Band gaps and lattice solitons for the higher-order nonlinear Schrödinger equation with a periodic potential

Localization and dynamics of the one-dimensional biharmonic nonlinear Schrödinger (NLS) equation in the presence of an external periodic potential is studied. The band-gap structure is determined using the Floquet-Bloch theory and the shape of its dispersion curves as a function of the fourth-order dispersion coupling constant $\beta$ is discussed. Contrary to the classical NLS equation ($\beta =0$) with an external periodic potential for which a gap in the spectrum opens for any nonzero potential, here it is found that for certain negative $\beta ,$ there exists a nonzero threshold value of potential strength below which there is no gap. For increasing values of potential amplitudes, the shape of the dispersion curves change drastically leading to the formation of localized nonlinear modes that have no counterpart in the classical NLS limit. A higher-order two-band tight-binding model is introduced that captures and intuitively explains most of the numerical results related to the spectral bands. Lattice solitons corresponding to spectral eigenvalues lying in the semi-infinite and first band gaps are constructed. In the anomalous dispersion case, i.e., $\beta <0$ (where for the self-focusing nonlinearity no localized nonradiating solitons exist in the absence of an external potential), nonlinear finite-energy stationary modes with eigenvalues residing in the first band gap are found and their properties are discussed. The stability of various localized lattice modes is studied via linear stability analysis and direct numerical simulation.

Figure 1

Floquet-Bloch bands for (a, b) $\beta =1/8$, (c) $\beta =0$ as a function of $U_0.$ Regions I, II, and III are the semi-infinite, first, and second band gaps, respectively. The solid (dashed) line shows $min\lambda \left(\mu \right)[max\lambda (\mu \left)\right]$.

Figure 2

From top to bottom: First FB band, group velocity, and curvature for $\beta =1/8$ at various values of $U_0$.

Figure 3

FB modes (unnormalized) for $\beta =1/8$ and $U_0=5.$ The first (second) row corresponds to the first (second) band.

Figure 4

(Top) Band structure for $\beta =-1/10$ where shaded regions indicate spectral bands. The solid (dashed) line gives $min\lambda \left(\mu \right)(max\lambda )$. Figs. (a),(b) and (c) depicts the first (solid line) and second (dashed line) bands at $U_0=0.5,2.0835,5,$ respectively.

Figure 5

Band-gap structure for $\beta =-1/8$ and various values of $U_0$. Regions I, II, and III correspond to the semi-infinite, first, and second band gaps, respectively. The solid and dashed lines indicate minimum and maximum values of $\lambda .$

Figure 6

From top to bottom: The second FB bands, group velocity, and group velocity dispersion for $\beta =-1/8$ and various values of $U_0.$

Figure 7

FB modes (unnormalized) for $\beta =-1/8$ and $U_0=10.$ The first (second) row corresponds to the first (second) band.

Figure 8

Schematic illustration of the diatomic lattice model. Shown are the coupling constants between different sites.

Figure 9

Dispersion relation (13) for $c=2,{\kappa }_1=1/2$, and ${\kappa }_2$ = (a) $1/10$, (b) $1/2$, (c) 1, and (d) 3.

Figure 10

Group velocity (20) for $c=2,{\kappa }_1=1/2$ and ${\kappa }_2$ = (a) $1/10$, (b) $1/2$, (c) 1, and (d) 3.

Figure 11

Group velocity dispersion (21) for $c=2,{\kappa }_1=1/2$, and ${\kappa }_2$ = (a) $1/10$, (b) $1/2$, (c) 1, and (d) 3.

Figure 12

Lattice solitons for $\beta =1/8$ and $U_0=5$ (see Fig. 13 for the location of the corresponding eigenvalues in the power curve). In panels (a), (b), (c), and (d), $g=+1$ at eigenvalues $\lambda =-1,2.2,3.3,5.4,$ respectively. Panels (e) and (f) correspond to $g=-1$ and eigenvalues $\lambda =3.5,5.6,$ respectively. Vertical lines indicate potential minima.

Figure 13

Power curves for $\beta =1/8$ and $U_0=5.$ Each point on the solid ($g=+1$) and dashed ($g=-1$) lines represents a soliton solution. Typical profiles of such solutions are shown in Fig. 12.

Figure 14

Lattice solitons for $\beta =1/8$ and $U_0=25$ (see Fig. 15 for the location of the corresponding eigenvalues in the power curve). In panels (a), (b), (c), and (d), $g=+1$ at eigenvalues $\lambda =3,7.6,9.1,19.8,$ respectively. Panels (e) and (f) correspond to $g=-1$ and eigenvalues $\lambda =9,19.9,$ respectively. Vertical lines indicate potential minima.

Figure 15

Power curves for $\beta =1/8$ and $U_0=25.$ Each point on the solid ($g=+1$) and dashed ($g=-1$) lines represents a soliton solution. Typical profiles of such solutions are shown in Fig. 14.

Figure 16

Lattice solitons for $\beta =-1/8$ and $U_0=10$ (see Fig. 17 for the location of the corresponding eigenvalues in the power curve). In panels (a), (b), (c), and (d), $g=-1$ at eigenvalues $\lambda =8.2,11,3.2,6.1,$ respectively. Panels (e) and (f) correspond to $g=+1$ and eigenvalues $\lambda =3.2,6.3,$ respectively. Vertical lines indicate potential minima.

Figure 17

Power curves for $\beta =-1/8$ and $U_0=10.$ Each point on the solid ($g=-1$) and dashed ($g=+1$) lines represents a soliton solution. Typical profiles of such solutions are shown in Fig. 16.

Figure 18

Lattice solitons for $\beta =-1/8$ and $U_0=45$ (see Fig. 19 for the location of the corresponding eigenvalues in the power curve). In panels (a), (b), (c), and (d), $g=-1$ at eigenvalues $\lambda =36.3,46,16.2,34.8,$ respectively. Panels (e) and (f) correspond to $g=+1$ and eigenvalues $\lambda =16.5,35.3,$ respectively. Vertical lines indicate potential minima.

Figure 19

Power curves for $\beta =-1/8$ and $U_0=45.$ Each point on the solid ($g=-1$) and dashed ($g=+1$) lines represents a soliton solution. Typical profiles of such solutions are shown in Fig. 18.

Figure 20

(a) Lattice soliton $\varphi \left(t\right)$ given in Fig. 14 with $max\left|\varphi \right|\approx 2.73$. (b) $\varphi \left(t\right)+\chi (t,0)$ where $\chi (t,0)$ is the perturbation given in Eq. (29) for one realization of randomness and $\varepsilon =0.1.$ (c) Spectrum of the linear stability eigenvalue problem (27) corresponding to the lattice soliton in panel (a). (d) Dynamic stability obtained by direct numerical simulation using Eq. (1) with initial condition shown in panel (b).

Figure 21

Dynamical evolution of the linear stability problem (25) with initial random perturbation given in Eq. (29) for $\beta =1/8.$ (a, b) correspond to soliton solutions depicted in Figs. 12 and 14, respectively, for $g=-1.$ (c, d) for solutions shown in Figs. 12 and 14, respectively, with $g=+1$. A linear least-squares fit (dashed line) is also shown with approximate slopes of (a) 0.15, (b) 2.3, (c) 0.041, and (d) 0.28. The logarithm is base 10.

Figure 22

Direct numerical simulation of the lattice soliton shown in Fig. 12 with $U_0=5,max\left|\varphi \right|\approx 2.01$ (left) and Fig. 14 with $U_0=25,max\left|\varphi \right|\approx 3.94$ (right). For both panels the random perturbation is given by Eq. (29) with $\varepsilon =0.1$ and $g=-1.$

Figure 23

Direct numerical simulation of the lattice soliton shown in Fig. 12 with $U_0=5,max\left|\varphi \right|\approx 2.27$ (left) and Fig. 14 with $U_0=25,max\left|\varphi \right|\approx 4.32$ (right). For both panels the random perturbation is given by Eq. (29) with $\varepsilon =0.1$ and $g=+1.$

Figure 24

(a) Lattice soliton $\varphi \left(t\right)$ given in Fig. 16 with $max\left|\varphi \right|\approx 2.26$. (b) $\varphi \left(t\right)+\chi (t,0)$ where $\chi (t,0)$ is the perturbation given in Eq. (29) for one realization of randomness and $\varepsilon =0.1.$ (c) Spectrum of the linear stability eigenvalue problem (27) corresponding to the lattice soliton in panel (a). (d) Dynamic stability obtained by direct numerical simulation using Eq. (1) with initial condition shown in panel (b).

Figure 25

Dynamical evolution of the linear stability problem (25) with initial random perturbation given in Eq. (29) for $\beta =-1/8.$ Panels (a, b) correspond to soliton solutions depicted in Figs. 16 and 18, respectively, for $g=+1.$ (c, d) Solutions shown in Figs. 16 and 18, respectively, with $g=-1$. A linear least-squares fit (dashed line) is also shown with approximate slopes of (a) 0.81, (b) 3.5, (c) 0.27, and (d) 0.15. The logarithm is base 10.

Figure 26

Direct numerical simulation of the lattice soliton shown in Fig. 16 with $U_0=10,max\left|\varphi \right|\approx 2.35$ (left) and Fig. 18 with $U_0=45,max\left|\varphi \right|\approx 5.11$ (right). For both panels the random perturbation is given by Eq. (29) with $\varepsilon =0.1$ and $g=+1.$

Figure 27

Direct numerical simulation of the lattice soliton shown in Fig. 16 with $U_0=10,max\left|\varphi \right|\approx 2.53$ (left) and Fig. 18 with $U_0=45,max\left|\varphi \right|\approx 5.31$ (right). For both panels the random perturbation is given by Eq. (29) with $\varepsilon =0.1$ and $g=-1.$