One-Dimensional Solitons in Fractional Schr\"{O} Dinger Equation with a Spatially Modulated Nonlinearity: Nonlinear Lattice

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2.4 3.3 2.2 2.4 (a) (b) (a) (b)

1.6 2.2 1.8 1.6 A3 R P

0.8 1.1 1.4 k=0 0.8 g =1 g =2 A2 0 0 A1 0 0 1 0 -10 0 10 -10 0 10 0 1 2 3 4 5 0 1 2 3 4 5 x x b b 3.6 3.6 4.8 1.2 (c) (d) (c) (d) 2.4 2.4 3.2 0.8 R P 1.2 1.2 1.6 k=0 0.4 B1 b=2 b=0.7 =1.1 =1.8 B2 B3 0 0 0 0 -10 0 10 -10 0 10 1 1.5 2 1 1.5 2 x x

FIG. 1. (a) Profile (blue) of 1D fundamental soliton (number of nodes FIG. 2. (a) Fundamental soliton power P and the relevant linear- k = 0), found in the fractional model with b = 1 and α = 1.5, and stability analysis results (b) shown as maximal real part of eigen- its comparison to the normal Gaussian solution (magenta dashed) values λR [obtained numerically from Eq. (5)] versus propagation given by Eq. (4) at Ag = 1.65 and W = 0.5. Profiles of fundamen- constant b at α = 1.15. (c) P and (d) λR versus α at b = 2. The tal solitons for different values of the nonlinearity strength g0 (b), of propagation simulations of the marked points (A1, A2, A3) and (B1, the propagation constant b (c), and of the Lévy index α (d). Other B2, B3) in panels (c) and (d), respectively, are displayed in Fig. 3. parameters are b = 2 for panels (b) and (d), α = 1.5 for (b) and α = 1.15 for (c). We set g0 = 1 through out this letter except for that in Fig. 1(b). Alternating shaded and blank regions here and in Here E and z denote the field amplitude and propagation dis- the Fig. 4 below are shown for lattice periods of the local nonlinear- tance respectively, while g(x) > 0 is the strength of self- ity in Eq. (2). focusing nonlinearity . Note that (−∂2/∂x2)α/2 represents the fractional Laplacian where α (1 < α ≤ 2) being the Lévy index. Particularly, Eq. (1) degenerates to the well-know non- and a focusing nonlinear lattice, numerically and theoretically. linear Schrödinger equation when α = 2. In this Letter, we Noteworthy results are obtained. We find that such model, un- define the form of periodic nonlinearity (nonlinear lattice) as der different initial conditions (solitons’ power), gives rise to abundant stable soliton family solutions, including 1D fun- 2 g(x)=g0cos (x). (2) damental solitons and multipole ones of forms of dipole and tripole solitons, verified by linear-stability analysis to com- We search the stationary field amplitude U by E = pute complex eigenvalues associated with each family of the U exp(ibz) (b is the propagation constant), which yields: 1D solitons and by direct simulations of their dynamics under sufficiently small initial perturbations. Profiles of the soliton 2 α/2 1 ∂ 2 families, together with their stability, are found to be deeply −bU = − U + g(x) |U| U. (3) 2 ∂x2 affected by the fractional-order diffraction (of the physical model), which is characterized by Lévy index α. Besides, 2 The soliton power is given by P = |U(x)| dx. By solving a threshold value of α, below which all the numerically found Eq. (3) with Gaussian beams as initial input , we can get the soliton families are unstable, is identified. R stationary solutions of solitons. We also compare the soliton solutions to the normal Gaussian function, which follows:

II. FRACTIONAL MODEL AND NUMERICAL WAYS 2 2 Ug = Agexp[−x /(2W )]. (4)

The propagation of paraxial laser beams under fractional- Here Ag, W (constant parameters) are the relevant amplitude order diffraction is described by the NLFSE, which is written and width, respectively. in the dimensionless form: The stability of the numerical found soliton solutions is a key issue. To this, we take the linear stability anal- 2 α/2 ∂E 1 ∂ 2 ysis method, by perturbing the ﬁeld amplitude as U = i = − E − g(x) |E| E. (1) ∗ ∗ ∂z 2 ∂x2 [U(x)+p(x)exp(λz)+q (x)exp(λ z)]exp(ibz), where U(x) 3

4.8 3.3 (a) (b) b=1 =1.8

0 0 U U

b=5 =1.2 -4.8 -3.3 -10 0 10 -10 0 10 x x 4.8 3.3 (c) (d) b=1 =1.9 0 0 U U

b=5 =1.1 FIG. 3. Top row: stable propagations of the 1D fundamental solitons -4.8 -3.3 against small perturbations at α = 1.15, b = 0.7 (a), at α = 1.5, b = -10 0 10 -10 0 10 2 (b), and at α = 1.9, b = 2 (c). Bottom row: unstable propagations x x of the perturbed 1D fundamental solitons at α = 1.1, b = 2 (d), at α = 1.15, b = 2.5 (e), and at α = 1.15, b = 4 (f). x ∈ [−6, 6] for all panels. FIG. 4. Top row: profiles of dipole solitons (number of nodes k = 1) for different values of the nonlinearity strength g0 at α = 1.5 (a) and of the Lévy index α at b = 2 (b). Bottom row: the same but for ∗ is undisturbed field amplitude, p(x) and q (x) are small per- tripole solitons with different values of b at α = 1.7 (c) and of α at turbations. This leads to the following eigenvalue equations b = 1.5 (d). from Eq. (1):

α/2 1 ∂2 The above issues are partially addressed in Fig. 1, where iλp =+ − p + bp +gU 2(2p + q),  2 ∂x2 shows the fundamental solitons under various constraint con- (5)  2 ditions with good convergence can be obtained numerically.  1 ∂2 α/ iλq = − − q − bq − gU 2(2q + p). Fig. 1(a) depicts the profiles of the numerical found solution 2 ∂x2  (blue solid line) and the normal Gaussian solution (magenta  dashed line) with the same soliton’s power and amplitude. If the real parts of the eigenvalues, λR, calculated by the One can see from the panel that the fundamental soliton starts eigenvalue problem (5), are all zero (that is λR = 0), the to expand at the waist (compared to its Gaussian counterpart), perturbed solitons are stable. Otherwise, they are unstable so- with decaying tails on both sides penetrating into nearest ad- lutions. The stationary states of Eq. (3) are calculated by the jacent cells, keeping its main lobe within a single cell of the modified squared-operator method [37], and their propagation lattice, and resting on the lattice maximum. A comparison dynamics are tested by linear-stability analysis [Eq. (5)] and of their profiles with different nonlinear strength g0 reveals in direct numerical simulations of Eq. (1) based on the well- that both amplitude and width of the fundamental soliton de- known split-step Fourier method. creases when increasing g0, accordingto Fig. 1(b). Under different propagation constants b, as displayed in Fig. 1(c), the soliton’s amplitude increases while its waist (width) shrinks III. NUMERICAL RESULTS with an increase of b. Conversely, Fig. 1(d) demonstrates that the fundamental soliton expands and gets shorter with an in- A. Fundamental solitons crease of Lévy index α that denotes diffraction order. We then investigate two possible binding dependencies of The first soliton family we are interested in is the funda- the power P of the fundamental soliton family at fixed Lévy mental solitons [the ones have number of nodes (zeros) k =0] index α (while changing b), and at fixed propagation constant that have a single peak, since in a general nonlinear physical b (while changing α), which are portrayed respectively as the model ground states is an important role they usually play. curves P (b) and P (α) in Figs. 2(a) and 2(c). One can see One is legitimate to wonder about the issues like what are the from the former panel that P initially decreases then increases conditions that such fundamental modes can exist, and how with gradually increasing b, while the P has linear relation- they can be made stable, and what are the differences between ship with the α for the latter panel—the power P largens lin- the ones generated in the considered fractional model and that early when the Lévy index α increases. Linear stability anal- from conventional model with second-order diffraction. ysis for both cases, in the forms of λR(b) and λR(α) [λR is 4

9 1.2 (a) (b)

6 0.8 R P C2 3 0.4 k=1 C1 0 0 1 2 3 4 5 b 1 2b 3 4 5 12 1.2 (c) (d)

8 0.8 R P

4 0.4 k=1 D2 D1 FIG. 6. Top row: stable propagations of 1D high-order (multipole) 0 0 solitons against small perturbations for dipole solitons at α = 1.3, 1 1.5 2 1 1.5 2 b = 1 (a) and α = 1.8, b = 2 (b), and for tripole soliton at α = 1.9, 15 1.2 b = 1.5 (c). Bottom row: unstable propagations of the perturbed 1D (e) (f) high-order solitons for dipole solitons at α = 1.3, b = 4 (d) and α = 1.15, b = 2 (e), and for tripole soliton at α = 1.15, b = 4 (f). 10 0.8 R

P E2 B. Multipole solutions: dipole and tripole solitons 5 0.4 k=2 E1 Besides the fundamental solitonic modes, the considered 0 0 fractional model—the Eq. (1) with a nonlinear lattice given 1 1.5 2 1 1.5 2 by (2)—also supports a vast variety of high-order localized modes, e.g., the solitons with different k (number of nodes/zeros). Examples of these modes appearing as dipole FIG. 5. (a) Dipole soliton power P and (b) maximal real part of solitons and tripole modes are displayed in the top and bottom eigenvalues λR [given by Eq. (5)] versus propagation constant b at rows of Fig. 4, where clearly shows that the spacing ∆s be- Lévy index α = 1.3. (c) P and (d) λR versus α for dipole solitons tween adjacent solitons is ∆s =2π (twice to that of the non- (number of nodes k = 1) at b = 2. (e) P and (f) λR versus α for linear lattice ∆NL = π) for the former modes, and ∆s = π tripole solitons (number of nodes k = 2) at b = 1.5. The propagation (equal to the nonlinear lattice period) for the latter. System- simulations of the marked points (C1, D1, E1) and (C2, D2, E2) in atic simulations of their propagations in Eq. (1) confirm that panels (b), (d) and (f) are portrayed in the top and bottom rows of the dipole solitons can be stable modes only if ∆ ≥ π, be- Fig. 6, respectively. s low such value (π ) repulsive interaction between each soliton destabilize their coherence; with regard to the tripole modes, identical solitons on both sides can balance the π phase shift for the central soliton thus keep their coherence as a bound state, making them stable localized modes. The profiles of these modes under different scenarios, such as fixing Lévy the maximal real part of the eigenvalues of the so found so- index α (while varying b) and fixing propagation constant b lutions], are presented in Figs. 2(b) and 2(d), which show (while varying α), exhibit typical features to that of their fun- that fundamental solitons can be completely stable modes pro- damental counterparts presented in Fig. 1. vided that the relevant propagation constant b is within certain The numerical found relation of P versus b for the dipole domain (not too small and not too big), that the Lévy index solitons (with number of zeros k = 1) is displayed in Fig. α above a threshold value—its existence may be explained by 5(a), it is seen that P increases linearly with the increase of b the breaking balance between the nonlinearity and fractional- [we remark that the curve P (b) for tripole solitons also obeys order diffraction (at small α). These linear-stability results are this trend, which is not shown, space limited]. It is naturally confirmed with numerical simulations, as depicted in Fig. 3, to know, from the Figs. 5(c) and 5(e), that the dependen- from which we can see that the stable solitons keep their co- cies P (α) for dipole and tripole solitons follows this linear herence over long propagation distance, while unstable soli- growth relation, resembling their fundamental counterpart in tons transform themselves into regular oscillating modes like Fig. 2(c). Linear-stability analysis demonstrates that the sta- breathers. ble multipole solitons at given α are restricted to limited re- 5 gion of b, just like that for fundamental ones, as compared tice) in the Kerr focusing optical or atomic media. No- the Fig. 5(b) with Fig. 2(b). With an increase of the number tably, interaction between the nonlinear potential landscape of solitons of which the soliton composites (multipole soli- and the fractional-order diffraction can reach a balance and tons) are composed, the stability regions for stable multipole make the model is unique and suitable for stabilizing soli- modes at fixed b reduce drastically, as prominently evidenced tons, confirmed by linear stability analysis and direct simu- by their curves λR(α) in Figs. 5(d) and 5(f). Systematic sim- lations of the perturbed propagation. We found that the Lévy ulations performed in stability and instability regions verified index α greatly affects the profiles of the thus found soliton that the stable multipole solitons are robustly modes, while families and their stability. Stability and instability regions unstable ones either hold regular oscillations or lose their co- of all the soliton families under different parameter spaces herence completely and disappear as radiating waves during were obtained. The results presented here are extended nat- propagation, examples of both cases are displayed in the top urally to two-dimensional geometry to construct stable local- and bottom lines of Fig. 6, respectively. ized modes—solitons and vortices, which is an open issue by solely using the smoothly modulated nonlinear lattices.

IV. CONCLUSION

To summarize, we have studied the existence and stabil- V. ACKNOWLEDGMENT ity of various 1D soliton families—including fundamental solitons and multipole ones (dipole and tripole solitons)—in This work was supported, in part, by the Natural Science the framework of nonlinear fractional Schr¨odinger equation Foundation of China (project Nos. 61690224, 61690222), by by combing the recently introduced fractional Schr¨odinger the Youth Innovation Promotion Association of the Chinese equation with a spatially periodic nonlinearity (nonlinear lat- Academy of Sciences (project No. 2016357).

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