Stability theory of flat band solitons in nonlinear wave systems

TL;DR

This paper develops a precise stability criterion for flat band solitons in discrete nonlinear Schrödinger systems on multi-lattices, with applications to various lattice types and methods to stabilize otherwise unstable states.

Contribution

It provides a novel stability criterion based on the nonlinearity and flat band eigenspace projections, applicable to multiple lattice geometries, and demonstrates how to engineer nonlinearities for stabilization.

Findings

Derived a sharp stability criterion for MCS states.

Applied results to diamond, Kagomé, and checkerboard lattices.

Showed how to stabilize unstable MCS states through nonlinearity engineering.

Abstract

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.