  Due to its broken spin and orbit rotation symmetries, superfluid $^3$He plays
a unique role for testing rotational quantum properties on a macroscopic scale.
In this system the orbital momentum forms textures that provide an effective
potential well for the creation of stationary spin waves. In the limit of the
lowest temperatures presently attainable, we observe by NMR techniques a
profound change in the spin dynamics. The NMR line shape becomes asymmetric,
strongly hysteretic and displays substantial frequency shifts. This behavior,
quantitatively described by an anharmonic oscillator model, indicates that the
parameters of the potential well depend on the spin waves amplitude, and
therefore that the orbital motion is not damped in this new regime, not
considered by the standard Leggett-Takagi theory. This regime of non-linear
stationary spin waves is shown to give rise to the pulsed NMR "Persistent
Signals" reported recently.
