Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Two-dimensional vortex dipole solitons in nonlocal nonlinearity with PT-symmetric Scarff-II potential

Peijun Chen and Hong Wang

Open Access Open Access

Abstract

We investigate the dynamics and stability of two-dimensional (2D) vortex dipole solitons in nonlocal nonlinearity with PT-symmetric Scarff-II potential. We analyze the solitons with single charge and higher-order charge using analytical and numerical methods. By the variational approach, we can obtain analytical solutions for the model. It is found that the nonlocality degree affects the evolution of the beams. We discover that the vortex dipole solitons will undergo stable deformation rather than maintaining their basic profile when the nonlocality is strong. Moreover, the stability of the vortex dipole solitons depends on the potential depth and there exists a threshold, below which the beams can keep their shapes and propagate stably whether the nonlocality is weak, intermediate, or strong. Numerical simulations are consistent with the analytical results.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Solitons are localized waves that remain stable when the nonlinear self-focusing effect balances linear diffraction [1,2]. They have been extensively studied in various media, such as nematic liquid crystals [3], thermo-optic materials [4], atomic vapors [5], and photorefractive crystals [6]. These media exhibit a novel type of nonlinear response: nonlocal nonlinearity, which means that the refractive index change of a material at a specific point is determined by the light incidence point and its neighborhood [7–9]. Nonlocality can profoundly influence the propagation dynamics of solitons [10]. It has been found that nonlocality can support many solitons, including vector solitons [11,12], competing cubic-quintic solitons [13–15], vortex solitons [16,17], and necklace solitons [18].

Vortex solitons, which carry angular momentum with phase singularities in the centers, have been demonstrated to be hardly stable and will break into multipole scalar fundamental solitons due to azimuth instability. The topological charge is one of the most critical characteristics of vortex solitons, influencing the spiral structures of the soliton phase [19]. Much attention has been paid to vortex solitons. Numerous researches have shown that nonlocal nonlinearity can stabilize many types of vortex solitons, such as vector vortex solitons, elliptical vortex solitons [20,21], annular vortex solitons [22–24], and necklace vortex-solitons [25–27].

Parity-time (PT) symmetry has also drawn considerable attention from the physics and mathematics community since Bender and Boettcher pointed out in 1998 that non-Hermitian Hamiltonians can exhibit entirely real spectra [28]. There exists a phase transition threshold in the PT system, above which the eigenvalue of the PT-symmetric potential will not be real, and PT symmetry will be broken. In order to satisfy the PT symmetry condition, the complex potential must obey $V({x,y} )= {V^ \ast }({x,y} )$, that is, the real part of a PT symmetry complex potential should be an even function, whereas the imaginary part is odd. Here the symbol “${\ast} $” stands for complex conjugation. Up till now, fundamental solitons, multipole solitons, vortex solitons, vector solitons, and bandgap solitons have been reported to exist in nonlinear media in the presence of PT-symmetric potential [29–32].

Although some studies have investigated vortex solitons in PT-symmetric systems, most studies focused on the evolution of single vortex soliton. Considering that the internal interactions of dipole solitons are the simplest of the multipole solitons, in this paper, we explore the dynamics and stability of two-dimensional (2D) vortex dipole solitons in nonlocal nonlinearity with PT-symmetric Scarff-II potential. We discuss not only single-charged solitons but also higher-order charged vortex dipole solitons. The variational approach was used to obtain approximate analytical solutions. Numerical simulations reveal that nonlocality helps the formation of stable deformation solitons but cannot stabilize vortex dipole solitons. Moreover, the stability of the beams is strongly influenced by the potential depth.

2. Model

We consider vortex dipole solitons propagating in nonlocal nonlinearity with 2D PT-symmetric Scarff-II potential. The slowly varying beam envelope $U({x,y,z} )$ can be described by the normalized 2D nonlinear Schrödinger coupled equation:

$$\left\{ \begin{array}{l} i\frac{{\partial U}}{{\partial z}} + \frac{{{\partial^2}U}}{{\partial {x^2}}} + \frac{{{\partial^2}U}}{{\partial {y^2}}} + V({x,y} )U + \sigma nU = 0\\ d\left( {\frac{{{\partial^2}n}}{{\partial {x^2}}} + \frac{{{\partial^2}n}}{{\partial {y^2}}}} \right) + {|U |^2} - n = 0, \end{array} \right.$$
where x and y are the scaled transverse coordinates and z is the scaled propagation distance. The parament d and n represent the degree of nonlocality and nonlinear refractive index, respectively. The symbol of $\sigma $ indicates that the corresponding nonlinearity is focusing (+) or defocusing (-). We suppose $\sigma = 1$, which means the corresponding nonlinearity is focusing. The PT-symmetric Scarff-II potential is set as $V({x,y} )= {V_0}({W + i{W_0}I} )$. Here $W({x,y} )$ and $I({x,y} )$ stand for the real part and imaginary part of the potential, which are assumed as
$$W = ({{{\textrm{sech}}^2}x} )({{{\textrm{sech}}^2}y} ),$$
$$I = ({\textrm{sech} (x )\tanh (x )} )({\textrm{sech} (y )\tanh (y )} ).$$

The parameter ${V_0}$ is the potential depth, while ${W_0}$ represents the gain-loss coefficient. As for the refractive index n, the nonlinearity can be given by

$$n = \int {\int_{ - \infty }^{ + \infty } {R({x - x^{\prime},y - y^{\prime}} )} {{|{U({x^{\prime},y^{\prime}} )} |}^2}dx^{\prime}dy^{\prime},}$$
where R is the 2D nonlocal response. For the sake of analytical simplicity and without loss of generality, we consider the Gaussian nonlocal response functions throughout this work, which can be described by
$$R({x,y} )= \frac{1}{{\pi {d^2}}}\exp \left( { - \frac{{{x^2} + {y^2}}}{{{d^2}}}} \right).$$

It is important to note that R is strongly dependent on the nonlocality degree d. In order to get a good understanding of the properties of vortex dipole solitons in nonlocal media in the presence of PT-symmetric Scarff-II potential, an analytical solution derived from the variational approach is necessary. According to the literature [33–38], the total Lagrangian density of Eq. (1) consists of a conservative part ${L_C}$ and a non-conservative part ${L_{NC}}$, i.e., $L = {L_C} + {L_{NC}}$, in which the imaginary part of the complex potential contributes to the non-conservative part. The conservative part is expressed as follows:

$$\begin{aligned} {L_c} &= \frac{i}{2}\left( {U\frac{{\partial {U^ \ast }}}{{\partial z}} - {U^ \ast }\frac{{\partial U}}{{\partial z}}} \right) + \left( {{{\left|{\frac{{\partial U}}{{\partial x}}} \right|}^2} + {{\left|{\frac{{\partial U}}{{\partial y}}} \right|}^2}} \right) - \frac{1}{2}W{|U |^2} - \frac{1}{2}n{|U |^2}\\ &\quad + \frac{1}{2}{|n |^2} + d\left( {{{\left|{\frac{{\partial n}}{{\partial x}}} \right|}^2} + {{\left|{\frac{{\partial n}}{{\partial y}}} \right|}^2}} \right). \end{aligned}$$

We assume a typical single annular vortex form for the amplitude of a vortex beam with angular momentum, which can be given by

$${U_{({x,y,z} )}} = A{\left( {\sqrt {{x^2} + {y^2}} } \right)^m}\exp \left( { - \frac{{{x^2} + {y^2}}}{{2{\omega^2}}}} \right)\exp \left( {ikz + im\left( {{{\tan }^{ - 1}}\left( {\frac{y}{x}} \right)} \right)} \right).$$

Here, A and $\omega$ stand for the amplitude and beam width, respectively, while k represents the propagation constant and m denotes the topological charge. By Substituting this form into Eq. (6), the effective Lagrangian for the conservative system can be simplified to

$$\begin{aligned} < {L_C} > &= \int {\int_{ - \infty }^{ + \infty } {{L_C}dxdy} } \\ &= k{A^2}{\omega ^4}\pi + \frac{7}{2}{\omega ^2}\pi {A^2} - \frac{1}{2}{V_0}{\omega ^4}\pi {A^2} - \frac{{{d^6}{\omega ^8}\pi }}{{2{{({{d^4} - {\omega^4}} )}^2}}}{A^4}\\ &\quad + \frac{{{d^6}{\omega ^8}\pi }}{{2{{({{d^2} - {\omega^2}} )}^2}}}{A^4} + \frac{{2{d^3}{\omega ^8}\pi }}{{{{({{d^2} - {\omega^2}} )}^4}}}{A^4}. \end{aligned}$$

The standard variational approach corresponding to the two-dimensional dissipative system is described as

$$\frac{\partial }{{\partial z}}\left( {\frac{{\partial < {L_C} > }}{{\partial {\eta_z}}}} \right) - \frac{{\partial < {L_C} > }}{{\partial \eta }} = 2\textrm{Re} \left( {\int {\int_{ - \infty }^{ + \infty } {Q\frac{{\partial {U^ \ast }}}{{\partial \eta }}dxdy} } } \right),$$
where $Q ={-} i{V_0}{W_0}IU$ is equal to the dissipative term in Eq. (1). $\eta$ corresponds to parameters that can vary freely in the variational ansatz. In this paper, we choose parameter $\omega$ and d, i.e., $\eta = \omega$ or d. $\eta$ can be substituted with parameter $\omega$ and d. The power of vortex solitons is defined as $P = m!\pi {A^2}{\omega ^{2({m + 1} )}}$.e.g. the total power is $P = \pi {A^2}{\omega ^4}$ for fundamental charged ($m = 1$) vortex beam, and for higher-order charged vortex beam with $m = 3$, the power is $P = 6\pi {A^2}{\omega ^8}$. In the case of $m = 1$, by substituting $\eta$ with $\omega$ in Eq. (9), the amplitude A as a function of $\omega$ is given by
$${A^2} = \frac{{[{4k{\omega^2} + 7 - 2{\omega^2}{V_0} - 2{\omega^2}{V_0}{W_0}} ]{{({{d^2} + {\omega^2}} )}^3}{{({{d^2} - {\omega^2}} )}^5}}}{{{T_1} + {T_2} - {T_3} - {T_4} - {T_5} - {T_6}}},$$
where
$${T_1} = 4{d^6}{\omega ^6}{({{d^2} - {\omega^2}} )^3}({{d^2} + {\omega^2}} );$$
$${T_2} = 4{d^6}{\omega ^{10}}{({{d^2} - {\omega^2}} )^2};$$
$${T_3} = 4{d^6}{\omega ^6}{({{d^2} + {\omega^2}} )^3}({{d^2} - {\omega^2}} );$$
$${T_4} = 4{d^6}{\omega ^8}{({{d^2} + {\omega^2}} )^3};$$
$${T_5} = 16{d^3}{\omega ^6}{({{d^2} + {\omega^2}} )^3}({{d^2} - {\omega^2}} );$$
$${T_6} = 16{d^3}{\omega ^8}{({{d^2} + {\omega^2}} )^3}.$$

Similarly, substituting $\eta$ with d, we can obtain the critical power P of the vortex dipole solitons as a function of nonlocality degree $d$

$$P = \frac{{\pi {{({{d^{^2}} + {\omega^2}} )}^{\frac{3}{2}}}{{({{d^{^2}} - {\omega^2}} )}^{\frac{5}{2}}}}}{{{{({{T_7} - {T_8} + {T_9} - {T_{10}} + {T_{11}} - {T_{12}}} )}^{\frac{1}{2}}}}},$$
where
$${T_7} = 4{d^{^9}}{({{d^2} - {\omega^2}} )^2};$$
$${T_8} = 3{d^{^5}}({{d^2} + {\omega^2}} ){({{d^2} - {\omega^2}} )^3};$$
$${T_9} = 3{d^{^5}}{({{d^2} + {\omega^2}} )^3}({{d^2} - {\omega^2}} );$$
$${T_{10}} = 4{d^{^7}}{({{d^2} + {\omega^2}} )^3};$$
$${T_{11}} = 6{d^{^2}}{({{d^2} + {\omega^2}} )^3}({{d^2} - {\omega^2}} );$$
$${T_{12}} = 16{d^{^4}}{({{d^2} + {\omega^2}} )^3}.$$

According to Eq. (17), the variational result for the critical power of vortex dipole solitons in nonlocal media in PT-symmetric Scarff-II potential with fundamental charge ($m = 1$) versus nonlocality degree d and the beam width $\omega$ is displayed in Fig. 1. It is clear that the critical power of the beams increases monotonously when the nonlocality degree increases, which results from the fact that the nonlocality is related to the power in nonlocal media. Increasing the power will enhance the nonlinearity. As nonlocality strengthens, a larger nonlinearity induced by the power is required to balance the diffraction of the beams. Moreover, one can see that the critical power of the solitons will increase when the beam width expands. On the contrary, from Eq. (10), it indicates that for a fixed k ($k = 1$), there is only critical power when the potential depth is at ${V_0} < 2.5$. As the potential depth increases, the critical power decreases to zero, indicating that there is a threshold. For higher-order charged vortex solitons, the effective Lagrangian as a function of the nonlocality degree d and beam width $\omega$ can be obtained by using the same approach discussed above. For simplicity, complicated mathematical analysis will no longer be exhibited.

 figure: Fig. 1.

Fig. 1. The critical power of the single charged ($m = 1$) vortex dipole solitons as a function of the degree of nonlocality d and the width $\omega$ of the soliton. The parameters are: potential depth ${V_0} = 8$, gain loss constant ${W_0} = 0.4$, propagation constant $k = 1$, respectively.

Download Full Size | PDF

 figure: Fig. 2.

Fig. 2. The critical power of the single charge ($m = 1$) vortex dipole solitons as a function of the propagation constant k and potential depth ${V_0}$. The parameters are: degree of nonlocality $d = 1$, gain loss constant ${W_0} = 0.4$, beam width $\omega = 1$.

Download Full Size | PDF

To illustrate the effect of nonlocality on the evolution of vortex dipole solitons, we display the transverse profiles of refractive index at different degrees of nonlocality in Fig. 3. The nonlinear refractive index is shown for weak, intermediate, and strong nonlocality with the beam width set as $\omega = 1$. The first column in Fig. 3 is weak nonlocality while the second is intermediate nonlocality. The third and Forth columns in Fig. 3 represent the strong nonlocality. The charges of vortex dipole solitons are $m = 1$ to $m = 5$ from the top row to the bottom row, respectively. From Eq. (5), it is apparent that nonlocality degree d plays an important role in the response function. As shown in Fig. 3, the corresponding widths increases as the vortex dipole solitons’ topological charge rises for the equivalent $A,\omega ,k$. Moreover, strong nonlocality induces a smooth and attractive potential that stabilizes the solitons.

 figure: Fig. 3.

Fig. 3. The profiles of the refractive index at different nonlocality degrees. The first, second, third and fourth columns correspond to the weak ($d = 0.5$), intermediate ($d = 1.5$), and strong ($d = 2.5$, $d = 5$) nonlocality, respectively. From the top to the bottom row, the topological charge of the vortex dipole solitons is $m = 1$, $m = 2$, $m = 3$, $m = 4$, and $m = 5$, respectively.

Download Full Size | PDF

3. Numerical analysis

The variational technique can only provide approximate analytical solutions. In what follows, we will numerically investigate the dynamics and stability of 2D vortex dipole solitons in nonlocal nonlinearity in the presence of PT-symmetric Scarff-II potential with the split-step Fourier transform method. We perturbed the stationary solutions with 5% noise and apply the variational results discussed above as the initial input of our two-dimensional numerical codes.

We first explore how the degree of nonlocality affects the propagation dynamics of the vortex dipole solitons in Fig. 4–6. There are three types of nonlocality in our discussion: weak nonlocality, intermediate nonlocality, and strong nonlocality. It is worth mentioning that even though we are concerned with only vortex dipole solitons with topological charge $m = 1,3,4$, our results apply to arbitrary values of m. The results indicate that the nonlocality degree strongly influences the propagation dynamics of vortex dipole solitons. For $m = 1$, when the nonlocality degree is weak, the solitons can maintain the annular profiles before $z = 200$. Then they break into quadrupole symmetry-breaking solitons (Fig. 4(a)). However, strong nonlocality induces an attractive force to stabilize the beams, which experience deformation and become symmetry-breaking solitons (Fig. 4(c)). The behavior of the solitons for $m = 3$ is similar, with the weak, intermediate nonlocality only providing temporary stabilization and strong nonlocality making the beams undergo stable deformation (Fig. 5). Due to the attractive force induced by strong nonlocality and soliton rotation generated by vortex soliton, the vortex dipole solitons with $m = 1$ and $m = 3$ experiences symmetry-breaking.

 figure: Fig. 4.

Fig. 4. The evolution of single charged vortex dipole solitons ($m = 1$) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different nonlocality degrees. The nonlocality degrees are: (a) $d = 0.5$, (b) $d = 1$, (c) $d = 3$. The first, second and third columns are the intensity profiles of solitons. The fourth and fifth columns correspond to solitons’ real parts and imaginary parts at $z = 0$, respectively. The parameters are fixed as ${W_0} = 0.4$, $\omega = 1$, $A = 2$.

Download Full Size | PDF

 figure: Fig. 5.

Fig. 5. The evolution of higher-order charged vortex dipole solitons ($m = 3$) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different nonlocality degrees. The nonlocality degrees are: (a) $d = 0.5$, (b) $d = 1$, (c) $d = 3$. The first, second and third columns are the intensity profiles of solitons. The fourth and fifth columns correspond to solitons’ real parts and imaginary parts at $z = 0$, respectively. The parameters are fixed as ${W_0} = 0.4$, $A = 2$.

Download Full Size | PDF

 figure: Fig. 6.

Fig. 6. The evolution of higher-order charged vortex dipole solitons ($m = 4$) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different nonlocality degrees. The nonlocality degrees are: (a) $d = 0.5$, (b) $d = 1$, (c) $d = 3$. The first, second and third columns are the intensity profiles of solitons. The fourth and fifth columns correspond to solitons’ real parts and imaginary parts at $z = 0$, respectively. The parameters are fixed as ${W_0} = 0.4$, $\omega = 1$, $A = 2$.

Download Full Size | PDF

As for the vortex dipole solitons with higher-order charges ($m = 4$), the numerical simulation is displayed in Fig. 6. In the case of weak nonlocality, the beams break up and scatter into twelve fundamental solitons. These solitons are stable within a relatively short distance $z = 28$, which is much shorter than the distances for $m = 1$ and $m = 3$, which are $z = 200$ and $z = 62$, respectively (Figs. 4(a),5(a),6(a)). At intermediate nonlocality, the stationary distances for the vortex soliton with $m = 1,3,4$ are $z = 350,98,40$, respectively (Figs. 4(b),5(b),6(b)). The more extended stationary propagation means the beam has better stability. From above, it is easy to discover that the vortex dipole solitons become more unstable with increasing topological charge. The numerical simulations also reveal that increasing the nonlocality degree will effectively improve the solitons’ stability and decrease splitting. However, completely stable vortex dipole solitons cannot exist regardless of the nonlocality degree. The results indicate that the nonlocality degree allows the vortex dipole solitons to undergo stable deformation. For the analytical results shown in Fig. 1, it is clear that for a fixed beam width, critical power only occurs when $d > 2$ (strong nonlocality). The simulations confirmed the existence of stable solitons under strong nonlocality, which is consistent with the variational results.

Next, we investigate the impact of the potential depth on the propagation of vortex dipole solitons with both single ($m = 1$) and higher ($m = 2$) topological charge. The relationship between the potential depth and the stability of the vortex dipole solitons with $m = 1$ is illustrated in Figs. 7. We find that there is a threshold for potential depth, below which the vortex dipole solitons can remain stable. Specifically, at ${V_0} = 1$ (Fig. 7(a)), the vortex solitons’ azimuthal instability and attractive force tend to be suppressed, allowing the beams to keep their original profile. At ${V_0} = 2.5$(Fig. 7(b)), the solitons exist again without splitting or scattering. However, in the case of ${V_0} = 3$ (Fig. 7(c)), the vortex dipole solitons can only be stabilized with a relatively long distance of $z = 350$. When the potential depth increases to ${V_0} = 5$, the beams will diffract at $z = 132$ (Fig. 7(d)).

 figure: Fig. 7.

Fig. 7. The evolution of single charged vortex dipole solitons ($m = 1$) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different potential depths. The potential depths are: (a) ${V_0} = 1$, (b) ${V_0} = 2.5$, (c) ${V_0} = 3$, (d) ${V_0} = 5$. The parameters are fixed as $d = 1$, $\omega = 1$, $k = 1$, ${W_0} = 0.4$.

Download Full Size | PDF

We also explore how the potential depth influences the stability of the higher-order charged ($m = 2$) vortex dipole solitons, as depicted in Fig. 8. It shows that vortex dipole solitons can only be stable when ${V_0} \le 3.5$ (Figs. 8(a),(b)). As the potential depth increases beyond this threshold, the stability of the beams deteriorates, as exhibited in (Figs. 8(c),(d)). The beam width is set as $\omega = 1$ in Figs. 7, 8. To further discuss this issue, we assume a higher-order charged ($m = 2$) vortex dipole soliton with a relatively large beam width $\omega = 1.5$, as shown in Fig. 9. The potential depth is particularly set to ${V_0} \le 3.5$. It is evident that the solitons propagate stably even though the vortex solitons deform into petal dipole solitons (Figs. 9(a),(b)). However, no completely stable soliton can be found when ${V_0} > 3.5$ (Figs. 9(c),(d)).

 figure: Fig. 8.

Fig. 8. The evolution of higher-order charged vortex dipole solitons ($m = 2$) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different potential depths. The potential depths are: (a) ${V_0} = 1$, (b) ${V_0} = 3.5$, (c) ${V_0} = 4$, (d) ${V_0} = 5$. The parameters are fixed as $d = 1$, $\omega = 1$, $k = 1$, ${W_0} = 0.4$.

Download Full Size | PDF

 figure: Fig. 9.

Fig. 9. The evolution of higher-order charged vortex dipole solitons ($m = 2$) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different potential depths. The potential depths are: (a) ${V_0} = 1$, (b) ${V_0} = 2.5$, (c) ${V_0} = 4.5$, (d) ${V_0} = 5$.The parameters are fixed as $d = 1$, $\omega = 1.5$, $k = 1$, ${W_0} = 0.4$.

Download Full Size | PDF

Besides, three points are worth noting. First, our results are applicable to arbitrary values of m, although we only discuss the effect of potential depth on the vortex dipole soliton with the values of $m = 1,2$. We find that there exists a threshold of potential depth ${V_0}$, above which, no absolutely stable vortex solitons can be supported. Moreover, the threshold of the potential depth for vortex dipole solitons with $m = 2$ is ${V_0} = 3.5$, which is larger than the beams with $m = 1$. Second, below the threshold of potential depth, the vortex solitons preserve their profile no matter whether the nonlocality is weak, intermediate, or strong. The solitons may deform or break into multipole solitons when other parameters are too large or small, but they can still be stabilized when the potential depth is below the stability threshold. The numerical simulation of the second point is analogous to Figs. 7–9, which will not be depicted again due to the limited space of this paper. Third, we find that the stability of the vortex dipole solitons depends solely on the potential depth. Moreover, there exists a threshold corresponding to the simulation results. Although the fundamental charged vortex dipole solitons can exhibit stable deformed solitons as the potential depth increases to near ${V_0} = 8$ (the numerical results similar to the simulation shown in Fig. 4(c)), the range of stable potential depth is too small to be ignored, which is consistent with the variational result in Fig. 2.

4. Conclusion

The dynamics and stability of the vortex dipole solitons with single and higher topological charge in nonlocal nonlinearity with PT-symmetric Scarff-II potential have been studied. Using the variational approach, we obtained the analytical solutions for the solitons. We also numerically demonstrated the dynamics and stability of the vortex dipole solitons using the split-step Fourier transform method, which showed that the nonlocality can decrease splitting and enhance stationary distances but cannot stabilize them. However, it enables the solitons to evolve into stable deforming solitons when the nonlocality degree is strong. Despite the deformation or splitting under certain conditions, it can also be stabilized when the potential depth is below the threshold. These numerical results are in good agreement with the results of the variational analysis.

Funding

Science and Technology Development Special Fund Projects of Zhongshan City (2019AG042, 2020AG023, 2019AG014); Guangzhou Municipal Science and Technology Bureau (2021ZD001); Science and Technology Planning Project of Guangdong Province (2020B010171001).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. A. Piccardi, S. Residori, and G. Assanto, “Nonlocal soliton scattering in random potentials,” J. Opt. 18(7), 07LT01 (2016). [CrossRef]  

2. D. Mihalache, “Localized structures in optical and matter-wave media: a selection of recent studies,” Rom. Rep. Phys. 73(2), 403 (2021).

3. P. S. Jung, Y. V. Izdebskaya, V. G. Shvedov, D. N. Christodoulides, and W. Krolikowski, “Formation and stability of vortex solitons in nematic liquid crystals,” Opt. Lett. 46(1), 62–65 (2021). [CrossRef]  

4. H. Zhang, Z. Weng, and J. Yuan, “Vector vortex breathers in thermal nonlocal media,” Opt. Commun. 492(5318), 126978 (2021). [CrossRef]  

5. Z. Shi, F. Badshah, and L. Qin, “Two-dimensional lattice soliton and pattern formation in a cold Rydberg atomic gas with nonlocal self-defocusing Kerr nonlinearity,” Chaos, Solitons Fractals 166, 112886 (2023). [CrossRef]  

6. C. Huang, L. Dong, H. Deng, X. Zhang, and P. Gao, “Fundamental and vortex gap solitons in quasiperiodic photonic lattices,” Opt. Lett. 46(22), 5691–5694 (2021). [CrossRef]  

7. O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66(4), 046619 (2002). [CrossRef]  

8. J. H. Guo and R. Guo, “Soliton solutions for two kinds of fourth-order nonlinear nonlocal Schrödinger equations,” Commun. Nonlinear Sci. Numer. Simul. 117, 106940 (2023). [CrossRef]  

9. H. Zhang, T. Zhou, and C. Dai, “Stabilization of higher-order vortex solitons by means of nonlocal nonlinearity,” Phys. Rev. A 105(1), 013520 (2022). [CrossRef]  

10. B. A. Malomed, “Two-dimensional solitons in nonlocal media: a brief review,” Symmetry 14(8), 1565 (2022). [CrossRef]  

11. L. Wu, L. Ge, Q. Kong, and M. Shen, “Nonlocal two-color vector dark solitons,” Results Phys. 48, 106433 (2023). [CrossRef]  

12. M. Shen, X. Chen, J. Shi, Q. Wang, and W. Krolikowski, “Incoherently coupled vector dipole soliton pairs in nonlocal media,” Opt. Commun. 282(24), 4805–4809 (2009). [CrossRef]  

13. L. Chen, Q. Wang, M. Shen, H. Zhao, Y. Y. Lin, C. C. Jeng, R. K. Lee, and W. Krolikowski, “Nonlocal dark solitons under competing cubic–quintic nonlinearities,” Opt. Lett. 38(1), 13–15 (2013). [CrossRef]  

14. W. Chen, M. Shen, Q. Kong, J. Shi, Q. Wang, and W. Krolikowski, “Interactions of nonlocal dark solitons under competing cubic–quintic nonlinearities,” Opt. Lett. 39(7), 1764–1767 (2014). [CrossRef]  

15. M. Mishra, S. K. Kajala, M. Sharma, S. Konar, and S. Jana, “Generation, dynamics and bifurcation of high power soliton beams in cubic-quintic nonlocal nonlinear media,” J. Opt. 24(5), 055504 (2022). [CrossRef]  

16. M. Shen, J. J. Zheng, Q. Kong, Y. Y. Lin, C. C. Jeng, R. K. Lee, and W. Krolikowski, “Stabilization of counter-rotating vortex pairs in nonlocal media,” Phys. Rev. A 86(1), 013827 (2012). [CrossRef]  

17. Q. Wang, M. R. Belić, D. Mihalache, L. Zeng, L. Zhang, and J. Lin, “Vortex chaoticons in thermal nonlocal nonlinear media,” Phys. Rev. E 106(5), 054214 (2022). [CrossRef]  

18. H. Q. Jin, J. C. Liang, Z. B. Cai, F. Liu, and L. Yi, “Three-Dimensional Hermite—Bessel—Gaussian Soliton Clusters in Strongly Nonlocal Media,” Chin. Phys. Lett. 29(12), 124207 (2012). [CrossRef]  

19. X. J. Lai, C. Q. Dai, X. O. Cai, and J. F. Zhang, “Azimuthally modulated vortex solitons in bulk dielectric media with a Gaussian barrier,” Opt. Commun. 353, 101–108 (2015). [CrossRef]  

20. G. Liang and Z. Dai, “Spiraling elliptic Hermite-Gaussian solitons in nonlocal nonlinear media without anisotropy,” Sci. Rep. 7(1), 3234 (2017). [CrossRef]  

21. G. Liang, “Spiraling elliptic breathers in nonlocal nonlinear media with linear anisotropy,” J. Mod. Opt. 65(13), 1609–1615 (2018). [CrossRef]  

22. X. Liu, Y. Qin, K. Ji, Y. Hu, H. Ren, and H. Zheng, “Rotation and oscillation of gap vortex solitons in optically induced square photonic lattices with a self-focusing nonlinearity,” Laser Phys. 31(4), 045401 (2021). [CrossRef]  

23. X. Huang, W. Tan, T. Jiang, S. Nan, Y. Bai, and X. Fu, “Propagation dynamics of modulated vortex beams in nonlocal nonlinear media,” Opt. Commun. 527, 128970 (2023). [CrossRef]  

24. M. Gao, Y. Chen, and M. Shen, “Elliptic fundamental, dipole and vortex solitons in nonlocal nonlinear media with linear anisotropic diffraction,” J. Opt. 22(2), 025502 (2020). [CrossRef]  

25. Y. Y. Lin and R. K. Lee, “Symmetry-breaking instabilities of generalized elliptical solitons,” Opt. Lett. 33(12), 1377–1379 (2008). [CrossRef]  

26. D. Buccoliero, S. Lopez-Aguayo, S. Skupin, A. S. Desyatnikov, O. Bang, W. Krolikowski, and Y. S. Kivshar, “Spiraling solitons and multipole localized modes in nonlocal nonlinear media,” Phys. B (Amsterdam, Neth.) 394(2), 351–356 (2007). [CrossRef]  

27. M. Shen, Y. Lin, C. C. Jeng, and R. K. Lee, “Vortex pairs in nonlocal nonlinear media,” J. Opt. 14(6), 065204 (2012). [CrossRef]  

28. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having P T symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998). [CrossRef]  

29. H. Wang, Z. Li, X. Ren, and Y. Weng, “Two-dimensional solitons in parity-time-symmetric optical lattices with nonlocal defocusing nonlinearity,” Opt. Express 24(20), 23063–23071 (2016). [CrossRef]  

30. H. Q. Zhang and M. Gao, “Rational soliton solutions in the parity-time-symmetric nonlocal coupled nonlinear Schrödinger equations,” Commun. Nonlinear Sci. Numer. Simul. 63, 253–260 (2018). [CrossRef]  

31. M. W. Chen, H. J. Hu, M. Zhu, Q. Q. Zhou, Z. C. Qiu, B. B. Li, Y. Zhao, L. Xue, and S. L. Xu, “Weak-light solitons and their active control in Rydberg-dressed parity-time symmetry moiré optical lattices,” Results Phys. 48, 106392 (2023). [CrossRef]  

32. X. Zhu, Z. Shi, and H. Li, “Gap solitons in parity-time-symmetric mixed linear-nonlinear optical lattices with fourth-order diffraction,” Opt. Commun. 382, 455–461 (2017). [CrossRef]  

33. M. Struwe and M. Struwe, Variational Methods (Springer-verlag, 2000).

34. T. P. Suneera and P. A. Subha, “Higher eigenmodes of nonlocal gap solitons in parity-time symmetric complex potential with a defocusing nonlinearity,” Chaos, Solitons Fractals 98, 183–188 (2017). [CrossRef]  

35. S. Hu, H. Chen, and W. Hu, “A variational solution to solitons in parity-time symmetric optical lattices,” Eur. Phys. J. Plus 132(9), 374 (2017). [CrossRef]  

36. N. B. Aleksić, M. S. Petrović, A. I. Strinić, and M. R. Belić, “Solitons in highly nonlocal nematic liquid crystals: variational approach,” Phys. Rev. A 85(3), 033826 (2012). [CrossRef]  

37. A. Khare, F. Cooper, and J. F. Dawson, “Exact solutions of a generalized variant of the derivative nonlinear Schrödinger equation in a Scarff II external potential and their stability properties,” J. Phys. A: Math. Theor. 51(44), 445203 (2018). [CrossRef]  

38. M. Shen, D. Wu, H. Zhao, and B. Li, “Vortex solitons under competing nonlocal cubic and local quintic nonlinearities,” J. Phys. B: At. Mol. Opt. Phys. 47(15), 155401 (2014). [CrossRef]  

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)
Fig. 1.
Fig. 1. The critical power of the single charged ( $m = 1$ ) vortex dipole solitons as a function of the degree of nonlocality d and the width $\omega$ of the soliton. The parameters are: potential depth ${V_0} = 8$ , gain loss constant ${W_0} = 0.4$ , propagation constant $k = 1$ , respectively.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. The critical power of the single charge ( $m = 1$ ) vortex dipole solitons as a function of the propagation constant k and potential depth ${V_0}$ . The parameters are: degree of nonlocality $d = 1$ , gain loss constant ${W_0} = 0.4$ , beam width $\omega = 1$ .

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. The profiles of the refractive index at different nonlocality degrees. The first, second, third and fourth columns correspond to the weak ( $d = 0.5$ ), intermediate ( $d = 1.5$ ), and strong ( $d = 2.5$ , $d = 5$ ) nonlocality, respectively. From the top to the bottom row, the topological charge of the vortex dipole solitons is $m = 1$ , $m = 2$ , $m = 3$ , $m = 4$ , and $m = 5$ , respectively.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. The evolution of single charged vortex dipole solitons ( $m = 1$ ) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different nonlocality degrees. The nonlocality degrees are: (a) $d = 0.5$ , (b) $d = 1$ , (c) $d = 3$ . The first, second and third columns are the intensity profiles of solitons. The fourth and fifth columns correspond to solitons’ real parts and imaginary parts at $z = 0$ , respectively. The parameters are fixed as ${W_0} = 0.4$ , $\omega = 1$ , $A = 2$ .

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. The evolution of higher-order charged vortex dipole solitons ( $m = 3$ ) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different nonlocality degrees. The nonlocality degrees are: (a) $d = 0.5$ , (b) $d = 1$ , (c) $d = 3$ . The first, second and third columns are the intensity profiles of solitons. The fourth and fifth columns correspond to solitons’ real parts and imaginary parts at $z = 0$ , respectively. The parameters are fixed as ${W_0} = 0.4$ , $A = 2$ .

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. The evolution of higher-order charged vortex dipole solitons ( $m = 4$ ) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different nonlocality degrees. The nonlocality degrees are: (a) $d = 0.5$ , (b) $d = 1$ , (c) $d = 3$ . The first, second and third columns are the intensity profiles of solitons. The fourth and fifth columns correspond to solitons’ real parts and imaginary parts at $z = 0$ , respectively. The parameters are fixed as ${W_0} = 0.4$ , $\omega = 1$ , $A = 2$ .

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. The evolution of single charged vortex dipole solitons ( $m = 1$ ) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different potential depths. The potential depths are: (a) ${V_0} = 1$ , (b) ${V_0} = 2.5$ , (c) ${V_0} = 3$ , (d) ${V_0} = 5$ . The parameters are fixed as $d = 1$ , $\omega = 1$ , $k = 1$ , ${W_0} = 0.4$ .

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. The evolution of higher-order charged vortex dipole solitons ( $m = 2$ ) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different potential depths. The potential depths are: (a) ${V_0} = 1$ , (b) ${V_0} = 3.5$ , (c) ${V_0} = 4$ , (d) ${V_0} = 5$ . The parameters are fixed as $d = 1$ , $\omega = 1$ , $k = 1$ , ${W_0} = 0.4$ .

View in Article | Download Full Size | PDF

Fig. 9.
Fig. 9. The evolution of higher-order charged vortex dipole solitons ( $m = 2$ ) in nonlocal nonlinearity with PT-symmetric Scarff-II potential at different potential depths. The potential depths are: (a) ${V_0} = 1$ , (b) ${V_0} = 2.5$ , (c) ${V_0} = 4.5$ , (d) ${V_0} = 5$ .The parameters are fixed as $d = 1$ , $\omega = 1.5$ , $k = 1$ , ${W_0} = 0.4$ .

View in Article | Download Full Size | PDF

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

{ i ∂ U ∂ z + ∂ 2 U ∂ x 2 + ∂ 2 U ∂ y 2 + V ( x , y ) U + σ n U = 0 d ( ∂ 2 n ∂ x 2 + ∂ 2 n ∂ y 2 ) + | U | 2 − n = 0 ,
W = ( sech 2 x ) ( sech 2 y ) ,
I = ( sech ( x ) tanh ⁡ ( x ) ) ( sech ( y ) tanh ⁡ ( y ) ) .
n = ∫ ∫ − ∞ + ∞ R ( x − x ′ , y − y ′ ) | U ( x ′ , y ′ ) | 2 d x ′ d y ′ ,
R ( x , y ) = 1 π d 2 exp ⁡ ( − x 2 + y 2 d 2 ) .
L c = i 2 ( U ∂ U ∗ ∂ z − U ∗ ∂ U ∂ z ) + ( | ∂ U ∂ x | 2 + | ∂ U ∂ y | 2 ) − 1 2 W | U | 2 − 1 2 n | U | 2 + 1 2 | n | 2 + d ( | ∂ n ∂ x | 2 + | ∂ n ∂ y | 2 ) .
U ( x , y , z ) = A ( x 2 + y 2 ) m exp ⁡ ( − x 2 + y 2 2 ω 2 ) exp ⁡ ( i k z + i m ( tan − 1 ( y x ) ) ) .
< L C > = ∫ ∫ − ∞ + ∞ L C d x d y = k A 2 ω 4 π + 7 2 ω 2 π A 2 − 1 2 V 0 ω 4 π A 2 − d 6 ω 8 π 2 ( d 4 − ω 4 ) 2 A 4 + d 6 ω 8 π 2 ( d 2 − ω 2 ) 2 A 4 + 2 d 3 ω 8 π ( d 2 − ω 2 ) 4 A 4 .
∂ ∂ z ( ∂ < L C > ∂ η z ) − ∂ < L C > ∂ η = 2 Re ( ∫ ∫ − ∞ + ∞ Q ∂ U ∗ ∂ η d x d y ) ,
A 2 = [ 4 k ω 2 + 7 − 2 ω 2 V 0 − 2 ω 2 V 0 W 0 ] ( d 2 + ω 2 ) 3 ( d 2 − ω 2 ) 5 T 1 + T 2 − T 3 − T 4 − T 5 − T 6 ,
T 1 = 4 d 6 ω 6 ( d 2 − ω 2 ) 3 ( d 2 + ω 2 ) ;
T 2 = 4 d 6 ω 10 ( d 2 − ω 2 ) 2 ;
T 3 = 4 d 6 ω 6 ( d 2 + ω 2 ) 3 ( d 2 − ω 2 ) ;
T 4 = 4 d 6 ω 8 ( d 2 + ω 2 ) 3 ;
T 5 = 16 d 3 ω 6 ( d 2 + ω 2 ) 3 ( d 2 − ω 2 ) ;
T 6 = 16 d 3 ω 8 ( d 2 + ω 2 ) 3 .
P = π ( d 2 + ω 2 ) 3 2 ( d 2 − ω 2 ) 5 2 ( T 7 − T 8 + T 9 − T 10 + T 11 − T 12 ) 1 2 ,
T 7 = 4 d 9 ( d 2 − ω 2 ) 2 ;
T 8 = 3 d 5 ( d 2 + ω 2 ) ( d 2 − ω 2 ) 3 ;
T 9 = 3 d 5 ( d 2 + ω 2 ) 3 ( d 2 − ω 2 ) ;
T 10 = 4 d 7 ( d 2 + ω 2 ) 3 ;
T 11 = 6 d 2 ( d 2 + ω 2 ) 3 ( d 2 − ω 2 ) ;
T 12 = 16 d 4 ( d 2 + ω 2 ) 3 .
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use