  We introduce a dynamic stabilization scheme universally applicable to
unidirectional nonlinear coherent waves. By abruptly changing the waveguiding
properties, the breathing of wave packets subject to modulation instability can
be stabilized as a result of the abrupt expansion a homoclinic orbit and its
fall into an elliptic fixed point (center). We apply this concept to the
nonlinear Schr\"odinger equation framework and show that an Akhmediev breather
envelope, which is at the core of Fermi-Pasta-Ulam-Tsingou recurrence and
extreme wave events, can be frozen into a steady periodic (dnoidal) wave by a
suitable variation of a single external physical parameter. We experimentally
demonstrate this general approach in the particular case of surface gravity
water waves propagating in a wave flume with an abrupt bathymetry change. Our
results highlight the influence of topography and waveguide properties on the
lifetime of nonlinear waves and confirm the possibility to control them.
