  We experimentally and theoretically demonstrate that nonlinear spin-wave
interactions suppress the hybrid magnon-photon quasiparticle or "magnon
polariton" in microwave spectra of an yttrium iron garnet film detected by an
on-chip split-ring resonator. We observe a strong coupling between the Kittel
and microwave cavity modes in terms of an avoided crossing as a function of
magnetic fields at low microwave input powers, but a complete closing of the
gap at high powers. The experimental results are well explained by a
theoretical model including the three-magnon decay of the Kittel magnon into
spin waves. The gap closure originates from the saturation of the ferromagnetic
resonance above the Suhl instability threshold by a coherent back reaction from
the spin waves.
