Solitons in nonlinear lattices

This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are also surveyed, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation. The solitons are considered in one, two, and three dimensions. Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of one-dimensional solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for theoretical and experimental studies alike. In both the one-dimensional and two-dimensional cases, the mechanism that creates solitons in NLs in principle is different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

Figure 1

(a) The norm of the stationary soliton solution of Eq. (17) vs its amplitude at $r_0=0$, in the fundamental model of the 1D nonlinear lattice. Circles connected by the continuous line are values found from the direct numerical solution, while the dashed curve shows the prediction of the variational approximation. (b) The chemical potential vs the number of atoms for numerically found solitons. From 254.

Figure 2

(Color online) Spatially localized solitons of the NLSE with inhomogeneous nonlinearity coefficient $R(x)=r_0/{cosh}^3(x)$, obtained by mapping of periodic solutions of Eq. (19), $W(X)=C(\mu X,k)/(\mu X,k)$, with specifically selected values of modulus $k$, in the model devised to produce exact soliton solutions. From 23.

Figure 3

(Color online) Profiles of (a) fundamental, (b) even, (c) dipole-mode, and (d) triple-mode solitons residing at the center of layered thermal sample and described by Eq. (23). (e) Profiles of fundamental solitons shifted from the center of the sample. (f) The distribution of the thermally induced perturbation of the refractive index $\sigma T$ for a fundamental soliton with $\mu =0.9$ residing at the center of the sample, and for a shifted fundamental soliton with $\mu =6.5$. In shaded regions $\sigma >0$, while in white regions $\sigma <0$. From 159.

Figure 4

(Color online) The number of bosons (norm) vs energy (chemical potential) $\mathcal{E}$ in the first gap ${\mathcal{E}}_1^{(+)}<\mathcal{E}<{\mathcal{E}}_2^{(-)}$ in the dynamical model of the ${}^{87}\mathrm{Rb}-{}^{40}\mathrm{K}$ (boson-fermion) mixture. (a) and (b) correspond to different signs of average nonlinearity coefficient $\chi$ near the ${\mathcal{E}}_1^{(+)}$ edge of the gap. (c), (d), and (e) show explicit shapes of the modes corresponding to different points on the solution branches, while (f) shows the dynamics of mode G (in the insets the initial and final shapes are shown by solid lines). From 32.

Figure 5

(Color online) The phase-space construction of asymptotic (solitary) solutions for odd $n$ (a) and even $n$ (b), in the model with the nonlinear lattice of the Kronig-Penney type. Black dots depict the transition at the boundary between linear and a nonlinear layers. Examples of solitons corresponding to $n=2$ (c) and 3 (d). Shaded areas indicate nonlinear layers. From 175.

Figure 6

(Color online) The critical angle vs propagation constant $\mu$ for (a) odd and (b) even solitons in lattices with out-of-phase modulation of linear refractive index and nonlinearity. (c) The propagation dynamics of odd solitons with $\mu =9.8$, launched into the lattice at two different angles. Distributions of the absolute value of the field corresponding to different input angles are superimposed. In all cases, $\sigma =0.4$. From 160.

Figure 7

(Color online) (a) A stable three-peak soliton in the second band gap at $\mu =0.4$, in the model of the 1D photonic-crystal waveguide of the Kronig-Penney type. (b) A weakly unstable nearly flat-top soliton in the second band gap corresponding to $\mu =0$. (c) An unstable counterpart of the three-peak soliton with the inverted shape, corresponding to $\mu =-0.22$. In all cases, the nonlinearity is defocusing and $D/L=0.5$. Left panels show the soliton shape, while right panels show the spectral plane of stability eigenvalues for the respective solitons, $\lambda ={\lambda }_r+i{\lambda }_i$ (the soliton is stable if ${\lambda }_i=0$). From 203.

Figure 8

Top panels: Effective potential $V(x_0)$ as a function of the soliton’s central coordinate $x_0$ with the trap’s strength $\Omega =0.05$, for a fundamental bright soliton of the unit amplitude, initially placed at the trap’s center $[x_0(0)=0]$, in the model of the collisionally inhomogeneous BEC. Different values of the gradient modify the character of the potential: in (a) it is purely attractive, while in (b) it is either purely gravitational $(\delta =\sqrt{3}\Omega )$ or expulsive $(\delta =2\Omega )$. Bottom panels: The motion of the center of the bright soliton inside (c) an attractive effective potential, and (d) gravitational or expulsive potentials. From 305.

Figure 9

(a) Three lowest branches of soliton solutions with $R_{\pm }=\pm 1$, in the model of the nonlinearity management for BEC. Shown is the number of particles $N$ vs chemical potential $\mu$. The bolder regions of curves ${\Gamma }^n$ correspond to stable solutions, while the lighter ones correspond to unstable ones. Regions of the multistability are shaded. The explicit shapes of stable modes close to the linear limit are shown in the insets. (b) The same as in (a) but for $R_{+}=5$, $R_{-}=-1$. From 339.

Figure 10

(a), (c) Chemical potential $\mu$ vs norm $N$ at different values of the commensurability factor, $q$, for ordinary and gap solitons, respectively, in the BEC model including mutually commensurate or incommensurate linear and nonlinear lattices. (b), (d) The stability boundary for ordinary and gap solitons, defined as per the VK criterion by the condition $d\mu /dN=0$. Note that the threshold value of the norm necessary for the existence of the solitons, of both the ordinary and gap types, vanishes at two points, $q=0$ and 2, which correspond, respectively, to the models with the constant nonlinearity coefficient, and with the direct commensurability between the linear and nonlinear lattices. From 259.

Figure 11

(a) An example of the stable evolution of a perturbed cn-type wave near the edge of its stability area at $g_0=-1$, $b=-0.7$, $k=0.95$, in the model of the nonlinear lattice admitting exact periodic solutions in terms of the elliptic functions. (b) An example of the stable evolution of a perturbed dn-type wave with $g_0=+1$, $g_1=-2$, $b=r=1$, and $k=0.9$. From 312.

Figure 12

(Color online) The profile of an even-dipole vector soliton at ${\mu }_1=2.67$, ${\mu }_2=3$, $C=1$, in the model of the two-component system with the nonlinear lattice. (b) Energy sharing between components of the even-dipole vector solitons vs ${\mu }_1$ at ${\mu }_2=3$, $C=1$. Domains of stability (white) and instability (shaded) in the (${\mu }_1,{\mu }_2$) plane at $C=1$ (c), and in the ($C,{\mu }_1$) plane at ${\mu }_2=3$ (d). From 156.

Figure 13

A set of bifurcation diagrams describing the symmetry breaking of pinned modes in the one-dimensional double-well pseudopotential at different values of the wells’ width $a$ ($a=0$ corresponds to the limit form of the model with the delta functions). From 204.

Figure 14

(Color online) (a), (b) Intensity profiles for typical examples of spatial subwavelength solitons of the TE and TM types in the model of the array of nanowires. Shaded and white areas correspond, severally, to the strips made of silicon and silica. (a) and (b) show solitons of the on-site and off-site types. (c), (d) The total power vs the nonlinear shift of the propagation constant of the subwavelength solitons, whose examples, corresponding to points $A$ and $B$, are shown in (a), (b). (c) and (d) show families of the TE and TM solitons, respectively. Curves correspond to the on-site (off-site) solitons, while solid (dashed) curves designate stable (unstable) soliton families. From 129.

Figure 15

The dashdotted curves show examples of stable exact solutions obtained for trapped solitons within the framework of the model of the nonlinear defect, based on one-dimensional equation (39) with $f(z)=\delta (z)$. The solid curves show counterparts of these states, generated by the numerical solution of the underlying 3D Gross-Pitaevskii equation, for the same values of parameters $\epsilon$ and $n_0$. From 7.

Figure 16

Typical examples of stationary discrete solutions corresponding to $W_2=0$, in the model of the discrete linear-nonlinear lattice based on Eq. (41): (a) bright solitons, (b) kinks (dark solitons), and (c) states sitting on top of a finite background (“antidark solitons”). Modes (a) and (b) may be stable, while those of type (c) are always unstable. From 4.

Figure 17

Stationary fundamental (a), (b) and dipole (c), (d) modes supported by the interface between discrete lattices with the attractive and repulsive on-site cubic nonlinearity. Modes shown in (a), (b) correspond to $C=0.1$, while modes shown in (c), (d) correspond to $C=0.135$. From 190.

Figure 18

(Color online) Examples of stable staggered discrete solitons in the lattice with interlaced focusing nonlinear and linear sites. These solitons were found in the finite band gap. (a), (b) The evolution of a stable soliton and antisymmetric bound state of two solitons, with random initial perturbations added to them. (c), (d) Profiles of the respective stationary solutions. From 142.

Figure 19

(Color online) Shapes of typical odd, even, and twisted two-component solitons in the semidiscrete model based on Eqs. (44). The values of the XPM coupling constant are indicated in (a), (b), and (c). Equal propagation constants are taken here for the discrete and continuous components, ${\lambda }_1={\lambda }_2\equiv \lambda$. From 221.

Figure 20

(Color online) The release of soliton trains in the model of the matter-wave laser. (a)–(f) pertain to the following values of the negative scattering length in Eq. (45): $R_0=0.9R_{\mathrm{cr}}$, $R_0=2.0R_{\mathrm{cr}}$, $R_0=3.3R_{\mathrm{cr}}$, $R_0=4.8R_{\mathrm{cr}}$, $R_0=6.5R_{\mathrm{cr}}$, and $R_0=8.7R_{\mathrm{cr}}$. Recall that the equilibrium position for the soliton exists at $|R_0|<|R_{\mathrm{cr}}|$. From 243.

Figure 21

(Color online) (a) The domain (dashed area) in the plane of the strength of the linear lattice $V$ and amplitude of the nonlinearity modulation, $G$, where the condition for the stability of the Bloch oscillations of the gap soliton in the framework of Eq. (47) is satisfied. In this case, the constant part of the nonlinearity coefficient is $g=-0.777$. Also shown are examples of stable (c) and unstable (d) Bloch oscillations of the gap solitons under the action of the constant driving force. Parameters in (b) and (c) correspond, respectively, to points $B$ and $A$ in (a). From 266.

Figure 22

(a) Examples of stable 2D soliton solutions supported by the circular modulation of the local nonlinearity, with ${\rho }_0=0$, $\mu =-0.0399$ (solid line) and ${\rho }_0=0.5$, $\mu =-0.0648$ (dashed line). (b) The chemical potential vs the norm for soliton families found at ${\rho }_0=0$, 0.2, and 0.5. (c) The stability diagram for the soliton solutions. In the regions between the two borders, the solitons are stable simultaneously according to the VK criterion (i.e., against radial perturbations) and against azimuthal modulations. From 256.

Figure 23

(a) The norm (total power) of 2D solitons vs propagation constant $\mu$ in the model of the 2D nonlinear lattice, built as an array of self-focusing circles, for several values of $w_s$ in the medium with the cubic nonlinearity. (b) The minimum norm vs the lattice spacing $w_s$. The horizontal dashed lines in (a) and (b) correspond to the critical norm, $N_T=5.85$. (c) The real part of the perturbation growth rate vs $\mu$ at $w_s=3$. (d) The norm vs $\mu$ in the saturable medium. From 155.

Figure 24

(Color online) (a) The geometry of the liquid-infiltrated PCF. (b) The intensity distribution in a numerically calculated gap soliton for power $P=4\times {10}^{-5}\mathrm{W}$ and $s=10$. (c), (d) Experimentally observed output diffraction pattern and soliton localization in the PCF at low (3 mW) and high (100 mW) input powers, respectively. From 240.

Figure 25

(Color online) (a) Intensity distributions for fundamental solitons created in the liquid-infiltrated PCF. The panels correspond to (from left to right) $\gamma \to 0$, $\gamma =0.0010$, 0.0015, and 0.0017, in the PCF with pitch (the spacing between parallel voids) $\Lambda =23\mu \mathrm{m}$, radius $a=4\mu \mathrm{m}$, and $\lambda =1.55\mu \mathrm{m}$. (b) Nodal solitons with different orientation of nodal lines in the PCF with $\Lambda =23\mu \mathrm{m}$, $a=8\mu \mathrm{m}$, and $\lambda =1.06\mu \mathrm{m}$, obtained for $\gamma =0.006$. The first two panels show distributions of the absolute value of the field, while the last two panels show the corresponding phase patterns. From 103, 102.

Figure 26

The lowest-order eigenfunctions of nonlinear operator $\mathcal{L}$ generated by the soliton solution in the fundamental representation of ${\mathcal{C}}_{6v}$, in the model of the PCF with the corresponding symmetry of the intrinsic structure. The symmetry of the full operator is ${\mathcal{C}}_{6v}$, i.e., $[\mathcal{L},{\mathcal{C}}_{6v}]=0$. Modes in the two middle rows correspond to vortices with charges $\pm 1$ and $\pm 2$. From 105.

Figure 27

(Color online) Examples of different types of vortexlike solitons in the dual-core PCF: (a) a double-vortex state, (b) a combined state of vortex and fundamental solitons, (c) a double-triple vortex state, and (d) a combined state of a triple vortex and fundamental soliton. From 268.

Figure 28

(Color online) The existence domain for the vector solitons in the PCF model, shown in the plane of (${\mu }_1,{\mu }_2$), for two values of coupling constant $C$. From 269.

Figure 29

(Color online) (a)–(c) The phase space for each part of the structure (in the one-dimensional model based on the Kronig-Penney lattice with the intrinsic surface). (a) The nonlinear part for $\mu >{\epsilon }_1$, (b) the linear part for $\mu <{\epsilon }_i$ ($i=2$ or 3), and (c) the linear part for $\mu >{\epsilon }_i$ ($i=2$ or 3). (d)–(f) The phase-space representation of the soliton solutions for $n$ even. (d) The nonlinear homogeneous part, (e) the linear homogeneous part at $\mu <{\epsilon }_3$, and (f) the linear homogeneous part at $\mu >{\epsilon }_3$. The dotted line denotes the solution in the lattice part, and the solid line denotes the solution in the uniform part. From 178.

Figure 30

(Color online) Profiles of (a) fundamental, (b) dipole-mode, and (c) tripole solitons with different values of $\mu$, in the model of the thermal layered medium with a surface. (d) Distributions of the refractive index for fundamental solitons with $\mu =5$ and $\mu =1$. In all cases, the sign of the nonlinearity alternates between different layers of the thermal medium. In gray regions, $\sigma >0$ holds. From 163.

Figure 31

(Color online) (a) The norm of the surface soliton residing at the interface of lattice with out-of-phase modulation of refractive index and nonlinearity vs the propagation constant; (b) the Hamiltonian vs the norm at $p=3$, $\sigma =0.6$. Darker and lighter curves show stable and unstable soliton families, respectively. Stability domains for surface solitons: (c) in the plane of ($\sigma ,N$) at $p=3$; (d) in the plane of ($p,N$) at $\sigma =0.7$. From 157.

Figure 32

Typical examples of stable on-site-centered (odd) discrete solitons generated by Eqs. (56), in the model of the lattice with the ${\chi }^{(2)}$ nonlinearity. Triangles and squares show numerical profiles of the fundamental and second harmonics, respectively. The dashed and continuous curves represent the respective profiles as predicted by the variational approximation. (a) Solitons with staggered fundamental and unstaggered second-harmonic components, (b) both components staggered, and (c) both unstaggered. From 293.

Figure 33

The top and bottom panels display typical examples of stable semidiscrete ${\chi }^{(2)}$ solitons generated by Eqs. (57, 58), respectively, whereas, the left and right panels show the odd and even species of the solitons. Vertical lines designate the location of the discrete waveguides in the system. These examples pertain to zero mismatch (e.g., From 222).

Figure 34

A soliton generated by Eqs. (59) in the case of the large wave number of the nonlinear grating, $\gamma =20$, and large mismatch, $\alpha =10$, in the model of the photonic crystal with the ${\chi }^{(2)}$ nonlinearity. The grating is represented by the gray pattern in the background. The profile of the SH component follows the form of the nonlinear grating, while the FF component features a sech-type profile, with only small distortions induced by the grating, as additionally shown in the inset, where the grating is indicated by the dashed sinusoid. From 224.

Figure 35

The norm vs the propagation constant for (a) different values of $S$ at $d=0.4$ and (b) different values of $d$ at $S=0.5$, in the model of the radial tandem structure supporting “light bullets.” In  (b), values of the domain’s width are $d=0.8$, 0.6, 0.4, 0.2, and 0.1, from the lower to upper curve. (c) The perturbation growth rate vs $\mu$, for the azimuthal perturbation index $k=0$ and $S=0.5$. In (a)–(c), the central domain is linear. (d) The growth rate vs $\mu$ at $k=1$, $d=1.2$, $S=0.5$ in the structure with linear (1) and nonlinear (2) central domains. From 309.