Stability theory of flat band solitons in nonlinear wave systems

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 Schrödinger 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{é} 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.

Figures (4)

• Figure 1: Lattices, $\mathbb L$, and band dispersion loci of $-\Delta_{\mathbb L}$ for three cases: (a) Diamond lattice ($\mathbb L = \mathbb D$) ; (b) Kagomé lattice ($\mathbb L = \mathbb K$); (c) Checkerboard lattice. ($\mathbb L = \mathbb Ch$). Vertices ("atoms") comprising a fundamental cell are encircled by a dashed curve. Dotted lines connect nearest neighbors. Non-zero connectivity matrix elements, $w_{xy}$, are labeled for the lattice $\mathbb{D}$. For the lattices $\mathbb{K}$ and $\mathbb{C}h$, $w_{xy}\equiv1$. The flat band energy occurs at maximum energy of the spectrum of $-\Delta_{\mathbb L}$, $E=E_F$; for $\mathbb{D}$ and $\mathbb{K}$, $E_F = 6$, and for $\mathbb{C}h$, $E_F = 8$. Vertices in the support of a minimal compact flat band eigenstate of energy, $E_F$, are colored red and blue, which denote equal and opposite values of the wavefunction.
• Figure 2: Universal character of stability transitions for minimal compact solitons (MCS states), which bifurcate from a flat band at $E=E_F$. Theorem \ref{['theorem: power law nonlinearity stability']} implies two possible scenarios: (a) when $\sigma < \sigma^{\rm cr}_{\mathbb L}$, the compact state is at first stable and then transfers its stability to a second branch in a symmetry breaking bifurcation, and (b) when $\sigma > \sigma^{\rm cr}_{\mathbb L}$, the compact state is always unstable. In general, the bifurcation can be sub- or super-critical.
• Figure 3: Motion of eigenvalues, $\lambda_n(|a|^2)$, Kagomé case. Right: Schematic showing $|a|$ small ($\mathcal{N}[\psi_a]$ small), $L_+=\bigoplus_{n\ne s/2}L_{+,\omega^n}$ has $s-1$ strictly negative eigenvalues in the interval $(-|a|^{2\sigma},0)$. As $a$ increases, the discrete eigenvalue of each $L_{+,\omega^n}$, zero of $\lambda\mapsto F_n(\lambda,|a|^{2\sigma})$, crosses zero energy from negative to positive values, and (may) induce successive bifurcations. The minimal compact soliton $\psi_a$ becomes unstable at the first such crossing. Left: Trajectory of $\lambda_{\frac{s}{2}\pm 1}(|a|^2)$ of $L_+$ as $a>0$ varies for cubic nonlinearity $\sigma = 1$.
• Figure 4: Bifurcation diagrams plotting power $\mathcal{N}$ vs. nonlinear frequency $E$ for diamond lattice, Kagomé lattice, and checkerboard lattice. (a) Diamond lattice $\mathbb D-$DNLS with $\sigma =1$. (b) and (c), Kagomé lattice $\mathbb K-$DNLS with $\sigma = 1 < \sigma_{\mathbb K}^{\rm cr}$, and $\sigma = 1.5 > \sigma_{\mathbb K}^{\rm cr}$, respectively. (d) and (e), Checkerboard lattice $\mathbb Ch-$DNLS with $\sigma = 0.6 < \sigma^{\rm cr}_{\mathbb Ch}$, and $\sigma = 1 > \sigma^{\rm cr}_{\mathbb Ch}$, respectively. Solid lines indicate spectral stability, dotted lines indicate spectral instability. (The vertical axis for (b) and (d) is the difference between the power of the solution and the power of the compact solution, which was chosen for ease of visualization).

Theorems & Definitions (1)

• Theorem 1