Truncated-Bloch-wave solitons in optical lattices

We study self-trapped localized nonlinear states in the form of truncated Bloch waves in one-dimensional optical lattices, which appear in the gaps of the linear band-gap spectrum. We demonstrate the existence of families of such localized states which differ by the number of intensity peaks. These families do not bifurcate from the band edge, and their power curves exhibit double branches. Linear-stability analysis demonstrates that in deep lattice potentials, the states corresponding to the lower branches are stable, whereas those corresponding to the upper branches are unstable, independently of the number of peaks.

Figure 1

(Color online) Nonlinear Bloch waves bifurcated from the lower edge of the first Bloch band under repulsive nonlinearity $\left(V_0=6\right)$. (a) The amplitude versus the chemical-potential diagram; [(b) and (c)] nonlinear Bloch waves at two chemical potentials marked in (a). The shaded stripes represent low-potential regions.

Figure 2

(Color online) Truncated-Bloch-wave solitons on the upper and lower power branches of the three-peak (left column) and seven-peak (right column) families in the first band gap $\left(V_0=6\right)$. The chemical potentials of these solitons are marked in Fig. 3. In (d), the corresponding nonlinear Bloch wave (dashed curve) is also shown for comparison [see also Fig. 1c].

Figure 3

(Color online) Power curves of the (a) three-peak and (b) seven-peak families of truncated-Bloch-wave solitons $\left(V_0=6\right)$. The solutions at marked points are displayed in Fig. 2.

Figure 4

(Color online) Linear-stability analysis of the seven-peak family of truncated-Bloch-wave solitons in Figs. 1, 2 $\left(V_0=6\right)$: (a) stability spectrum of the lower-branch soliton shown in Fig. 2d; (b) stability spectrum of the upper-branch soliton shown in Fig. 2c.

Figure 5

(Color online) Nonlinear evolution of the seven-peak truncated-Bloch-wave soliton of Fig. 2d under 10% perturbations: (a) intensity $\left({\left|u\right|}^2\right)$ contours in the $\left(x,t\right)$ plane; (b) initial perturbed state of $\left|u\right|$; (c) final state of $\left|u\right|$ after 60 time-unit evolution.

Figure 6

(Color online) Truncated-Bloch-wave solitons in a shallow lattice potential $\left(V_0=1\right)$: (a) the power diagram; [(b) and (c)] solitons on the upper and lower branches as marked in (a); (d) linear-stability spectrum for the lower-branch soliton in (c).