Figure 1

A comparison of the first band-gap boundary of Mathieu’s equation obtained from Eq. (6) and the spectral renormalization method for the NLS Eq. (1).Reuse & Permissions

Figure 2

Contour images of the lattices, all with $V_0=2$, $K=2\pi$.Reuse & Permissions

Figure 3

(a) Band-gap boundaries for NLS equation with the external potentials, $N=2$–10.Reuse & Permissions

Figure 4

(Top) On-axis mode profiles for $N=4,5,7$ (a) and (b) soliton centered at a potential minimum profile along the $x$ and $y$ axes. [(c) and (d)] Soliton centered at a potential maximum profile along the $x$ and $y$ axes.Reuse & Permissions

Figure 5

The mode profile for periodic ($N=4$) and Penrose ($N=5$) potentials.Reuse & Permissions

Figure 6

Soliton centered at the global maximum of Penrose lattice; (a) soliton intensity in a 3D view showing the dimple; (b) cross section along $y$ axis of a Penrose soliton (solid line) superimposed on the underlying lattice (dashed line).Reuse & Permissions

Figure 7

Soliton centered at the local minimum of Penrose lattice; (a) soliton intensity in 3D (no dimple); (b) cross section along $y$ axis of a Penrose soliton (solid line) superimposed on the underlying lattice (dashed line).Reuse & Permissions

Figure 8

The power versus $\mu$ for (a) periodic potential ($N=4$) with $V_0=12.5$; (b) Penrose potential ($N=5$) with $V_0=12.5$ for solitons located on the lattice maxima.Reuse & Permissions

Figure 9

Evolution of soliton situated at the periodic potential with $V_0=12.5$ and $\mu =-3$. (a) Peak amplitude $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution of Eq. (1) as a function of the propagation distance. The initial condition is taken as the fundamental soliton with a 0.01 noise in the amplitude and phase; (b) center-of-mass evolution in $x$ and $y$ coordinates; (c) cross section along the diagonal axis of a fundamental soliton at the maximum superimposed on the periodic potential at $z=0$; (d) cross section along the diagonal axis of the fundamental soliton superimposed on the periodic potential after the propagation ($z=10$).Reuse & Permissions

Figure 10

Evolution of soliton situated at the Penrose potential with $V_0=12.5$ and $\mu =-3$. (a) Peak amplitude $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution of Eq. (1) as a function of the propagation distance. The initial condition is taken as the fundamental soliton with a 0.01 noise in the amplitude and phase; (b) center-of-mass evolution in $x$ and $y$ coordinates; (c) cross section along the diagonal axis of a fundamental soliton at the maximum superimposed on the periodic potential at $z=0$; (d) cross section along the diagonal axis of the fundamental soliton superimposed on the periodic potential after the propagation ($z=10$).Reuse & Permissions

Figure 11

The power versus $\mu$ for (a) periodic potential with $V_0=12.5$; (b) Penrose potential with $V_0=12.5$; (c) Penrose potential with $V_0=50$.Reuse & Permissions

Figure 12

Evolution of soliton situated at the periodic potential with $V_0=12.5$ and $\mu =-3$. (a) Peak amplitude $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution of Eq. (1) as a function of the propagation distance. The initial condition is taken as the fundamental soliton with a 0.01 noise in the amplitude and phase; (b) center-of-mass evolution in $x$ and $y$ coordinates; (c) cross section along the diagonal axis of a fundamental soliton at the minimum superimposed on the periodic potential at $z=0$; (d) cross section along the diagonal axis of the fundamental soliton superimposed on the periodic potential after the propagation ($z=10$).Reuse & Permissions

Figure 13

Evolution of soliton situated at the Penrose potential with $V_0=12.5$ and $\mu =-3$. (a) Peak amplitude $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution of Eq. (1) as a function of the propagation distance. The initial condition is taken as the fundamental soliton with a 0.01 noise in the amplitude and phase; (b) center-of-mass evolution in $x$ and $y$ coordinates; (c) cross section along the diagonal axis of a fundamental soliton at the minimum superimposed on the periodic potential at $z=0$; (d) cross section along the diagonal axis of the fundamental soliton superimposed on Penrose potential after the propagation ($z=10$).Reuse & Permissions

Figure 14

Evolution of soliton situated at the Penrose potential with $V_0=12.5$ and $\mu =-1$. (a) Peak amplitude $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution of Eq. (1) as a function of the propagation distance. The initial condition is taken as the fundamental soliton with a 0.01 noise in the amplitude and phase; (b) center-of-mass evolution in $x$ and $y$ coordinates; (c) cross section along the diagonal axis of a fundamental soliton at the minimum superimposed on the periodic potential at $z=0$; (d) cross section along the diagonal axis of the fundamental soliton superimposed on Penrose potential after the propagation ($z=10$).Reuse & Permissions

Figure 15

Evolution of soliton situated at the Penrose potential with $V_0=50$ and $\mu =-1$; (a) peak amplitude $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution of Eq. (1) as a function of the propagation distance. The initial condition is taken as the fundamental soliton with a 0.01 noise in the amplitude and phase; (b) center-of-mass evolution in $x$ and $y$ coordinates; (c) cross section along the diagonal axis of a fundamental soliton at the minimum superimposed on the periodic potential at $z=0$; (d) cross section along the diagonal axis of the fundamental soliton superimposed on Penrose potential after the propagation ($z=10$).Reuse & Permissions

Figure 16

Dynamics of solutions of Eq. (13) with the periodic and Penrose potentials ($N=5$). (a) Peak amplitude $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution as the function of $z$ for periodic potential with $V_0=12.5$ and $\mu =-1$; (b) $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution as the function of $z$ for Penrose potential with $V_0=12.5$ and $\mu =-1$; (c) $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution as the function of $z$ for Penrose potential with $V_0=50$ and $\mu =-1$.Reuse & Permissions

Figure 17

(a) Eigenvalues in the stability spectrum of the fundamental soliton centered at the lattice maximum for periodic potential with $V_0=12.5$ and $\mu =-1$; (b) eigenvalues in the stability spectrum of the fundamental soliton centered at the lattice maximum for Penrose potential ($N=5$) with $V_0=12.5$ and $\mu =-1$.Reuse & Permissions

Figure 18

(a) Eigenvalues in the stability spectrum of the fundamental soliton centered at the lattice minimum for periodic potential with $V_0=12.5$ and $\mu =-1$; (b) eigenvalues in the stability spectrum of the fundamental soliton centered at the lattice minimum for Penrose potential ($N=5$) with $V_0=12.5$ and $\mu =-1$; (c) eigenvalues in the stability spectrum of the fundamental soliton centered at the lattice minimum for Penrose potential ($N=5$) with $V_0=50$ and $\mu =-1$.Reuse & Permissions

Figure 19

Dynamics of solutions of Eq. (13) with the periodic and Penrose potentials ($N=5$). (a) Peak amplitude $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution as the function of $z$ for periodic potential with $V_0=12.5$ and $\mu =-1$; (b) peak amplitude $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution as the function of $z$ for Penrose potential with $V_0=12.5$ and $\mu =-1$; (c) peak amplitude $A(z)={\max }_{x,y}|\mathrm{ũ}(x,y,z)|$ of the solution as the function of $z$ for Penrose potential with $V_0=50$ and $\mu =-1$.Reuse & Permissions