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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 α2 or nonlinear gain-loss coefficient β2, we do find stable exact solitons in the second quadrant of the (α2, β2) 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]
where A = A(x, z) is a complex field, α1,2, β1,2, 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]
where 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 α1 and Kerr-nonlinearity coefficient β1 are fixed as α1 = β1 = 1 in the paper; the real parameter α2 can be used to describe the spectral filtering or linear parabolic gain (α2 > 0), and the real constant β2 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 α2 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 iAz = δℋ(A)/δA*, where the Hamiltonian and the asterisk denotes the complex conjugate. If we define the optical power of Eq. (2) as , then one can elicit immediately that the power evolves by . 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)eiqz, 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
In general, exact nonlinear localized modes of Eq. (3) can be attainable only for some special parameters [84,85]. Therefore, some numerical techniques are necessary to find its stationary nonlinear localized modes [82,86].To investigate the linear stability of stationary solutions ϕ(x)eiqz, we perturb them in the vicinity of the singular point
where |∊| ≪ 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 where L̂1 = (α1 + iα2)∂xx +V(x) + iW(x) + 2(β1 + iβ2)|ϕ|2 − q and L̂2 = (β1 + iβ2)ϕ2. 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
where , the real parameters V0 and W0 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 α2 = β2 = 0, meanwhile Eq. (2) becomes the well-known 𝒫𝒯-symmetric NLSE. However, when α2 or β2 is perturbed around the origin of the (α2, β2) 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 (α2, β2) parameter space. In addition, it is also apparent that the aforementioned complex potential possesses even symmetry if W0 = 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
where 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 (V0, W0) space, unless (α2, β2) = (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 V0 = 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 W0 = 1.25. However, only if α2 or β2 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 W0 grows. Hence a useful conclusion can be reached that nonzero α2, β2, and large values of |W0| are all extremely adverse to the generation of a full-real spectrum, which leads to the breaking of phases.
Fig. 2 Real and imaginary components of the first two lowest energy eigenvalues λ of the linear spectral problem (7) as a function of W0 at V0 = 1. (a1, a2) (α2, β2) = (0, 0), (b1, b2) (α2, β2) = (0, 0.1), (c1, c2) (α2, β2) = (−0.1, 0), (d1, d2) (α2, β2) = (−0.1, 0.1), in the near 𝒫𝒯-symmetric potential (6).
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 = α1 can be found in the form
It is noteworthy that the exact soliton is isolated and always keeps invariant, no matter how α2 and β2 change in the potential (6). However, the variation of α2 and β2 can dramatically change the stability of the soliton solution (8), which will be demonstrated in the following.When α1 and β1 are fixed, we can regulate the potential parameters V0 and W0 to control the profiles of the complex potential (6) and soliton solution (8). For convenience, we always fix V0 = 1 in the following discussion. When we choose W0 = 0.1, the potential (6) looks almost even symmetric (see Fig. 3(a)); if we further increase W0 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 𝒫𝒯symmetry, 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 W0 in the above-mentioned the phase-transition point.
Fig. 3 Profiles of the near 𝒫𝒯-symmetric potential (6) and soliton solutions with α2 = −1, β2 = 1. (a, b) W0 = 0.1, (c, d) W0 = 1.5. Evolutions of the exact solitons (8) with W0 = 0.1 in the second row while W0 = 1.5 in the third row: (a1, b1) (α2, β2) = (0, 0), (a2, b2) (α2, β2) = (0, 1), (a3, b3) (α2, β2) = (−1, 0), (a4, b4) (α2, β2) = (−1, 1), (a5, b5) (α2, β2) = (−0.2, −0.01). Unstable evolutions with W0 = 0.1 in the last row: (c1) (α2, β2) = (1, 1), (c2) (α2, β2) = (1, 0), (c3) (α2, β2) = (1, −1), (c4) (α2, β2) = (0, −1), (c5) (α2, β2) = (−1, −1).
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 (α2, β2) = (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 β2 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 (α2, β2) space, the growth of W0 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 W0, the soliton (8) is usually stable in the second quadrant of the (α2, β2) 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 W0 on soliton stability in the whole (α2, β2) space. We can observe apparently from Figs. 4(a) and 4(b) that when W0 is small to some extent, the stable domain of the soliton (8) is located in the second quadrant of the (α2, β2) space (including the corresponding axes). As W0 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 (α2, β2) = (0, 0), the soliton is unstable though the potential is 𝒫𝒯-symmetric. However, we can regulate the parameter α2 or β2 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.
Fig. 4 Linear-stability maps [cf. Eq. (5)] of the exact solitons (8) in the (α2, β2) space [only black and dark regions denote stable solitons]: (a) W0 = 0, (b) W0 = 0.1, (c, d) W0 = 1.5, where (d) clearly indicates the concrete linear-stability situation around the original point in (c). Linear-stability spectra with W0 = 0.1: (a1) (α2, β2) = (0, 1), (a2) (α2, β2) = (0, −1), (b1) (α2, β2) = (−1, 0), (b2) (α2, β2) = (1, 0), (c1) (α2, β2) = (−1, 1), (c2) (α2, β2) = (1, −1), (d1) (α2, β2) = (−1, −1), (d2) (α2, β2) = (1, 1).
Another intriguing phenomenon is closely related to the concrete linear-stability spectrum. It is well-known that if (α2, β2) = (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 β2 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 α2 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 α2 and β2 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 α2 or β2 is nonzero; only the non-negative β2 and non-positive α2 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 (α2, β2) and that at (−α2, −β2) are symmetric with regard to the imaginary axis, that is, the centrosymmetric two points in the (α2, β2) 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) e4ix. For illustration, we set V0 = 1,W0 = 0.1, and first choose the bright soliton (8) with (α2, β2) = (0, 0) and the initial condition as A(x, 0) = ϕ(x) + sech(x + 20) e4ix 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 β2 or decrease α2 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 β2 and decreasing α2 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 α2 and β2 when W0 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.
Fig. 5 Collisions between the bright soliton (8) and boosted sech-shaped or rational solitary pulse, produced by the simulation of Eq. (2), with the initial input A(x, 0) = ϕ(x) + sech(x + 20) e4ix. (a) (α2, β2) = (0, 0), (b) (α2, β2) = (0, 1), (c) (α2, β2) = (−0.01, 0), (d) (α2, β2) = (−0.01, 1). Here ϕ(x) is given by Eq. (8) with V0 = 1,W0 = 0.1.
4.5. Energy flow across the exact soliton
We now examine the transverse energy flow intensity of the exact soliton (8), defined by . Based on the celebrated continuity relation of the GL equation, , where ρ = |A|2 denotes the energy density, we can attain the density of energy gain or loss
which determines the gain or loss distribution of energy. If these complex coefficients in Eq. (2) disappear, i.e., α2 = β2 = 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 W0 = 0.1 without loss of generality, the variation of the parameters α2 and β2 basically does not change the gain and loss distributions of energy and flux (see Figs. 6(a) and 6(b)). However, when W0 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 α1 = β1 = V0 = 1, and substitute the exact solution (8) into the aforementioned formulas with respect to E and j, then we can obtain and , both only related to W0 and independent on α2 and β2. In brief, α2 and β2 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. W0 can regulate their magnitudes whereas keep their shapes and the flow direction.Fig. 6 The density of energy generation E and energy flux j: (a, b) The same parameters are used as Figs. 3(a) and 3(b); (c, d) the same parameters as Figs. 3(c) and 3(d). Here ‘G’ (‘L’) denotes the gain (loss) region.
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: α2 → α2(z) or β2 → β2(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
where V(x, z), W(x, z) are given respectively by Eqs. (6) with α2 → α2(z) and β2 → β2(z). For convenience, both α2(z) and β2(z) are selected as the following unified form where ∊1,2 respectively represent the real initial-state and final-state parameters. One can easily examine that the soliton (8) with α2 → α2(z) or β2 → β2(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 β2(z) given by Eq. (11) and α2(z) ≡ α2. 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 α2 (see Fig. 7(b)). However, when the two-parameter simultaneous excitation is carried out with both V0(z) and W0(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 W0 → W0(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)).
Fig. 7 Excitations of exact nonlinear localized modes [cf. Eq. (10)]. (a) α21 = β2 = 0, α22 = −1, (b) α2 = β21 = 0, β22 = 1, (c) α21 = β21 = 0, α22 = −1, β22 = 1, other parameter is W0 = 1.5; (d) W01 = 0.1, W02 = 1.5, other parameters are α2 = −0.2, β2 = −0.01.
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 α2 or nonlinear gain-loss coefficient β2. Nonlinear-propagation dynamics and linear-stability analysis reveal that the overwhelming majority of stable solitons are located in the second quadrant of the (α2, β2) parameter space. Moreover, by adiabatically changing α2 and β2, 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.
References
1. I. S. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99–143 (2002). [CrossRef]
2. M. Ipsen, L. Kramer, and P. G. Sørensen, “Amplitude equations for description of chemical reaction–diffusion systems,” Phys. Rep. 337, 193–235 (2000). [CrossRef]
3. M. van Hecke, “Coherent and incoherent structures in systems described by the 1d cgle: Experiments and identification,” Phys. D 174, 134–151 (2003). [CrossRef]
4. M. F. Ferreira, M. M. Facao, and S. C. Latas, “Stable soliton propagation in a system with spectral filtering and nonlinear gain,” Fiber & Integr. Opt. 19, 31–41 (2000). [CrossRef]
5. P. Mandel and M. Tlidi, “Transverse dynamics in cavity nonlinear optics (2000–2003),” J. Opt. B 6, R60 (2004). [CrossRef]
6. N. Rosanov, S. Fedorov, and A. Shatsev, “Two-dimensional laser soliton complexes with weak, strong, and mixed coupling,” Appl. Phys. B 81, 937–943 (2005). [CrossRef]
7. C. Weiss and Y. Larionova, “Pattern formation in optical resonators,” Rep. Prog. Phys. 70, 255–335 (2007). [CrossRef]
8. N. Akhmediev, J. Soto-Crespo, and P. Grelu, “Spatiotemporal optical solitons in nonlinear dissipative media: From stationary light bullets to pulsating complexes,” Chaos 17, 037112 (2007). [CrossRef] [PubMed]
9. Y. He and D. Mihalache, “Soliton dynamics induced by periodic spatially inhomogeneous losses in optical media described by the complex Ginzburg-Landau model,” J. Opt. Soc. Am. B 29, 2554–2558 (2012). [CrossRef]
10. D. Mihalache, “Localized structures in nonlinear optical media: a selection of recent studies,” Rom. Rep. Phys. 67, 1383–1400 (2015).
11. N. Akhmediev, A. Ankiewicz, and J. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047 (1997). [CrossRef]
12. N. Akhmediev and J. M. Soto-Crespo, “Exploding solitons and Shil’nikov’s theorem,” Phys. Lett. A 317, 287–292 (2003). [CrossRef]
13. J.-M. Soto-Crespo and N. Akhmediev, “Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation,” Math. Comput. Simul 69, 526–536 (2005). [CrossRef]
14. E. N. Tsoy and N. Akhmediev, “Bifurcations from stationary to pulsating solitons in the cubic–quintic complex Ginzburg–Landau equation,” Phys. Lett. A 343, 417–422 (2005). [CrossRef]
15. N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001). [CrossRef]
16. V. Skarka, N. Aleksić, H. Leblond, B. Malomed, and D. Mihalache, “Varieties of stable vortical solitons in Ginzburg-Landau media with radially inhomogeneous losses,” Phys. Rev. Lett. 105, 213901 (2010). [CrossRef]
17. D. Mihalache, D. Mazilu, F. Lederer, B. Malomed, Y. V. Kartashov, L.-C. Crasovan, and L. Torner, “Stable spatiotemporal solitons in Bessel optical lattices,” Phys. Rev. Lett. 95, 023902 (2005). [CrossRef] [PubMed]
18. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. Malomed, “Stability of dissipative optical solitons in the three-dimensional cubic-quintic Ginzburg-Landau equation,” Phys. Rev. A 75, 033811 (2007). [CrossRef]
19. Y. He and B. A. Malomed, “Accessible solitons in complex Ginzburg-Landau media,” Phys. Rev. E 88, 042912 (2013). [CrossRef]
20. 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]
21. Y. He, B. A. Malomed, and D. Mihalache, “Localized modes in dissipative lattice media: an overview,” Phil. Trans. R. Soc. A 372, 20140017 (2014). [CrossRef] [PubMed]
22. C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80, 5243 (1998). [CrossRef]
23. C. M. Bender, D. C. Brody, and H. F. Jones, “Must a Hamiltonian be Hermitian?” Am. J. Phys 71, 1095–1102 (2003). [CrossRef]
24. C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70, 947–1018 (2007). [CrossRef]
25. Z. Ahmed, “Real and complex discrete eigenvalues in an exactly solvable one-dimensional complex PT-invariant potential,” Phys. Lett. A 282, 343–348 (2001). [CrossRef]
26. Z. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Optical solitons in PT periodic potentials,” Phys. Rev. Lett. 100, 030402 (2008). [CrossRef] [PubMed]
27. Z. Yan, Z. Wen, and C. Hang, “Spatial solitons and stability in self-focusing and defocusing Kerr nonlinear media with generalized parity-time-symmetric Scarf-ii potentials,” Phys. Rev. E 92, 022913 (2015). [CrossRef]
28. Z. Yan, “Complex PT-symmetric nonlinear Schrödinger equation and Burgers equation,” Philos. Trans. R. Soc. London, Ser. A 371, 20120059 (2013). [CrossRef]
29. Z.-C. Wen and Z. Yan, “Dynamical behaviors of optical solitons in parity–time (PT) symmetric sextic anharmonic double-well potentials,” Phys. Lett. A 379, 2025–2029 (2015). [CrossRef]
30. Z. Yan, Z. Wen, and V. V. Konotop, “Solitons in a nonlinear Schrödinger equation with PT-symmetric potentials and inhomogeneous nonlinearity: Stability and excitation of nonlinear modes,” Phys. Rev. A 92, 023821 (2015). [CrossRef]
31. Y. Lumer, Y. Plotnik, M. C. Rechtsman, and M. Segev, “Nonlinearly induced PT transition in photonic systems,” Phys. Rev. Lett. 111, 263901 (2013). [CrossRef]
32. S. Nixon, L. Ge, and J. Yang, “Stability analysis for solitons in PT-symmetric optical lattices,” Phys. Rev. A 85, 023822 (2012). [CrossRef]
33. V. Achilleos, P. Kevrekidis, D. Frantzeskakis, and R. Carretero-González, “Dark solitons and vortices in PT-symmetric nonlinear media: From spontaneous symmetry breaking to nonlinear PT phase transitions,” Phys. Rev. A 86, 013808 (2012). [CrossRef]
34. Z. Shi, X. Jiang, X. Zhu, and H. Li, “Bright spatial solitons in defocusing kerr media with PT-symmetric potentials,” Phys. Rev. A 84, 053855 (2011). [CrossRef]
35. 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] [PubMed]
36. 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, 192–195 (2010). [CrossRef]
37. A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488, 167–171 (2012). [CrossRef] [PubMed]
38. G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013). [CrossRef] [PubMed]
39. A. Regensburger, M.-A. Miri, C. Bersch, J. Näger, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Observation of defect states in PT-symmetric optical lattices,” Phys. Rev. Lett. 110, 223902 (2013). [CrossRef] [PubMed]
40. B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, “Parity-time-symmetric whispering-gallery microcavities,” Nat. Phys. 10, 394–398 (2014). [CrossRef]
41. A. A. Zyablovsky, A. P. Vinogradov, A. A. Pukhov, A. V. Dorofeenko, and A. A. Lisyansky, “PT-symmetry in optics,” Phys. Usp. 57, 1063 (2014). [CrossRef]
42. P.-Y. Chen and J. Jung, “PT Symmetry and Singularity-Enhanced Sensing Based on Photoexcited Graphene Metasurfaces,” Phys. Rev. Appl 5, 064018 (2016). [CrossRef]
43. K. Takata and M. Notomi, “PT-Symmetric Coupled-Resonator Waveguide Based on Buried Heterostructure Nanocavities,” Phys. Rev. Appl 7, 054023 (2017). [CrossRef]
44. E. A. Ultanir, G. I. Stegeman, and D. N. Christodoulides, “Dissipative photonic lattice solitons,” Opt. Lett. 29, 845–847 (2004). [CrossRef] [PubMed]
45. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100, 103904 (2008). [CrossRef] [PubMed]
46. K. Makris, R. El-Ganainy, D. Christodoulides, and Z. H. Musslimani, “PT-symmetric periodic optical potentials,” Int. J. Theor. Phys. 50, 1019–1041 (2011). [CrossRef]
47. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “PT-symmetric optical lattices,” Phys. Rev. A 81, 063807 (2010). [CrossRef]
48. D. A. Zezyulin and V. V. Konotop, “Nonlinear modes in the harmonic PT-symmetric potential,” Phys. Rev. A 85, 043840 (2012). [CrossRef]
49. Z. H. Musslimani, K. G. Makris, R. El-Ganainy, and D. N. Christodoulides, “Analytical solutions to a class of nonlinear Schrödinger equations with PT-like potentials,” J. Phys. A: Math. Theor. 41, 244019 (2008). [CrossRef]
50. C.-Q. Dai, X.-G. Wang, and G.-Q. Zhou, “Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials,” Phys. Rev. A 89, 013834 (2014). [CrossRef]
51. B. Midya and R. Roychoudhury, “Nonlinear localized modes in PT-symmetric Rosen-Morse potential wells,” Phys. Rev. A 87, 045803 (2013). [CrossRef]
52. S. Hu, X. Ma, D. Lu, Z. Yang, Y. Zheng, and W. Hu, “Solitons supported by complex PT-symmetric Gaussian potentials,” Phys. Rev. A 84, 043818 (2011). [CrossRef]
53. J. Yang, “Symmetry breaking of solitons in one-dimensional parity-time-symmetric optical potentials,” Opt. Lett. 39, 5547–5550 (2014). [CrossRef] [PubMed]
54. C. P. Jisha, L. Devassy, A. Alberucci, and V. Kuriakose, “Influence of the imaginary component of the photonic potential on the properties of solitons in PT-symmetric systems,” Phys. Rev. A 90, 043855 (2014). [CrossRef]
55. 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]
56. N. Moiseyev, “Crossing rule for a PT-symmetric two-level time-periodic system,” Phys. Rev. A 83, 052125 (2011). [CrossRef]
57. C. P. Jisha, A. Alberucci, V. A. Brazhnyi, and G. Assanto, “Nonlocal gap solitons in PT-symmetric periodic potentials with defocusing nonlinearity,” Phys. Rev. A 89, 013812 (2014). [CrossRef]
58. H. Wang and D. Christodoulides, “Two dimensional gap solitons in self-defocusing media with PT-symmetric superlattice,” Commun. Nonlinear Sci. Numer. Simul. 38, 130–139 (2016). [CrossRef]
59. S. V. Suchkov, A. A. Sukhorukov, J. Huang, S. V. Dmitriev, C. Lee, and Y. S. Kivshar, “Nonlinear switching and solitons in pt-symmetric photonic systems,” Laser Photonics Rev. 10, 177–213 (2016). [CrossRef]
60. H. Cartarius and G. Wunner, “Model of a PT-symmetric Bose-Einstein condensate in a δ-function double-well potential,” Phys. Rev. A 86, 013612 (2012). [CrossRef]
61. F. Single, H. Cartarius, G. Wunner, and J. Main, “Coupling approach for the realization of a PT-symmetric potential for a Bose-Einstein condensate in a double well,” Phys. Rev. A 90, 042123 (2014). [CrossRef]
62. G. Burlak and B. A. Malomed, “Stability boundary and collisions of two-dimensional solitons in PT-symmetric couplers with the cubic-quintic nonlinearity,” Phys. Rev. E 88, 062904 (2013). [CrossRef]
63. Y. V. Bludov, V. V. Konotop, and B. A. Malomed, “Stable dark solitons in PT-symmetric dual-core waveguides,” Phys. Rev. A 87, 013816 (2013). [CrossRef]
64. R. Fortanier, D. Dast, D. Haag, H. Cartarius, J. Main, G. Wunner, and R. Gutöhrlein, “Dipolar Bose-Einstein condensates in a PT-symmetric double-well potential,” Phys. Rev. A 89, 063608 (2014). [CrossRef]
65. D. Dizdarevic, D. Dast, D. Haag, J. Main, H. Cartarius, and G. Wunner, “Cusp bifurcation in the eigenvalue spectrum of PT- symmetric Bose-Einstein condensates,” Phys. Rev. A 91, 033636 (2015). [CrossRef]
66. C.-Q. Dai, X.-F. Zhang, Y. Fan, and L. Chen, “Localized modes of the (n+1)-dimensional Schrödinger equation with power-law nonlinearities in PT-symmetric potentials,” Commun. Nonlinear Sci. Numer. Simul. 43, 239–250 (2017). [CrossRef]
67. Z. Yan, Y. Chen, and Z. Wen, “On stable solitons and interactions of the generalized Gross-Pitaevskii equation with PT-and non-PT-symmetric potentials,” Chaos 26, 083109 (2016). [CrossRef]
68. Y. Chen and Z. Yan, “Solitonic dynamics and excitations of the nonlinear Schrödinger equation with third-order dispersion in non-Hermitian PT-symmetric potentials,” Sci. Rep. 6, 23478 (2016). [CrossRef]
69. Y. Chen, Z. Yan, D. Mihalache, and B. A. Malomed, “Families of stable solitons and excitations in the PT-symmetric nonlinear Schrödinger equations with position-dependent effective masses,” Sci. Rep. 7, 1257 (2017). [CrossRef]
70. Y. Chen and Z. Yan, “Stable parity-time-symmetric nonlinear modes and excitations in a derivative nonlinear Schrödinger equation,” Phys. Rev. E 95, 012205 (2017). [CrossRef]
71. Z. Yan and Y. Chen, “The nonlinear schrödinger equation with generalized nonlinearities and PT-symmetric potentials: Stable solitons, interactions, and excitations,” Chaos 27, 073114 (2017). [CrossRef]
72. Z. Wen and Z. Yan, “Solitons and their stability in the nonlocal nonlinear schrödinger equation with pt-symmetric potentials,” Chaos 27, 053105 (2017). [CrossRef]
73. Y. Chen, Z. Yan, and X. Li, “One-and two-dimensional gap solitons and dynamics in the PT-symmetric lattice potential and spatially-periodic momentum modulation,” Commun. Nonlinear Sci. Numer. Simul. 55, 287–297 (2018). [CrossRef]
74. J. Shen, Z. Wen, Z. Yan, and C. Hang, “Effect of PT symmetry on nonlinear waves for three-wave interaction models in the quadratic nonlinear media,” Chaos 28, 043104 (2018). [CrossRef]
75. E. N. Tsoy, I. M. Allayarov, and F. K. Abdullaev, “Stable localized modes in asymmetric waveguides with gain and loss,” Opt. Lett. 39, 4215–4218 (2014). [CrossRef] [PubMed]
76. V. V. Konotop and D. A. Zezyulin, “Families of stationary modes in complex potentials,” Opt. Lett. 39, 5535–5538 (2014). [CrossRef] [PubMed]
77. S. D. Nixon and J. Yang, “Bifurcation of soliton families from linear modes in non-PT-symmetric complex potentials,” Stud. Appl. Math. 136, 459–483 (2016). [CrossRef]
78. Y. Kominis, “Soliton dynamics in symmetric and non-symmetric complex potentials,” Opt. Commun. 334, 265–272 (2015). [CrossRef]
79. Y. Kominis, “Dynamic power balance for nonlinear waves in unbalanced gain and loss landscapes,” Phys. Rev. A 92, 063849 (2015). [CrossRef]
80. 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]
81. V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88, 035002 (2016). [CrossRef]
82. J. M. Soto-Crespo, N. Akhmediev, and K. S. Chiang, “Simultaneous existence of a multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115–123 (2001). [CrossRef]
83. N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005). [CrossRef]
84. N. Akhmediev and V. Afanasjev, “Novel arbitrary-amplitude soliton solutions of the cubic-quintic complex ginzburg-landau equation,” Phys. Rev. Lett. 75, 2320 (1995). [CrossRef] [PubMed]
85. N. Akhmediev, V. Afanasjev, and J. Soto-Crespo, “Singularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. E 53, 1190 (1996). [CrossRef]
86. J. Yang, Nonlinear waves in integrable and nonintegrable systems (SIAM, 2010). [CrossRef]
