1. Introduction

Spatial soliton could maintain its profile during propagating in nonlinear media, due to the balance between spontaneous diffraction of light beams and nonlinear self-focusing effect (induced by inherent nonlinearity of some media). Nonlocality is one type of nonlinearities, and it means that the nonlinear response at a specific point depends on the light intensity in both the point and its neighborhood. This is quite different from the local response (depends on the single incident point) in Kerr medium. There are various materials that exhibit nonlocal response, such as nematic liquid crystals [1–3], photorefractive crystals [4,5] and thermo-optical materials [6]. In recent years, nonlocality is demonstrated to support many types of soliton, including bright solitons [1,2,4,7], dark solitons [8,9], surface solitons [10], multipole solitons [11,12] and breather solitons [13]. It is worth emphasizing that solitons with complex structure (such as multipole type) are more likely to generate in nonlocal media, which have been observed experimentally [12]. Among types of multipole solitons, the dipole solitons is the basic type.

Parity-time (PT) symmetric system was proven to have entirely real spectra even if its Hamiltonian is a non-Hermitian one [14]. In optics, the two-dimensional (2D) PT symmetric systems should meet condition $n(x,y)=n*(-x,-y)$, that is, the real part of the complex refractive-index distribution is even while the imaginary part is odd. Practically, the imaginary part of refractive-index can be achieved by inducing gain and loss [15,16]. One of the peculiar features of PT symmetric systems is the threshold for PT symmetry, above which PT symmetry breaking will occur, and the eigenvalues of the PT potential will no longer be all real [17,18]. Recently, the theoretical foundation and the relevant experiments of PT symmetric systems [19,20] have been reviewed, and the potential applications in photonic devices have been revealed. Furthermore, nonlinear media with PT symmetric potential are found to stabilize spatial solitons (including fundamental solitons, vector solitons, gray solitons) when the nonlinearities are local [21–25].

Spatial solitons were found to be generated and stabilized in nonlocal nonlinear lattices imprinted with PT symmetry [26–29]. Most references focused on one-dimensional(1D) solitons because their theoretical models are simpler and intuitive to exhibit the solitons propagation. Comparing with 1D case, 2D solitons can provide more details of the internal interaction, transverse profiles and phase structures. For instance, our previous work has demonstrated that the 2D nonlocal fundamental solitons in PT symmetric optical lattices tend to split and shift in the strong nonlocal nonlinearity [29]. In fact, the interaction between solitons or the internal interaction of solitons plays a crucial role in all optical switches and optical logic gates [30,31]. The multipole solitons in such specific lattices reveal more novel dynamics of the internal interactions of solitons,which is a problem to be further studies. From a practical point of view, the nonlocal nonlinear lattices with PT symmetry can be fabricated by interfering two ordinarily polarized broad laser beams in isotropic nonlinear media [32,33]. Multipole solitons can be generated by launching a proper laser in a nonlocal nonlinear medium. For instance, C. Rotschild et al. used an 1.8 W laser beam at 488 nm to induce the nonlocal nonlinearity of lead glass samples. They observed the scalar multipole solitons [12].

In this paper, we investigated the existence,stability and internal interaction of 2D multipole solitons in the PT symmetric medium with defocusing nonlocal nonlinearity. Considering the internal interaction of dipole solitons is the simplest of multipole solitons, we first simulated the propagation of in-phase and out-of-phase dipole solitons with numerical methods, and obtained their existence and stable parameter ranges. We also calculated the change of the gravity center of the dipole soliton, which reveals the joint influence of the nonlocality and the defocusing nonlinearity. Finally, two more complex multipole solitons, such as the quadrupole and the 8-hump solitons, have been studied.

2. Theoretical model

We consider a two dimensional phenomenological model, in which the light beam propagates through a nonlocal nonlinear medium imprinted with PT symmetry. In practice, the introduction of PT symmetry into optical systems requires the use of optical amplifiers. However, the amplifier will add ASE (amplifier spontaneous emission) noise into optical signals (i.e. solitons), which may deteriorate the propagation of solitons [34]. In this model, we only consider the effect of nonlinearity and PT symmetric potential on the internal interaction of soliton.Therefore,we assure that the ASE is ignored.The model can be described by the normalized 2D nonlinear Schrödinger coupled equation as follow [29,35,36]:

The refractive index n, i.e. the nonlinearity, is strongly associated with the intensity $I(x,y)$. Its general form is:

Physically, x and y are scaled to the width of input beam w,z is scaled to the diffraction length $L_{diff}=2n_0{k}_0{w}^2$. Where $n_0$ is the background refractive index and $k_0=2\pi /{\lambda }_0$ is the wavenumber.

It is clear from the Eq. (3) that the degree of nonlocality d plays a key role in the response function, equally the distribution of refractive index. We obtain the transverse profiles of the refractive index at different nonlocality degree, as shown in Fig. 1. The weak nonlocality ($d=0.05$) is equivalent to locality, as shown in Fig. 1(a). With the increase of the degree of nonlocality $d$, the nonlocality becomes intermediate ($d=0.5$) or strong ($d=3$), and the profiles of the corresponding refractive index also expand, as shown in Figs. 1(a) and 1(b).

 

Fig. 1 The profiles of the refractive index of (a) the weak nonlocality, (b) the intermediate nonlocality and (c) the strong nonlocality.

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Soliton solution of Eq. (1) can be sought in the form $U(x,y,z)=q(x,y,z)e^{i\mu z}$, where $q(x,y)$ is the complex function and $\mu$ is the propagation constant. Then Eq. (1) become:

Equation (4) can be numerically solved by the developed modified squared-operator method (MSOM) [30]. The power of solitons is defined as $P={\displaystyle \int {\displaystyle {\int }_{-\infty }^{+\infty }{\left|q\right|}^2}}dxdy$.

We investigate the stability of solitons by adding a small perturbation:

Substituting Eq. (5) into Eq. (1) and reducing the equations, we obtain the eigenvalue equations:

3. Numerical analysis

3.1 Out-of-phase dipole solitons

We first investigate out-of-phase dipole solitons in the PT symmetric defocusing nonlocal nonlinear medium. We find that dipole solitons can only exist in the first gap ($6.82<\mu <9.72$). There are three types of nonlocalities can support these solitons, i.e. weak, intermediate and strong nonlocality (Fig. 2). We obtain the power curves and find that the power of the dipole solitons will decrease with the increase of $\mu$. Moreover, when the degree of nonlocality d increases, the power of soliton increases as well, as shown in Fig. 2. It is worth mentioning that the power curve of the dipole soliton with the local nonlocality (d=0) is very close to the power curve with the weak nonlocality (d=0.05). But in our model these local dipole solitons are all unstable, whether it is in-phase or out-of-phase. They are unable to maintain their profiles during propagation, and their gravity centers suffer strong oscillation.

 

Fig. 2 The power curves of the dipole solitons with different nonlocalities in the first gap. The blue lines indicate the in-phase dipole solitons and the red lines indicate the out-of-phase dipole solitons. The dash lines represent the unstable solitons and the green solid lines represent the stable solitons. The gray areas represent the energy bands. ${\sigma }_1=-1,$ $W_{0}0.3$ ,$V_0=8$.

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The main characteristic of the out-of-phase soliton is that the imaginary parts of each hump are different, as shown in Fig. 3(b). To investigate their stability, we simulate the propagation of these dipole solitons in the PT symmetric lattices. Taking the weak-nonlocal (d=0) dipole soliton with $\mu =9.1$ as an example, it corresponds to point A in Fig. 2. The change of its gravity center is illustrated in Fig. 3(c). Here, the gravity center of an optical field is defined as $(x_0{y}_0)=\frac{1}{P}{\displaystyle \int {\displaystyle {\int }_{+\infty }^{+\infty }(x,y)}}{\left|u\right|}^{2}dxdy$. At the beginning of the propagation ($z=0$), the gravity center is $(x_0{y}_0)=(\pi /2,0)$. Thus the change of gravity center can be defined as $\Delta =\pm \sqrt{{(x^{\prime }-x_0)}^2+{(y^{\prime }-y_0)}^2}$, where $\pm$ represent $(x^{\prime },y^{\prime })$ is on the right (+) or left (-) of $(x_0{y}_0)$. As shown in Fig. 3(c), the gravity center of this dipole soliton first moves to the left and then suffers an oscillation during propagation. The intensity profiles of the soliton at $z=0,100,160$ are shown in Figs. 3(a), 3(d) and 3(c). It is clear that the right hump decays quickly, and it acts as a “breather” [13]. At $z=100$, corresponding to one trough of the oscillation curve, the right hump of the soliton is quite weak, thus the gravity center is close to the position of the left hump $(0,0)$. At $z=160$, corresponding to one ridge of the oscillation curve, the right hump is still weak, but it is strong enough to make the gravity center close to its original position $(\pi /2,0)$. The oscillation is mainly because of the joint influence of the nonlinearity and the nonlocality. First, one hump of the out-of-phase dipole soliton decays, so the gravity center moves close to another hump. Then due to the nonlocality, the waveguide (caused by this dipole soliton) forces the right hump focusing, leading to an increase in the power of the decaying hump. At the same time, the defocusing nonlinearity causes the dipole soliton diffusion. Both nonlinearity and nonlocality affect the dipole soliton, which leads to the vibration of the power of one hump and the oscillation of the gravity center. The linear stability spectra of this dipole soliton is shown in Fig. 3(f), and the real part of the growth rate $\lambda$ is zero, which means this dipole soliton is linearly stable during propagation. In fact, we find that the out-of-phase dipole solitons are stable during propagation in a small range in the weak nonlocality case ($(8.6\le \mu \le 9.2,d=0.05)$). They are found to be unstable (i.e. they dissipates quickly) in the intermediate or strong nonlocality case, as shown as the red dash lines in Fig. 2.

 

Fig. 3 The out-of-phase dipole solitons in a PT symmetric weak-nonlocal medium with $\mu =9.1$, $d=0.05$, corresponding to point A in Fig. 2. (a), (d) and (e) are the profiles of the dipole soliton at z = 0, 100 and 160. (b) is the imaginary part of the dipole soliton. (c) is the change of the gravity center during propagation. (f) is the spectrum of perturbation growth rate.

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3.2 In-phase dipole solitons

Different from the out-of-phase dipole solitons, the in-phase dipole solitons are more stable, and their power is larger (blue lines in Fig. 2). The imaginary parts of each hump are same, as shown in Fig. 4(h). We find that in-phase dipole solitons can exist in almost the whole first gap (_{$6.82<\mu <9.72$}), and some of them can be stable when the nonlocality is weak (_{$8.4<\mu <9.0$}) and intermediate (_{$8.6<\mu <9.3$}). As the propagation constant $\mu$ increases, the power of the in-phase dipole solitons decreases, which is same as out-of-phase dipole solitons.

 

Fig. 4 The in-phase dipole solitons in a PT symmetric medium with intermediate nonlocality ($d=0.5$). The upper row represents the stable dipole soliton with _{$\mu =9.2$}, corresponding to point B in Fig. 2. The lower row represents the unstable dipole soliton with $\mu =8.4$, corresponding to point C in Fig. 2. (a), (b), (e) and (f) are the profiles of the corresponding dipole solitons. (c) and (g) are the spectrums of perturbation growth rate. (d) is the change of the gravity center of the dipole soliton with _{$\mu =9.2$} during propagation. (h) is the imaginary part of the in-phase solitons.

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In intermediate nonlocality case, in-phase dipole solitons can be stable during propagation. For example, when _{$\mu =9.2$}, the dipole soliton is found (corresponding to point B in Fig. 2). It can maintain its intensity profile and power in a long distance (more than $z=500$), as shown in Figs. 4(a) and 4(b). But the gravity center of this soliton suffer an oscillation, and the change of the gravity center is illustrated in Fig. 4(d). The oscillation is very small, that is, the gravity center is slightly vibrating during propagation. Thus the “breather-like” behavior of the right hump is imperceptible. The reason for the oscillation of the gravity center is the joint influence of nonlinearity and defocusing nonlocality, which is same as the out-of-phase dipole solitons. Moreover, we obtain its linear stability spectra, of which the real part of $\lambda$ is zero, as shown in Fig. 4(c). That is, this dipole soliton is linearly stable during propagation. We also elucidate another dipole soliton with _{$\mu =8.5$}, corresponding to point C in Fig. 2. Its intensity profiles at _$z=0,40$ are shown in Figs. 4(e) and 4(f). It is clear that this soliton diffuses quickly, which means that this soliton is unstable during propagation. And its linear stability spectra also verifies the instability [_{$\mathrm{Re}(\lambda )\ne 0$}], as shown in Fig. 4(g).

We also investigate the stability of in-phase dipole solitons in the strong nonlocality. Figure 5 represents the dipole soliton with $\mu =9.4$,$d=3$. The gravity center of this dipole soliton is illustrated in Fig. 5(d). We can see that this soliton can keep its basic profile before _$z=130$, as shown in Fig. 5(d). After that (_$z>130$), the soliton diffuses into the whole space. The intensity profiles of this soliton are shown in Figs. 5(a) and (b). When _$z=40$, the right hump of the dipole soliton becomes weak, causing the gravity center moves left. When _$z=80$, the left hump becomes weak and the right hump recovers, thus the gravity center moves right. However, the “competition” between the two humps is slight, and the major result of the “competition” is the oscillation of the gravity center. From the linear stability spectra in Fig. 5(c), the real part of the instability growth rate is nonzero, which indicates that the dipole soliton with $\mu =9.4$, $d=3$ is linearly unstable. Moreover, the further calculation demonstrates that the media with strong nonlocality are unable to support stable in-phase dipole solitons.

 

Fig. 5 The in-phase soliton in a PT symmetric strong-nonlocal medium with $\mu =9.4$,$d=3$. (a) and (b) is the profile of the dipole soliton at z = 40 and 80. (c) is the spectrum of perturbation growth rate. (d) is the change of the gravity center of the dipole soliton during propagation.

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To investigate the influence of nonlocality and PT symmetric potential on the existence and stability of dipole solitons, we carry out further calculations by varying the degree of nonlocality $d$ and the gain-loss component $W_0$. The existence and stability of dipole solitons with variable $d$ in defocusing PT symmetric lattices ($\sigma =-1$, $V_0=8$, $W_0=0.3$) is illustrated in Fig. 6. The gray areas in Fig. 6 represent the energy bands, where we cannot find a dipole soliton. In order to illustrate clearly, we take the x-axis as the logarithmic coordinate. It is worth emphasizing that there is no definite boundary between three kinds of nonlocalities. The difference between weak/intermediate nonlocalities or intermediate/ strong nonlocalities is about one order of magnitude. Therefore we set “weak”, “intermediate” and “strong” areas in Fig. 6. We find dipole solitons can exist inside the blue lines, and can be stable in the green area. In weak-nonlocality area, the stable range shrinks with the decrease of $d$. When $d\to 0$, we cannot find a stable dipole soliton. In intermediate-nonlocality area, the stable range shrinks with the increase of $d$. In strong-nonlocality area, dipole solitons can still exist but they are unstable. The exist area is $6.85\le \mu \le 9.70,d\le 3.8$, and the stable area is $7.80\le \mu \le 9.30,d\le 1.4$. In one word, PT symmetric dipole solitons tend to be stable in intermediate nonlocal media, and they exist when $d$ is not too large.

 

Fig. 6 The existence and stable range of the in-phase solitons with variable degree of nonlocality ($\sigma =-1$,$V_0=8$,$W_0=0.3$). The gray areas indicate the energy bands. The area inside the blue lines is the existence range and the green area is the stable range. The difference between different nonlocalities is about one order of magnitude. The x-axis is a logarithmic coordinate.

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The optical system considered in this paper is PT symmetric when the strength of the gain-loss component $W_0$, i.e. the imaginary part of PT symmetric $V(x,y)$, meets $W_0<0.5$. If _{$W_0\ge 0.5$}, the upper energy band and the lower energy band would merge together, and the PT symmetry would be broken. The influence of $W_0$ on the existence and stability of dipole solitons is illustrated in Figs. 7(a), 7(b) and 7(c), corresponding to weak ($d=0.05$), intermediate ($d=0.5$) and strong ($d=3$) nonlocality, respectively. The gray areas represent the energy bands, and the first bandgap is between the energy bands. We can see that when the nonlocality is weak [Fig. 7(a)], the dipole solitons almost exist in the whole bandgap. The stable area is $7.3\le \mu \le 10.5,W_0\le 0.356$, and it shrinks as $W_0$ becomes larger. When the nonlocality is intermediate, It is found that dipole solitons do not exist or stabilize near the lower energy band with larger $W_0$,which lead to a little “tail” of the existence and stable areas. The stable area is $7.3\le \mu \le 10.6,W_0\le 0.385$, which is wider than that of weak nonlocality. When the nonlocality becomes stronger, the existence range becomes narrow and the “tail” is apparent, which means the dipole solitons with large $W_0$ can only exist in a restricted range. They are found to be stable in a small range of parameters (_{$7.3\le \mu \le 10.7$},_{$W_0\le 0.102$}). In summary, $W_0$ has different effects on the existence and stability of dipole solitons with different nonlocality. Dipole solitons are easier to be stable in a PT symmetric nonlocal medium when the medium is intermediate-nonlocal ($d\sim {10}^{-1}$), and its gain-loss component is not too large ($W_0<0.2$).

 

Fig. 7 The existence and stability of the in-phase solitons with varied gain-loss components. (a) is the weak nonlocality case. (b) is the intermediate nonlocality case. (c) is the strong nonlocality case. The gray areas is the energy bands. The upper energy band and the lower energy band merges when $W_0=0.5$. The area inside the blue lines represents the existence range and the green area represents the stable range.

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3.3 Multipole solitons

Since the dipole solitons can be treated as the basic type of multipole solitons, a natural supposition is that the “breather-like” or “competition” behavior is possible to be found in the propagation of multipole solitons. To confirm our supposition, we make a further calculation on two kinds of symmetrical multipole solitons, i.e., quadrupole solitons and 8-hump solitons. They are found in the first gap, but are more unstable than dipole solitons.

Quadrupole solitons exist when the nonlocality is not too strong, that is $d<2.8$. With specific parameters, quadrupole solitons can propagate stably in a relatively long distance (_$z>150$). The quadrupole soliton with intermediate nonlocality ($d=0.5,\mu =9$) is shown in Figs. 8(a) to 8(e). It can maintain its profile until the energy is concentrated in one hump, leading to the destruction of the profile and its symmetry [Figs. 8(a) and (b)]. Before that (_$z<290$), the energy of the quadrupole soliton keeps transferring from one hump to another, resulting in the “breather-like” behavior and the oscillation of its gravity center, as shown in Fig. 8(c). Moreover, the imaginary part and the phase structure of this quadrupole soliton are presented in Figs. 8(d) and (e). As the nonlocality turns stronger, few quadrupole solitons can be stabilized, even their intensity profiles and the phase structures are similar to those soltions under intermediate nonlocalities. We consider the quadrupole soliton with strong nonlocality ($d=2,\mu =9.65$), and obtain the change of its gravity center, as shown in Fig. 8(f). It is worth noting that the periods of the gravity center oscillations become longer with the enhancement nonlocality. It is mainly because the stronger nonlocality expands the profiles of the refractive index more significantly (Fig. 1), and the differences between the refractive indexes of adjacent humps become smaller, which causes the efficiencies of energy transferring decrease.

 

Fig. 8 The quadrupole solitons in a PT symmetric nonlocal medium with (a-e) $d=0.5$,$\mu =9$, , (f) $d=3$,$\mu =9.65$. (a) and (b) are the profile of the quadrupole soliton with intermediate nonlocality at z = 0 and 290. (c) is the change of the gravity center of the quadrupole soliton with intermediate nonlocality. (d) is the imaginary part of quadrupole solitons. (e) is the phase structure of quadrupole solitons. (f) is the change of the gravity center of the quadrupole soliton with strong nonlocality.

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For 8-hump solitons, they exist in the weak- or intermediate-nonlocal medium with PT symmetry. However, neither of them can keep their profiles beyond the distance of $z=100$, according to our numerical simulation. The 8-hump solitons suffer great oscillations of gravity center, and diffuse into the whole lattices quickly. The profiles of these 8-hump solitons resemble the stable vortex solitons with higher topological charge ($m=4$) and the supervortex with unit charge ($m=1$) [39]. The main difference lies in their phase structures. The 8-hump solitons that we studied are of in-phase structure, and they are very unstable compared with vortex solitons. We choose the most stable 8-hump soliton ($d=0.5,\mu =9$) to illustrate in detail, as shown in Fig. 9. The 8-hump soliton maintains its basic profile at the beginning of propagation, but the intensity of each hump varies greatly [Figs. 9(a) and 9(b)]. Then the instability of the intensity leads to the eventual collapse of the basic 8-hump profile [Fig. 9(c)]. A detailed calculation on the gravity center of this soliton is presented in Fig. 9(d). It is obvious that the gravity center of the 8-hump soliton has experienced a great oscillation. In fact, the oscillation is nearly stochastic, due to the drastic interaction between the humps of the soliton. In contrast to dipole solitons and quadrupole solitons, the 8-hump solitons have too many components, i.e. humps, to balance with the nonlocal nonlinear medium, which makes them very difficult to stabilize with the internal interaction.

 

Fig. 9 The 8-hump solitons in a PT symmetric intermediate-nonlocal medium with $d=0.5$,$\mu =9$. (a), (b) and (c) are the intensity profile of the 8-hump soliton at z = 0, 50 and 100. (d) is the change of the gravity center of the 8-hump soliton.

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4. Conclusion

We investigated the existences and stabilities of dipole, quadrupole and 8-hump solitons in defocusing nonlocal nonlinear media with PT symmetry. According to our numerical calculations, the intermediate nonlocality is the most promising type to stabilize the multipole solitons. Since dipole solitons are the basic types of multipole solitons, we compared the in-phase and out-of-phase dipole solitons in detail, and demonstrated that in-phase dipole solitons are more stable. Moreover, we obtained the existence and stable range of dipole solitons under different nonlocalities and gain-loss components.

We also considered the gravity centers of dipole solitons and found that there are the behaviors of “breather-like” or “competition” in different nonlinearities, which are the results of the periodic oscillation of the gravity centers. Essentially, this is due to the internal interaction between the humps of dipole solitons. We numerically simulated the propagation of quadrupole and 8-hump solitons. Therefore, the quadrupole solitons do have the “breather-like” behavior, which leads to the periodic oscillations of the gravity centers. And the 8-humps solitons have the strong “competition” behavior, which make them difficult to stabilize.