## Abstract

We theoretically report the influence of a class of near-parity-time-(*𝒫𝒯*-) symmetric potentials on solitons in the complex Ginzburg-Landau (CGL) equation. Although the linear spectral problem with the potentials does not admit entirely-real spectra due to the existence of spectral filtering parameter *α*_{2} or nonlinear gain-loss coefficient *β*_{2}, we do find stable exact solitons in the second quadrant of the (*α*_{2}, *β*_{2}) space including on the corresponding axes. Other fascinating properties associated with the solitons are also examined, such as the interactions and energy flux. Moreover, we study the excitations of nonlinear modes by considering adiabatic changes of parameters in a generalized CGL model. These results are useful for the related experimental designs and applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The cubic complex Ginzburg-Landau (CGL) equation [1]

*A*=

*A*(

*x*,

*z*) is a complex field,

*α*

_{1,2},

*β*

_{1,2}, and

*γ*are all real parameters, is one of the most universal and significant nonlinear wave models in many areas of the physics community, describing all kinds of nonlinear phenomena, such as superfluidity, superconductivity, hydrodynamics, plasmas, reaction-diffusion systems, quantum field theory and Bose-Einstein condensation (BEC), liquid crystals, and strings in the field theory and other physical contexts [1–3]. The CGL equation can be regarded as a dissipative extension of the conservative nonlinear Schrödinger equation describing nonlinear optics, BEC, and waves on the deep water. The CGL equation can support stable spatial patterns on account of the simultaneous balance of gain and loss, as well as nonlinearity versus dispersion or diffraction. Intriguingly, a vast variety of applications and physical properties in the CGL equations are well elaborated in nonlinear optics [4–10], where various types of dissipative solitons emerge and are analyzed in detail, including multi-peak solitons [11], exploding solitons [12, 13], pulsating solitons [14], chaotic solitons [15], two-dimensional vortical solitons [16], three-dimensional spatiotemporal optical solitons [8,17,18], accessible solitons [19], and lattice solitons [20,21].

The *𝒫𝒯*-symmetry due to Bender and his coworker in 1998 [22–24], is an extremely crucial property and widely applied to complex potentials to possibly support all-real linear spectra [22,25] and stable nonlinear modes [26–34]. Many fascinating features and properties related to *𝒫𝒯* behaviors such as the celebrated *𝒫𝒯*-symmetry breaking phenomenon have been observed or demonstrated in optical experiments [35–43]. Indeed, the *𝒫𝒯*-symmetric structures can be easily achieved in optics by including a combination of optical gain and loss regions in the refractive-index guiding geometry [26, 44]. Particularly, in the periodic optical lattice potentials, a great number of novel *𝒫𝒯*-symmetric behaviors have also been experimentally observed such as the double refraction, secondary emissions, power oscillation, and phase singularities [45–47]. In the last few years, much attention has been concentrated on exploring one- and multi-dimensional solitons and their stability in all stripes of optical potentials, including the harmonic potential [48], Scarf-II potential [26–28,49,50], Rosen-Morse potential [51], Gaussian potential [33, 52, 53], super-Gaussian potential [54], optical lattices or super lattices [31, 32, 55–58], photonic systems [59], time-dependent harmonic-Gaussian potential [30], sextic anharmonic double-well potential [29], the double-delta potential [60,61], and etc. [62–66]. Recently, *𝒫𝒯*-symmetric stable nonlinear localized modes and dynamics were also elucidated in the generalized Gross-Pitaevskii (GP) equation with a variable group-velocity coefficient [67], the third-order nonlinear Schrödinger equation (NLSE) [68], the NLSE with position-dependent effective masses [69], the derivative NLSE [70], the NLSE with generalized nonlinearities [71], the nonlocal NLSE [72], the NLSE with spatially-periodic momentum modulation [73], and the three-wave interaction models [74].

Besides, stable solitons have been investigated theoretically in the NLSE with some non-*𝒫𝒯*-symmetric potentials [67, 75–81]. Notice that the dynamical behaviors of spatial dissipative solitons have been discussed in the cubic-quintic CGL equation with the *𝒫𝒯*-symmetric periodic potential [20, 21]. However, the non-*𝒫𝒯*-symmetric potentials have scarcely been studied in the CGL equation. Therefore, we, in this paper, aim to explore that a broad class of *𝒫𝒯*-symmetric stable exact solitons can exist in the cubic CGL model with non-*𝒫𝒯*-symmetric potentials. We also find that the non-*𝒫𝒯*-symmetric potentials can be bifurcated out from the *𝒫𝒯*-symmetric potential by regulating the related potential parameters, which thus are called the near *𝒫𝒯*-symmetric potentials. Furthermore, in the context of CGL model, various dynamical properties associated with the exact solitons are also analyzed and elucidated in detail under the near *𝒫𝒯*-symmetric potentials. These results are beneficial for applying them in the related experimental designs.

## 2. *𝒫𝒯*-symmetric nonlinear physical model

When an optical pulse with white noise goes through a planar slab waveguide, the upper part will suffer energy gain while the lower one experience energy loss (see Fig. 1). We predicate that the propagation of the pulse will be unstable if the gain and loss are unbalanced (which can be realized by regulating the spectral filtering parameter and nonlinear gain-loss coefficient), because the linear system with a complex diffraction coefficient always has no purely-real spectra. However, such a system can support a wide range of stable *𝒫𝒯*-symmetric solitons in the complex-coefficient Kerr medium, even though the complex refractive index distribution is non-*𝒫𝒯*-symmetric. To verify the above-mentioned idea, we begins by considering the spatial beam transmission in a cubic-nonlinear optical medium described by the following CGL equation with complex potentials [9,20]

*A*≡

*A*(

*x*,

*z*) is the normalized envelope of the complex light field,

*z*denotes the propagation distance, and

*x*represents the scaled spatial coordinate; for the convenience of study, both the diffraction coefficient

*α*

_{1}and Kerr-nonlinearity coefficient

*β*

_{1}are fixed as

*α*

_{1}=

*β*

_{1}= 1 in the paper; the real parameter

*α*

_{2}can be used to describe the spectral filtering or linear parabolic gain (

*α*

_{2}> 0), and the real constant

*β*

_{2}accounts for the nonlinear gain/loss processes. Different from the traditional GL equations [11–15, 82], we introduce the complex potential

*V*(

*x*) +

*iW*(

*x*) instead of the constant linear gain-loss coefficient. Compared with those discussed in [9, 20], the spectral filtering coefficient

*α*

_{2}is added such that it is possible to exhibit some distinct behaviors. The complex potential

*V*(

*x*) +

*iW*(

*x*) is

*𝒫𝒯*-symmetric provided that

*V*(

*x*) =

*V*(−

*x*) and

*W*(−

*x*) = −

*W*(

*x*). Physically, the real-valued external potential

*V*(

*x*) is closely related to the refractive index waveguide while

*W*(

*x*) characterizes the amplification (gain) or absorption (loss) of light beam in the optical material. On account of the occurrence of complex coefficients, equation (2) is not

*𝒫𝒯*-symmetric, where the operators

*𝒫*and

*𝒯*are defined by

*𝒫*:

*x*→ −

*x*;

*𝒯*:

*i*→ −

*i*,

*z*→ −

*z*, respectively. Besides, equation (2) can also be rewritten as another variational form

*iA*=

_{z}*δℋ*(

*A*)/

*δA*

^{*}, where the Hamiltonian $\mathscr{H}(A)={\int}_{-\infty}^{+\infty}\left\{\left({\alpha}_{1}+i{\alpha}_{2}\right){\left|{A}_{x}\right|}^{2}-\left[V(x)+iW(x)\right]{\left|A\right|}^{2}-\frac{1}{2}({\beta}_{1}+i{\beta}_{2}){\left|A\right|}^{4}\right\}dx$ and the asterisk denotes the complex conjugate. If we define the optical power of Eq. (2) as $P(z)={\int}_{-\infty}^{+\infty}{\left|A(x,z)\right|}^{2}dx$, then one can elicit immediately that the power evolves by ${P}_{z}={\int}_{-\infty}^{+\infty}\left[2{\alpha}_{2}{\left|{A}_{x}\right|}^{2}-{\alpha}_{2}{\left({\left|A\right|}^{2}\right)}_{xx}-2W(x){\left|A\right|}^{2}-2{\beta}_{2}{\left|A\right|}^{4}\right]dx$. Moreover, when setting

*z*→

*z*(cavity round-trip number) and

*x*→

*t*(retarded time) in Eq. (2), the aforementioned model may be used to describe the passively mode-locked lasers too [83].

## 3. Theoretical analysis

#### 3.1. Stationary solitons and linear-stability theory

Stationary solutions of Eq. (2) are explored in the form *A*(*x*, *z*) = *ϕ*(*x*)*e ^{iqz}*, where

*q*is a real propagation constant. Substituting it into Eq. (2), one can derive at once that the complex localized field-amplitude function

*ϕ*(

*x*) (lim

_{|x|→∞}

*ϕ*(

*x*) = 0 for

*ϕ*(

*x*) ∈

*C*[

*x*]) satisfies the following second-order ordinary differential equation (ODE) with complex coefficients

To investigate the linear stability of stationary solutions *ϕ*(*x*)*e ^{iqz}*, we perturb them in the vicinity of the singular point

*∊*| ≪ 1,

*f*(

*x*) and

*g*(

*x*) are the perturbation eigenfunctions, and

*δ*reveals the perturbation growth rate. Inserting this perturbed solution (4) into Eq. (2) and linearizing with respect to

*∊*yield the following linear-stability eigenvalue problem

*L̂*

_{1}= (

*α*

_{1}+

*iα*

_{2})

*∂*+

_{xx}*V*(

*x*) +

*iW*(

*x*) + 2(

*β*

_{1}+

*iβ*

_{2})|

*ϕ*|

^{2}−

*q*and

*L̂*

_{2}= (

*β*

_{1}+

*iβ*

_{2})

*ϕ*

^{2}. It is more than evident that the nonlinear localized modes are linearly unstable if

*δ*possesses a positive real part, otherwise they are linearly stable. In practice, the linear stability is determined by the maximal value of real parts of the linearized eigenvalues

*δ*, i.e., max [ℜ(

*δ*)]. The full stability spectrum of

*δ*can be numerically computed by the Fourier collocation method (see [86]).

#### 3.2. Near *𝒫𝒯*-symmetric Scarf-II potential

In what follows, we initiate our analysis by introducing the following near *𝒫𝒯*-symmetric Scarf-II potential

*V*

_{0}and

*W*

_{0}can be used to modulate the strength of the real and imaginary parts of the complex potential. It is evident that the aforementioned complex potential

*V*(

*x*) +

*iW*(

*x*) reduces to the usual

*𝒫𝒯*-symmetric Scarf-II potential at once if

*α*

_{2}=

*β*

_{2}= 0, meanwhile Eq. (2) becomes the well-known

*𝒫𝒯*-symmetric NLSE. However, when

*α*

_{2}or

*β*

_{2}is perturbed around the origin of the (

*α*

_{2},

*β*

_{2}) space, equation (2) turns into the complex cubic GL equation, and the corresponding complex potential is not

*𝒫𝒯*-symmetric. We call such a complex potential near

*𝒫𝒯*-symmetric in the (

*α*

_{2},

*β*

_{2}) parameter space. In addition, it is also apparent that the aforementioned complex potential possesses even symmetry if

*W*

_{0}= 0, due to

*V*(

*x*) =

*V*(−

*x*) and

*W*(

*x*) =

*W*(−

*x*).

## 4. Spectral problems and linear stability of nonlinear modes

#### 4.1. Unbroken or broken near *𝒫𝒯*-symmetric phases

Next we turn to investigate the unbroken or broken phases in the near *𝒫𝒯*-symmetric potential (6) by considering the linear eigenvalue problem

*λ*and Φ(

*x*) stand for the eigenvalue and eigenfunction, respectively. Unluckily, abundant numerical results indicate that unbroken-phase regions barely exist in the potential parameter (

*V*

_{0},

*W*

_{0}) space, unless (

*α*

_{2},

*β*

_{2}) = (0, 0) which means the linear operator

*L*is

*𝒫𝒯*-symmetric. It fully reveals that the

*𝒫𝒯*symmetrility of a complex potential in a Hamiltonian is of great importance to ensure the real property of spectra. For illustration, we take

*V*

_{0}= 1 in Eq. (6) to illustrate the spontaneous symmetry-breaking process, which stems from the collision of the first few lowest energy levels. Figures 2(a1) and 2(a2) display the classical situation of

*𝒫𝒯*-symmetric Scarf-II potential, with the phase-transition point

*W*

_{0}= 1.25. However, only if

*α*

_{2}or

*β*

_{2}is not equal to zero, there always exist at least an imaginary eigenvalue in the linear spectra (see the last three columns of Fig. 2). It is easy to observe that the absolute value of the imaginary part of these complex eigenvalues tends to increase monotonically as

*W*

_{0}grows. Hence a useful conclusion can be reached that nonzero

*α*

_{2},

*β*

_{2}, and large values of |

*W*

_{0}| are all extremely adverse to the generation of a full-real spectrum, which leads to the breaking of phases.

#### 4.2. Analytical solitons and dynamical stability

In the current section, we turn to discuss the stationary soliton solutions of Eq. (3) under the near *𝒫𝒯*-symmetric potential (6). Similar to the analytical theory in the NLS equation [26,49], the exact nonlinear localized mode of Eq. (3) with the propagation constant *q* = *α*_{1} can be found in the form

*α*

_{2}and

*β*

_{2}change in the potential (6). However, the variation of

*α*

_{2}and

*β*

_{2}can dramatically change the stability of the soliton solution (8), which will be demonstrated in the following.

When *α*_{1} and *β*_{1} are fixed, we can regulate the potential parameters *V*_{0} and *W*_{0} to control the profiles of the complex potential (6) and soliton solution (8). For convenience, we always fix *V*_{0} = 1 in the following discussion. When we choose *W*_{0} = 0.1, the potential (6) looks almost even symmetric (see Fig. 3(a)); if we further increase *W*_{0} to 1.5, the asymmetric phenomenon of the potential (6) begins to become obvious (see Fig. 3(c)). Nonetheless, the corresponding two solitons are *𝒫𝒯*-symmetric (see Figs. 3(b) and 3(d)), which indicates that at this moment, just the eigenstate of the system no longer meet the *𝒫𝒯*symmetry, the system still shows the characteristics of the conserved system. One of the possibly physical explanations we believe is that in the case of self-focusing nonlinearities, the increase of the nonlinear refractive index and real part of the potential function work together, resulting in the *𝒫𝒯*-symmetric soliton even for *W*_{0} in the above-mentioned the phase-transition point.

To explore the stability of the soliton (8), we take the soliton (8) with some 2% white noise as an initial condition to simulate the wave transmission. First, we show that the soliton in Fig. 3(b) is stable while that in Fig. 3(d) is unstable as (*α*_{2}, *β*_{2}) = (0, 0) (see Figs. 3(a1) and 3(b1)). An important reason is that the former lies in the parameter region with the unbroken *𝒫𝒯*-symmetric phase, whereas the latter lies in one with the broken *𝒫𝒯*-symmetric phase. Second, increasing *β*_{2} to the positive value or decreasing *α* to the negative value is more favorable to check the stability of the soliton (see Figs. 3(a2)–3(a4) and 3(b2)–3(b4)). Third, at some exceptional points in the (*α*_{2}, *β*_{2}) space, the growth of *W*_{0} can also change the soliton stability, which is a novel phenomenon and breaks the traditional mindset (comparing Fig. 3(a5) with Fig. 3(b5)). Moreover, we test out that for some small value of *W*_{0}, the soliton (8) is usually stable in the second quadrant of the (*α*_{2}, *β*_{2}) space (including the nonnegative vertical axis and nonpositive horizontal axis), beyond which the soliton immediately becomes extremely unstable (see Figs. 3(c1)–3(c5)). More importantly, these nonlinear-propagation stability results can be predicted and validated by the forthcoming linear stability analysis.

#### 4.3. Linear stability and spectral property

According to the above-mentioned linear-stability theory, we investigate that the influence of *W*_{0} on soliton stability in the whole (*α*_{2}, *β*_{2}) space. We can observe apparently from Figs. 4(a) and 4(b) that when *W*_{0} is small to some extent, the stable domain of the soliton (8) is located in the second quadrant of the (*α*_{2}, *β*_{2}) space (including the corresponding axes). As *W*_{0} rises, the stable region still remain in the second quadrant (see Fig. 4(c)); meanwhile, the unstable region also begins to emerge in the vicinity of the origin, which can be observed more clearly in Fig. 4(d). Noting that at the origin point (*α*_{2}, *β*_{2}) = (0, 0), the soliton is unstable though the potential is *𝒫𝒯*-symmetric. However, we can regulate the parameter *α*_{2} or *β*_{2} to make the soliton keep stable, although the potential may not be *𝒫𝒯*-symmetric. In addition, figure 4(d) also exhibits that, below and near the negative horizontal axis, stable solitons can be found too (see Fig. 3(b5)). This is possible because the beam can change the refractive index profile through optical nonlinearity and further adjust the amplitude to maintain the stable transmission.

Another intriguing phenomenon is closely related to the concrete linear-stability spectrum. It is well-known that if (*α*_{2}, *β*_{2}) = (0, 0) (which means the potential becomes *𝒫𝒯*-symmetric), the linear-stability spectrum is generally symmetric with respect to the real and imaginary axes, with the final (or tail) eigenvalues distributed on the imaginary axis. However, the positive (negative) values of *β*_{2} can generate several or finite pairs of complex-conjugate eigenvalues on the left (right) side of the imaginary axis (see Figs. 3(a1) and 3(a2)). In contrast, the negative (positive) values of *α*_{2} can lead to infinite pairs of complex-conjugate eigenvalues on the left (right) side of the imaginary axis (see Figs. 3(b1) and 3(b2)). The combined-action effect of *α*_{2} and *β*_{2} has also been displayed in Figs. 3(c1), 3(c2), 3(d1), and 3(d2). In brief, the linear-stability spectrum is only symmetric with respect to the real axis, if *α*_{2} or *β*_{2} is nonzero; only the non-negative *β*_{2} and non-positive *α*_{2} make real parts of the spectra admit the non-positive maximum value, which contributes to the generation of a stable soliton (see Figs. 3(a1) and 3(c1)); more importantly, through lots of numerical tests, one can summarize that the linear-stability spectrum at (*α*_{2}, *β*_{2}) and that at (−*α*_{2}, −*β*_{2}) are symmetric with regard to the imaginary axis, that is, the centrosymmetric two points in the (*α*_{2}, *β*_{2}) parameter space enjoy imaginary-axis symmetric (or even-symmetric) linear-stability spectra.

#### 4.4. Influence of exotic solitary wave on the stable exact soliton

To further examine the robustness of the exact nonlinear mode (8), we explore their interactions with boosted sech-shaped solitary pulses. Without loss of generality, we assume that the exotic solitary wave is always in the form sech(*x* + 20) *e*^{4ix}. For illustration, we set *V*_{0} = 1,*W*_{0} = 0.1, and first choose the bright soliton (8) with (*α*_{2}, *β*_{2}) = (0, 0) and the initial condition as *A*(*x*, 0) = *ϕ*(*x*) + sech(*x* + 20) *e*^{4ix} to simulate the wave propagation governed by Eq. (2). The result of interaction reveals that the bright soliton can remain stable without any change of the shape before and after collision, only with mild dissipation of the exotic wave (see Fig. 5(a)). When we increase *β*_{2} or decrease *α*_{2} a little, the shape of the exact soliton does not change at all, whereas the amplitude of the exotic solitary wave declines rapidly (see Figs. 5(b) and 5(c)). The combined action of increasing *β*_{2} and decreasing *α*_{2} only aggravates the rapid-decline process of the amplitude of the exotic solitary wave, while has no influence on the stable propagation of the exact soliton (see Fig. 5(d)). That can be explained by considering the relationship between the coefficients *α*_{2} and *β*_{2} when *W*_{0} is chosen as the phase-transition point. The nonlinear gain/loss of the exact soliton is greater than that of the linear parabolic gain, so the exotic solitary wave is continuously diffused in the transmission process, and the larger difference between the two parameters is, the more serious the diffusion is.

#### 4.5. Energy flow across the exact soliton

We now examine the transverse energy flow intensity of the exact soliton (8), defined by $j(x)=\frac{i}{2}\left(A{A}_{x}^{*}-{A}^{*}{A}_{x}\right)$. Based on the celebrated continuity relation of the GL equation, $\frac{\partial \rho}{\partial z}+\frac{\partial j}{\partial x}=E$, where *ρ* = |*A*|^{2} denotes the energy density, we can attain the density of energy gain or loss

*α*

_{2}=

*β*

_{2}= 0 and

*W*(

*x*) ≡ 0, the system is conservative because of

*E*= 0, otherwise it is dissipative. The energy of the optical field can be transported laterally from the gain to loss regions through the effect of phase gradient, so that the whole system maintains the balance of gain and loss effects, which corresponds to a passive system and therefore exhibits Hermitian properties. However, when the eigenvalues enter the complex region, the

*𝒫𝒯*symmetry of the system is broken, and the whole gain-loss effect is no longer balanced. The system shows a dissipative effect. For a fixed

*W*

_{0}= 0.1 without loss of generality, the variation of the parameters

*α*

_{2}and

*β*

_{2}basically does not change the gain and loss distributions of energy and flux (see Figs. 6(a) and 6(b)). However, when

*W*

_{0}rises, the strength of the gain-loss distribution and the corresponding flux will grow too, but their respective shapes and the flow direction still remain unchanged, by comparing Figs. 6(a)–6(b) with Figs. 6(c)–6(d). In fact, these findings can be proved by the analytical calculation. For convenience, we still fix

*α*

_{1}=

*β*

_{1}=

*V*

_{0}= 1, and substitute the exact solution (8) into the aforementioned formulas with respect to

*E*and

*j*, then we can obtain $E=-\frac{2}{9}{W}_{0}\left({W}_{0}^{2}+9\right)\text{sinh}(x){\text{sech}}^{4}(x)$ and $j=\frac{1}{27}{W}_{0}\left({W}_{0}^{2}+9\right){\text{sech}}^{3}(x)$, both only related to

*W*

_{0}and independent on

*α*

_{2}and

*β*

_{2}. In brief,

*α*

_{2}and

*β*

_{2}can not change the gain or loss distributions of energy and flux including the magnitude and direction which always flows from gain to loss regions at all.

*W*

_{0}can regulate their magnitudes whereas keep their shapes and the flow direction.

## 5. Generalized model and excitations of solitons

In this section we turn to elaborate the excitations of the exact soliton (8) by making the parameters rely on the propagation distance *z*: *α*_{2} → *α*_{2}(*z*) or *β*_{2} → *β*_{2}(*z*) (cf. [30, 70]). It requires that the simultaneous adiabatic switching is imposed on the near-*𝒫𝒯*-symmetric potential (6) and complex coefficients of Eq. (2), regulated by

*V*(

*x*,

*z*),

*W*(

*x*,

*z*) are given respectively by Eqs. (6) with

*α*

_{2}→

*α*

_{2}(

*z*) and

*β*

_{2}→

*β*

_{2}(

*z*). For convenience, both

*α*

_{2}(

*z*) and

*β*

_{2}(

*z*) are selected as the following unified form

*∊*

_{1,2}respectively represent the real initial-state and final-state parameters. One can easily examine that the soliton (8) with

*α*

_{2}→

*α*

_{2}(

*z*) or

*β*

_{2}→

*β*

_{2}(

*z*) do not satisfy Eq. (10) any longer, nevertheless the bright soliton (8) do solve Eq. (10) for both the initial state

*z*= 0 and excited states

*z*≥ 1000.

We first execute a single-parameter excitation of the soliton *A*(*x*, *z*) controlled by Eq. (10) via the initial condition determined by Eq. (8), with *β*_{2}(*z*) given by Eq. (11) and *α*_{2}(*z*) ≡ *α*_{2}. Figure 7(a) displays that the excitation or dynamical transformation of the nonlinear mode is unstable due to the unstable initial state, though the final state (8) is stable in Eq. (2). The similar situation happens for the excitation of the single-parameter *α*_{2} (see Fig. 7(b)). However, when the two-parameter simultaneous excitation is carried out with both *V*_{0}(*z*) and *W*_{0}(*z*) determined by Eq. (11) concurrently, we can excite an initially unstable exact nonlinear localized mode given by Eq. (8) to another stable exact nonlinear mode (see Fig. 7(c)). It can be obviously observed from the amplitude of the intensity that the final stable state in the process of excitation is not regulated by Eq. (8) any more, which is a novel finding. Moreover, only by modulating *W*_{0} → *W*_{0}(*z*) determined by Eq. (11), an initially unstable exact nonlinear mode given by Eq. (8) can also be transformed into another stable nonlinear localized mode, where the finally stable state satisfies Eq. (2) (see Fig. 7(d)).

## 6. Conclusions and discussions

In conclusion, we present a class of *𝒫𝒯*-symmetric solitons residing in the complex Kerr-nonlinear GL equation with a novel category of near-*𝒫𝒯*-symmetric potentials, where the phase in the linear regime is always symmetry-breaking because of the occurrence of spectral filtering parameter *α*_{2} or nonlinear gain-loss coefficient *β*_{2}. Nonlinear-propagation dynamics and linear-stability analysis reveal that the overwhelming majority of stable solitons are located in the second quadrant of the (*α*_{2}, *β*_{2}) parameter space. Moreover, by adiabatically changing *α*_{2} and *β*_{2}, we can excite an unstable nonlinear mode to another stable one. The interactions and energy flow with respect to solitons are also examined.

Before closing we would like to mention that the exact nonlinear localized modes are attained at some special fixed propagation-constant points. One can further investigate numerical solitons for other points, and their stability analysis and other significant properties. In addition, our analysis and methods can also be used to study some more general modes by adding competing nonlinearities, the higher-order dispersive terms, or other complex *𝒫𝒯*-symmetric (or near *𝒫𝒯*-symmetric) potentials into the GL equation, such as the well-known complex cubic-quintic GL equation and Swift-Hohenberg equation. Finally, it is an open problem that our results presented here may provide the related physical researchers with several helpful theoretical guidance to design relevant experiments in optics or other fields.

## Funding

National Natural Science Foundation of China (11571346, 11731014); the CAS Interdisciplinary Innovation Team.

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