Arc-shaped solitons on a gain-loss ring

Changming Huang; Chunyan Li; Zhenfen Huang

doi:10.1364/OE.27.015898

1. Introduction

The effect of amplification or dissipation on the dynamics of nonlinear physical systems is an essential issue in both classical and quantum theories. During the last decade, a more particular species of parity-time(𝒫𝒯)-symmetric system associated with gain and loss has been identified [1]. This concept of 𝒫𝒯 symmetry was firstly introduced by Bender and Boettcher in 1998 [2,3], and in such system the eigenvalue spectra are entirely real within parameter regimes. To meet 𝒫𝒯 symmetry, external potential V need satisfy the relationship: V(x) = V^*(−x) in one-dimensional (1D), and V(x, y) = V^*(−x, −y) in two-dimensional (2D) potential. More specifically, the external potential is a complex function, in which the real part must be a symmetric function of position, and the imaginary one should be antisymmetric.

However, it is difficult to realize an actual 𝒫𝒯-symmetric scheme in quantum mechanical systems. In 2007, El-Ganainy and his collaborators first introduced 𝒫𝒯-symmetry into optics [4], and in which they developed a formalism suitable for describing coupled optical 𝒫𝒯-symmetric systems. Also in 2008, 1D and 2D optical solitons in 𝒫𝒯 periodic potentials have been studied theoretically [5]. Subsequently, the fact that optics with delicately balanced gain and loss can provide a platform to realize 𝒫𝒯 symmetry has been demonstrated experimentally [6, 7]. For this reason, it has since stimulated intensive research on 𝒫𝒯-symmetric operators in optics [8–17], and its ramifications have been spawned, including non-𝒫𝒯-symmetry [18–21] and partially-𝒫𝒯-symmetry [22–24].

According to [22], 2D partially 𝒫𝒯-symmetric potentials are usual either V(x, y) = V^*(−x, y) or V(x, y) = V^*(x, −y), and this class of potentials can still allow for bound states with entirely real spectra when the imaginary component is finite. From [23,24], it can know that stable optical solitons can localize on a compound azimuthal potential or a ring-shaped partially-𝒫𝒯-symmetric waveguide, and the field modulus distributions preserve the radial symmetry. Thus, a natural question arises: Do arc-shaped solitons also appear on a partially-𝒫𝒯-symmetric ring? If yes, what is the stability domain for the possible soliton families and how does the lattice influence on them?

To answer these questions, we study the existence of arc-shaped solitons on a partially-𝒫𝒯-symmetric ring. Four types of arc-shaped solitons are found, and their corresponding power-flow characteristics are discussed. Systematic direct simulations have revealed the stability of arc-shaped solitons. Also, we see that robust nonlinear arc-shaped states with four different bright spots can be excited by Gaussian beams.

2. Theoretical model

We consider the model based on the nonlinear Schrödinger equation (NLSE) describing propagation of two-dimensional (2D) spatial optical solitons supported by defocusing Kerr-type nonlinearity on a gain-loss ring, namely

i \frac{\partial Ψ}{\partial z} = - \frac{1}{2} (\frac{\partial^{2} Ψ}{\partial x^{2}} + \frac{\partial^{2} Ψ}{\partial y^{2}}) - V Ψ + {| Ψ |}^{2} Ψ,

where

(x, y) = (X r_{0}^{- 1}, Y r_{0}^{- 1})

, and

z = Z L_{dif}^{- 1}

are the normalized transverse coordinate and propagation distance, respectively, and r₀ is the characteristic transverse scale,

L_{dif} = k r_{0}^{2}

is the diffraction length, k is the wave number, Ψ(x, y) is the dimensionless amplitude of the beam. The function V = p_rV_re + ip_iV_im describes the profile of a complex azimuthal potential, where p_pr and p_im characterize the actual depth of the refractive index modulation and gain-loss amplitude, respectively. And we introduce the parameter ϱ = max(p_iV_im)/max(p_rV_re) to define the relative magnitude of the gain-loss term.

Similarly to [23], we consider the distribution of refractive index and gain-loss profile in the following forms:

V_{re} = \sum_{k = 1}^{N} exp [- {(x - x_{k})}^{2} / d^{2} - {(y - y_{k})}^{2} / d^{2}],

V_{im} = \sum_{k = 1}^{N} exp [- {(x - x_{k})}^{2} / d^{2} - {(y - y_{k})}^{2} / d^{2}] \times [y cos (ϕ_{k}) - x sin (ϕ_{k})],

where N and d are the number and the width of Gaussian waveguides, respectively. The centers of Gaussian waveguides (x_k, y_k) = (R cos(ϕ_k), R sin(ϕ_k)) with ϕ_k = 2π(k − 1)/N and the radius of the ring R = NR₀/2. Without loss of generality, in what follows we set N = 18, d = 0.5, R₀ = 0.6 and p_r = 16. The representative example of the complex potential is displayed in Fig. 1. From which we can find that V_re(x, y) is mirror-symmetric [Fig. 1(b)], while V_im(x, y) is mirror-antisymmetric [Fig. 1(c)]. Obviously, the lattice distribution V(x, y) satisfies partially-parity-time symmetry.

Fig. 1 Profiles of modulus (a), real (b) and imaginary parts (c) of a ring-shaped gain-loss potential. p_r = 16, p_i = 7.466, ϱ = 0.1 and x, y ∈ [−9, +9] in all panels.

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Stationary solutions of Eq. (1) can be found in the form Ψ(x, y, z) = ϕ(x, y) exp(iβz), where ϕ(x, y) is a complex function and β is the propagation constant. After substituting the expression into Eq. (1), it yields the following ordinary differential equation:

\frac{1}{2} (\frac{\partial^{2} ϕ}{\partial x^{2}} + \frac{\partial^{2} ϕ}{\partial y^{2}}) + V ϕ - β ϕ - {| ϕ |}^{2} ϕ = 0 .

Eq. (4) can be solved numerically by the Newton-conjugate-gradient method [25]. The power of solitons can be defined as

U = \int_{- \infty}^{+ \infty} \int_{- \infty}^{+ \infty} {| ϕ (x, y) |}^{2} d x d y

.

3. Results and discussions

One of the central results of this work is that the arc-shaped solitons can be localized on one or several Gaussian waveguides on a gain-loss ring. Due to the characteristics of partially-parity-time symmetry, arc-shaped solitons with an odd number of spots are placed in y-axis-symmetry, while that with even number of spots are distributed in x-axis-symmetry. The typical field moduli of arc-shaped solitons with 1 ∼ 4 spots are depicted in Fig. 2. From which we can find the fundamental field modulus is localized at (0, −R) and characterized by a bright spot and two very small sidelobes [Fig. 2(a)]. Field modulus with two bright spots is also called even bound state, which corresponds to the center of symmetry being (+R, 0) [Fig. 2(b)]. For arc-shaped solitons with three or four bright spots, they can be viewed as fundamental solitons or even bound states that add a bright spot to the left and another bright spot to the right of their primary maxima [Figs. 2(c) and 2(d)]. Although the field moduli of the fundamental arc-shaped solitons and the even bound states are similar to the gap solitons in the conservative system [26], we believe that their essence is different because the arc-shaped soliton solutions are complex and the imaginary profiles depend on the distribution of gain and loss.

Fig. 2 Field moduli of arc-shaped solitons with 1 ∼ 4 spots are plotted in (a–d). p_r = 16, p_i = 7.466, ϱ = 0.1, β = 4.0 in all panels.

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Corresponding to the field moduli in Fig. 2, the real and imaginary parts of the arc-shaped soliton solutions are shown in Fig. 3. One can see clearly from Figs. 3(a1)–3(a4) that the real parts with an odd number of spots are mirror-symmetric at x = 0, and that with an even number of spots is mirror-symmetric at y = 0. However, we can find from Figs. 3(b1)–3(b4) that the imaginary parts with one or three spots are mirror-antisymmetric at x = 0, and that with two or four spots is mirror-antisymmetric at y = 0. This characteristic is associated with the distribution of the ring-shaped gain-loss potential.

Fig. 3 The real parts (a1–a4), imaginary parts (b1–b4), and absolute value of the imaginary parts (c1–c4) of the arc-shaped solitons with β = 4.0. The modulus (a5) of a arc-shaped soliton at a small propagation constant β = 1.1 and the absolute value of the linear mode (b5) are depicted. p_r = 16, ϱ = 0.1 in all panels.

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Also, we plotted the absolute value of the imaginary parts of the arc-shaped solitons in Figs. 3(c1)–3(c4). For the real part with the number of spots greater than or equal to 2, the amplitude distribution of the imaginary part in some single Gaussian waveguide varies greatly. It implies that the distribution of the transverse power-flow density in the Gaussian waveguides may be different within multiple spots.

Here we must stress that for the panels (a1) and (b1) in Fig. 3, the real and imaginary parts of the fundamental arc-shaped solitons with β = 4.0 are locally confined to a single Gaussian waveguide and a gain-loss channel, respectively. However, for a small b, the profile of the fundamental soliton is usually distributed in multiple Gaussian waveguides [Fig. 3(a5)]. A similar distribution is also found for the small b of arc-shaped solitons with multiple spots. These internal complex distributions may affect the stability of arc-shaped solitons.

Figure 4 summarizes the main properties of the arc-shaped solitons supported by the gain-loss ring. In defocusing medium prominent decrease in U is observed with increasing β, as shown in Figs. 4(a) and 4(b). In which we can also find there is no power threshold for the existence of fundamental solitons (the power tends to zero when β → β_upp), while for arc-shaped solitons with two or more bright spots, a power threshold exists. Numerically, the power thresholds of arc-shaped solitons with two, three and four bright spots at ϱ = 0.1 are 0.3486, 0.5553 and 0.7481, respectively, and the proportions of which are approximately 2 : 3 : 4. Moreover, the upper cutoff value β_upp of solitons with two or more bright spots is the same and is slightly smaller in value than that of fundamental solitons. Similar to [26], we also defined the abrupt change point of the slope of the U(β) curve corresponds to the lower cut-off value β_low. When β ≥ β_low, the distribution of solitons is arc-shaped, while for β < β_low, these solitons feature one or more spots on top of a circular pedestal, a detailed discussion on these families is beyond the aim of this work as we are concerned with arc-shaped solitons. From Figs. 4(a) and 4(b), we can find arc-shaped solitons with one or more spots share the same β_low at a fixed ϱ. With the increase of ϱ, the U(b) curve goes down as a whole for some type of arc-shaped solitons.

Fig. 4 Dependencies of power of arc-shaped solitons with 1 ∼ 4 bright spots on β for the relative magnitude of gain-loss term ϱ = 0.1 (a) and 0.3 (b). The black and red segments represent stable and unstable regions, respectively. In the green segments, the corresponding solitons radiate some energy and convert to another robust solutions. Existence domains of the fundamental arc-shaped solitons (c) in the (ϱ, β) plane. (d) Effective width of arc-shaped solitons versus β for ϱ = 0.1. Power versus propagation distance z for the fundamental arc-shaped solitons at β = 4.8 (e) and β = 5.0 (f), for the even bound states at β = 2.5 (g) and β = 5.2 (h). ϱ = 0.3 in (e–h), p_r = 16 in all panels.

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The existence domains of arc-shaped solitons can be changed by the relative magnitude of the gain-loss term ϱ [Figs. 4(a) and 4(b)]. As depicted in Fig. 4(c), the upper cutoff value β_upp of fundamental solitons decreases as ϱ increases, while the lower cutoff value β_low increases gradually with the growth of ϱ. At ϱ ≈ 0.46, these two cutoff curves are closed together. Therefore, such arc-shaped solitons mentioned above can not be found when ϱ > 0.46 (i.e., p_i > 34.344 at p_r = 16).

Arc-shaped solitons are trapped around the radius of the ring, featuring several bright spots and oscillating tails. The localization of solitons can be characterized by their effective widths. According to [27], the effective width is defined as w_eff = P^−1/2, here P ≡ [∬|ϕ(x, y)|⁴dxdy]/[∬|ϕ(x, y)|²dxdy]². We plotted the effective widths of four types of arc-shaped solitons in Fig. 4(d). Starting from β = 3.7, the effective width of arc-shaped solitons increases gradually as β decreases. This feature is due to the filed modulus develops from two sidelobes to multiple sidelobes, and the width and the amplitude of bright spots also increases. However, with the increase of β, arc-shaped solitons become more and more localized until dw_eff/dβ > 0 occurs. For the region of dw_eff/dβ > 0, the filed modulus again develops some sidelobes arrays on multiple Gaussian waveguides. When β rises to the upper cutoff value, the amplitude of the bright spots of the arc-shaped solitons decreases, and then the distribution of the field modulus of the fundamental soliton is similar to the fundamental linear mode supported by the gain-loss ring [Fig. 3(b5)].

To shed meaningful light on the physics of arc-shaped solitons on a partially-parity-time symmetric ring, we explore the transverse power flow density or Poynting vector [defined as S⃗ = (i/2)(ϕ∇ϕ^* − ϕ^*∇ϕ)] of the arc-shaped solitons for four different families, as plotted in Fig. 5. In which, the corresponding profiles of soliton solutions are shown in Figs. 3(a1–a4) and 3(b1–b4). Surprisingly, we found that the transverse power flow does not always flow from the gain to the loss region by detecting the direction of S⃗ in each Gaussian waveguide. Specifically, for the arc-shaped solitons that be well localized with an odd number of bright spots, the S⃗ always flows from the gain toward the loss region in a single Gaussian waveguide [Fig. 5(a) and 5(c)]. While for β close to the upper or the lower cutoff value, the long sidelobes of soliton solutions are arranged on the Gaussian waveguides in the region of y > 0, then the power flow from the loss to gain is observed at x > 0 although the power-flow density is small. The characteristics of the power-flow directions described above are shown more intuitively in the arc-shaped solitons with an even number of bright spots [Fig. 5(b) and 5(d)]. We believe that these properties discussed above are related to the soliton solutions featured by arc-shaped on a gain-loss ring, and the directions of global currents across the waveguide channels can be considered to be clockwise.

Fig. 5 Transverse power-flow vector S⃗ of arc-shaped solitons with 1 ∼ 4 bright spots. The arrow indicates the direction of S⃗. β = 4.0, p_r = 16, ϱ = 0.3 in all panels.

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Next, we address the critical question of the stability of these arc-shaped solitons. We have performed systematic simulations of Eq. (1) with the input corresponding to perturbed solutions, i.e., Ψ|_z=0 = ϕ(x, y)[1 + τ(x, y)], where τ is a small perturbation. The stable and unstable regions corresponding to p_i = 7.4662 and p_i = 22.3985 are depicted in Figs. 4(a) and 4(b), respectively. Figure 6 illustrates representative propagation dynamics (|Ψ(x, y, z)|) for four types of solutions with 10% noises added into the initial input. Stable arc-shaped solitons are usually localized in the middle of their existence domain and are maintaining their shapes over indefinitely long distances [Figs. 6(a)–6(d)]. For unstable fundamental arc-shaped solitons, there are two different cases: (i) When β is close to the upper cut-off value, the corresponding fundamental solution can maintain its field modulus sturdily for a moderate distance (z = 700), and then it disperses quickly for z > 1000 among the Gaussian waveguides [Figs. 4(f) and 6(e)]. (ii) When β is close to the lower cut-off value, the bright spot and side lobe of the soliton are coupled among Gaussian waveguides along the direction of the global current. The direct evidence is that the amplitude of the bright spot of the fundamental soliton decreases and that of a side lobe increases from z = 0 to z = 2500, and the transmission speed of the side lobe is faster than that of the bright spot.

Fig. 6 Stable (a–d) and unstable (e–h) propagation dynamics of arc-shaped solitons with 1 ∼ 4 bright spots. β = 4.8 (a), 5.2 (b), 3.5 (c), 4.7 (d), 5.0 (e), 2.7 (f), 2.5 (g), and 0.1 (h). ϱ = 0.3 in panels (a), (b), (e), (f) and (g). ϱ = 0.1 in panels (c), (d), and (h). p_r = 16 in all panels.

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In addition, for unstable arc-shaped solitons with two or three bright spots localized in the green regions of Fig. 4(b), their corresponding propagation characteristics should also be mentioned. At β = 2.5 and ϱ = 0.3, the amplitude of the bright spots decreases and the power of the even bound states radiates along the annular waveguide for z < 1000, after that the distribution of field modulus of arc solitons remains approximately constant [Figs. 4(g) and 6(g)]. At both ends of the existence region for arc-shaped soliton with 2 ∼ 4 bright spots for ϱ = 0.1 or with 2 ∼ 3 bright spots for ϱ = 0.3, the characteristics of unstable propagation are similar to those of the fundamental solitons [Fig. 6(h)]. This instability characteristic can be considered to be related to the power-flow distribution of arc-shaped soliton solutions mentioned above. Also, we found that arc-shaped solitons with four bright spots are completely unstable in their existence domain due to the high gain-loss level (ϱ = 0.3, p_i = 22.3985) [Fig. 4(b)].

Arc-shaped solitons possess complex field moduli and power-flow structures, yet they can be excited by Gaussian beams. The input beam is incident along one or more waveguides, hence the expression of the light beam in the mth waveguide is given by A exp(−((x − x_m)² +(y − y_m)²)/d²), where A is the amplitude, x_m and y_m are the coordinates of the centers of Gaussian beams. As shown in Fig. 7, robust nonlinear arc-shaped states with four different bright spots can be obtained at sufficiently large propagation distances for the fixed A and d.

Fig. 7 The dependencies of power U on the propagation distance z are plotted in (a–d), and four different types of nonlinear arc-shaped states at z = 5000 are illustrated in (A–D). (x_k ; y_k)=(0; −5.4) in (a), (x_k ; y_k)=(−5.318, −5.318; −0.9377, 0.9377) in (b), (x_k ; y_k)=(−1.847, 0, 1.847; −5.074, −5.4, −5.074) in (c), (x_k ; y_k)=(−4.677, −5.318, −5.318, −4.677; −2.7, −0.9377, 0.9377, 2.7) in (d), A = 2 in all panels.

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4. Conclusion

In conclusion, we have studied the existence and stability of four different families of arc-shaped solitons supported by defocusing nonlinearity on a partially-parity-time symmetric ring. Field moduli of arc-shaped solitons are characterized by one or more bright spots as well as arc-shaped sidelobes, and the profiles of their real and imaginary parts strongly depend on the distribution of the external potential. With the growth of the gain and loss level, the existence region of arc-shaped solitons shrinks, and when the level exceeds a certain value, arc-shaped solitons will not be found. Surprisingly, we have seen that the transverse power flow does not always flow from the gain to the loss region by detecting the direction in each Gaussian waveguide, and then the corresponding arc-shaped solitons’ sidelobes are usually distributed in multiple Gaussian waveguides. Stable arc-shaped solitons for four different families are generally localized in the middle of their existence domains at moderate gain and loss levels. These stability characteristics mentioned above for arc-shaped solitons are obtained by systematic simulations. Also, we have found that robust nonlinear arc-shaped states for four different families can be excited by Gaussian beams.

Funding Information

National Natural Science Foundation of China (NSFC) (Grant Nos. 11704339 and 11805145); Natural Science Foundation of Shaanxi Province (Grant No. 2019JQ-089).

References

1. V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in 𝒫𝒯-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016). [CrossRef]

2. C. M. Bender and S. Boettcher, “Real spectra in non-hermitian hamiltonians having 𝒫𝒯 symmetry,” Phys. Rev. Lett. 80, 5243–5246 (1998). [CrossRef]

3. C. M. Bender, S. Boettcher, and P. N. Meisinger, “𝒫𝒯-symmetric quantum mechanics,” J. Math. Phys. 40, 2201–2229 (1999). [CrossRef]

4. R. El-Ganainy, K. Makris, D. Christodoulides, and Z. H. Musslimani, “Theory of coupled optical 𝒫𝒯-symmetric structures,” Opt. Lett. 32, 2632–2634 (2007). [CrossRef] [PubMed]

5. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in 𝒫𝒯 periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008). [CrossRef]

6. A. Guo, G. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. Siviloglou, and D. Christodoulides, “Observation of 𝒫𝒯-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103, 093902 (2009). [CrossRef]

7. C. E. Rüter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, “Observation of parity-time symmetry in optics,” Nat. Phys. 6, 192 (2010). [CrossRef]

8. S. Longhi, “Bloch oscillations in complex crystals with 𝒫𝒯 symmetry,” Phys. Rev. Lett. 103, 123601 (2009). [CrossRef]

9. F. K. Abdullaev, Y. V. Kartashov, V. V. Konotop, and D. A. Zezyulin, “Solitons in 𝒫𝒯-symmetric nonlinear lattices,” Phys. Rev. A 83, 041805 (2011). [CrossRef]

10. A. E. Miroshnichenko, B. A. Malomed, and Y. S. Kivshar, “Nonlinearly 𝒫𝒯-symmetric systems: Spontaneous symmetry breaking and transmission resonances,” Phys. Rev. A 84, 012123 (2011). [CrossRef]

11. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in 𝒫𝒯-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012). [CrossRef]

12. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Lattice solitons in 𝒫𝒯-symmetric mixed linear-nonlinear optical lattices,” Phys. Rev. A 85, 013831 (2012). [CrossRef]

13. Y. He, X. Zhu, D. Mihalache, J. Liu, and Z. Chen, “Solitons in 𝒫𝒯-symmetric optical lattices with spatially periodic modulation of nonlinearity,” Opt. Commun. 285, 3320–3324 (2012). [CrossRef]

14. Y. He and D. Mihalache, “Lattice solitons in optical media described by the complex Ginzburg-Landau model with 𝒫𝒯-symmetric periodic potentials,” Phys. Rev. A 87, 013812 (2013). [CrossRef]

15. C. Huang, C. Li, and L. Dong, “Stabilization of multipole-mode solitons in mixed linear-nonlinear lattices with a 𝒫𝒯 symmetry,” Opt. Express 21, 3917–3925 (2013). [CrossRef] [PubMed]

16. C. Huang, F. Ye, Y. V. Kartashov, B. A. Malomed, and X. Chen, “𝒫𝒯 symmetry in optics beyond the paraxial approximation,” Opt. Lett. 39, 5443–5446 (2014). [CrossRef]

17. Y. V. Kartashov and V. A. Vysloukh, “Edge and bulk dissipative solitons in modulated 𝒫𝒯-symmetric waveguide arrays,” Opt. Lett. 44, 791–794 (2019). [CrossRef] [PubMed]

18. J. Yang and S. Nixon, “Stability of soliton families in nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials,” Phys. Lett. A 380, 3803–3809 (2016). [CrossRef]

19. S. Nixon and J. Yang, “All-real spectra in optical systems with arbitrary gain-and-loss distributions,” Phys. Rev. A 93, 031802 (2016). [CrossRef]

20. J. Yang, “Classes of non-parity-time-symmetric optical potentials with exceptional-point-free phase transitions,” Opt. Lett. 42, 4067–4070 (2017). [CrossRef] [PubMed]

21. X. Zhu and Y. He, “Vector solitons in nonparity-time-symmetric complex potentials,” Opt. Express 26, 26511–26519 (2018). [CrossRef] [PubMed]

22. J. Yang, “Partially 𝒫𝒯 symmetric optical potentials with all-real spectra and soliton families in multidimensions,” Opt. Lett. 39, 1133–1136 (2014). [CrossRef] [PubMed]

23. Y. V. Kartashov, V. V. Konotop, and L. Torner, “Topological states in partially-𝒫𝒯-symmetric azimuthal potentials,” Phys. Rev. Lett. 115, 193902 (2015). [CrossRef]

24. C. Huang and L. Dong, “Stable vortex solitons in a ring-shaped partially-𝒫𝒯-symmetric potential,” Opt. Lett. 41, 5194–5197 (2016). [CrossRef] [PubMed]

25. J. Yang, Nonlinear waves in integrable and nonintegrable systems (SIAM, 2010). [CrossRef]

26. Y. V. Kartashov, B. A. Malomed, V. A. Vysloukh, and L. Torner, “Gap solitons on a ring,” Opt. Lett. 33, 2949–2951 (2008). [CrossRef] [PubMed]

27. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52 (2007). [CrossRef] [PubMed]

Arc-shaped solitons on a gain-loss ring

Abstract

1. Introduction

2. Theoretical model

3. Results and discussions

4. Conclusion

Funding Information

References

Cited By

Figures (7)

Equations (4)

Optics Express