Stability theory of flat band solitons in nonlinear wave systems

We establish a sharp criterion for the stability of a class of compactly supported, homogeneous, symmetric states, "minimal compact solitons" or MCS states, of the time-dependent discrete nonlinear Schrödinger equation on a multilattice, L (L-DNLS). MCS states arise for multilattices where a nearest neighbor, a Laplace-type operator on L, has a flat band. Our stability criterion is in terms of the explicit form of the nonlinearity and the projection of a distinguished vector onto the flat band eigenspace. We apply our general results to MCS states of DNLS with a power-law nonlinearity for the diamond, Kagome, and checkerboard lattices. In lattices where MCS states are unstable, we demonstrate how the variation of the nonlinearity exponent enables the stabilization of small amplitude MCS states.