Discrete flat-band solitons in the kagome lattice

We consider a model for a two-dimensional kagome lattice with defocusing nonlinearity, and show that families of localized discrete solitons may bifurcate from localized linear modes of the flat band with zero power threshold. Each family of such fundamental nonlinear modes corresponds to a unique configuration in the strong-nonlinearity limit. By choosing well-tuned dynamical perturbations, small-amplitude, strongly localized solutions from different families may be switched into each other, as well as moved between different lattice positions. In a window of small power, the lowest-energy state is a symmetry-broken localized state, which may appear spontaneously.

Figure 1

(a) Kagome lattice structure with a unit cell of three fundamental sites, including their interactions. (b) Band structure. Linear modes profiles at $\lambda =3.91$ (c) and $\lambda =-1.98$ (d) for a lattice of 205 sites with rigid boundary conditions.

Figure 2

(a) $P$ vs $\lambda$, (b) $\Delta H$ vs $P$, and (c) $G$ vs $P$ diagrams. The one-peak, ring, and intermediate solutions are shown with black, blue, and red-dashed lines, respectively. Insets show profiles of the stationary modes for $P=0.43$.

Figure 3

Linear spectra $g$ vs $\lambda$ for the one-site (blue dots) and the ring (black dots) solutions in the regime of weak nonlinearity. Insets (a1)–(a3) show the profiles of the indicated linear modes.

Figure 4

Vertical center of mass evolution of an unstable one-peak solution for $P=0.4655$, $k_x=0$, $k_y=0.009$. Insets: Zoom of different profiles $|u_{\vec{n}}{\left(z\right)|}^2$. Horizontal full (dashed) lines represent the one- (six-)peaks solutions.