore extend our analysis to this system. However, our conclusions are
general. When the system is damped by the viscosity of the water, it is pulled
outside the separatrix, which in the real space corresponds to a phase-shift of
the envelope and therefore doubles the period of the Fermi-Pasta-Ulam-Tsingou
recurrence cycle. When the system is forced by the wind, it is pulled inside
the separatrix. Furthermore, for modulation frequencies outside the
conventional MI-band, we experimentally demonstrate that contrary to the linear
prediction, we do observe a growth and decay cycle of the plane-wave
modulation. Finally, we give a theoretical demonstration that forcing the NLSE
system can induce symmetry breaking during the evolution.
