When Linear Stability Does Not Exclude Nonlinear Instability

We describe a mechanism that results in the nonlinear instability of stationary states even in the case where the stationary states are linearly stable. This instability is due to the nonlinearity-induced coupling of the linearization’s internal modes of negative energy with the continuous spectrum. In a broad class of nonlinear Schrödinger equations considered, the presence of such internal modes guarantees the nonlinear instability of the stationary states in the evolution dynamics. To corroborate this idea, we explore three prototypical case examples: (a) an antisymmetric soliton in a double-well potential, (b) a twisted localized mode in a one-dimensional lattice with cubic nonlinearity, and (c) a discrete vortex in a two-dimensional saturable lattice. In all cases, we observe a weak nonlinear instability, despite the linear stability of the respective states.

Figure 1

The top panel shows the space-time evolution of a perturbed antisymmetric soliton in the NLS model (2) with the double-well potential (12). The slow growth over time is further illustrated in the bottom panel of the figure containing the evolution of the maximal amplitude in the left and right potential wells by a solid (blue) and dashed (red) line. Inset: detail of the growth dynamics.

Figure 2

Evolution of twisted localized modes in the discrete NLS model (13). Top: time evolution of the four centermost sites (i.e., the two central ones 0 and 1, with a square amplitude of $\approx 1.3$ shown also by solid and dashed line in the left inset, and their immediate neighbors, sites $-1$ and 2, with a square amplitude of $\approx 0.01$ shown also by solid and dashed line in the right inset) of the structure for $C=0.01$; no instability is manifested. The insets show a zoom in of the relevant sites revealing their oscillatory behavior. Bottom: the evolution of one of the central sites for $C=0.07$ (the inset shows its envelope illustrating the unstable evolution). In both panels $n_0=0$.

Figure 3

The top panel shows the modulus time evolution of $|u_{n=0,m=0,\pm 1}|$, while the bottom panel illustrates the corresponding evolution of $|u_{n=1,m=0,\pm 1}|$ [i.e., four sites constituting the discrete vortex [(0, 0), (0, 1), (1, 0), and (1, 1)] and two immediately adjacent ones [$(0,-1)$ and $(1,-1)$]]. Among the four central sites, only one (dashed red in the top panel) grows in power, while the other three (solid blue in the top and bottom and dashed red in the bottom) decrease in amplitude. The neighboring sites also remain at low amplitudes (green dash dotted).