Nonlinear compact periodic solutions in flat band networks

Linear wave equations on translationally invariant flatband (FB) networks exhibit one or more dispersionless bands in their Bloch spectrum. These macroscopically degenerate bands exist due to local symmetries and destructive interference on the lattice. Short-range hopping FB networks host compact localized (eigen)states (CLS) with nonzero amplitudes restricted to a finite volume. We consider the presence of local nonlinear terms in the wave equations. We study the continuation of CLS into the nonlinear domain while keeping their compactness and renormalizing their frequency. We then study the stability of these nonlinear CLS in terms of resonances with extended and compact localized states.