  Axions predicted in string theory may have a scalar potential which has a
much shallower potential region than the conventional cosine potential. We
first show that axions which were located at such shallow potential regions
generically undergo prominent resonance instabilities: the well-known narrow
resonance and/or the flapping resonance, which has not been well investigated.
We also study non-linear dynamics of axions caused by these resonance
instabilities based on lattice simulation. We find that string axions in
various mass ranges generate gravitational waves (GWs) with peaks at various
frequencies determined by the mass scales, dubbed the GW forest. This may allow
us to explore string axiverse through future multi-frequency GW observations.
We also investigate GWs produced by the axion which accounts for present dark
matter component.
