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 density``minimal compact solitons''or MCS states, of the time-dependent discrete nonlinear Schr\"odinger equation on a multi-lattice, $\mathbb L$ ($\mathbb L$-DNLS). MCS states arise for multi-lattices where a nearest neighbor Laplace-type operator on $\mathbb L$ has a flat band. Our stability criterion is in terms of the explicit form of the nonlinearity and the projection of distinguished vectors onto the flat band eigenspace. We apply our general results to MCS states of DNLS for the diamond, Kagom{\'e} and checkerboard lattices. In lattices where MCS states are unstable, we demonstrate how to engineer the nonlinearity to stabilize small amplitude MCS states. Finally, via systematic numerical computations, we put our analytical results in the context of global bifurcation diagrams.