⇦Chinese Physics Letters, 2021, Vol. 38, No. 9, Article code 094201 Soliton Rectangular Pulses and Bound States in a Dissipative System Modeled by the Variable-Coefficients Complex Cubic-Quintic Ginzburg–Landau Equation Yuan-Yuan Yan (闫园园) and Wen-Jun Liu (刘文军)* Affiliations State Key Laboratory of Information Photonics and Optical Communications, School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China Received 3 June 2021; accepted 30 June 2021; published online 2 September 2021 Supported by the National Natural Science Foundation of China (Grant Nos. 11875008 and 12075034), and the Beijing University of Posts and Telecommunications Excellent Ph.D. Students Foundation (Grant No. CX2021129).
*Corresponding author. Email: jungliu@bupt.edu.cn Citation Text: Yan Y Y and Liu W J 2021 Chin. Phys. Lett. 38 094201    Abstract The complex cubic-quintic Ginzburg–Landau equation (CQGLE) is a universal model for describing a dissipative system, especially fiber laser. The analytic one-soliton solution of the variable-coefficients CQGLE is calculated by a modified Hirota method. Then, phenomena of soliton pulses splitting and stable bound states of two solitons are investigated. Moreover, rectangular dissipative soliton pulses of the variable-coefficients CQGLE are realized and controlled effectively in the theoretical research for the first time, which breaks through energy limitation of soliton pulses and is expected to provide theoretical basis for preparation of high-energy soliton pulses in fiber lasers. DOI:10.1088/0256-307X/38/9/094201 © 2021 Chinese Physics Society Article Text Solitons have been applied in fields such as optics, fluid mechanics, plasma physics and the Bose–Einstein condensates.[1–7] As one of the momentous parts of solitons, optical solitons have valuable applications and prospects in optical communication, all-optical network, signal processing and other systems because of their long-distance stable transmission and low loss.[8–12] In an optical fiber, which is an energy balanced system (Hamiltonian system), the formation and transmission mechanism of optical solitons can be simulated by the non-linear Schrödinger equation (NLSE).[13–17] Those solitons appear as hyperbolic secant pulses, which are called traditional solitons, and there are abundant results in the experiment and theory about them.[18–23] However, fiber lasers and most systems in nature are dissipative systems with energy changing at any time.[24–26] Thus, the Ginzburg–Landau equation (GLE), which is frequently used to simulate a dissipative system, has been a wide concern for scholars from some different fields.[27–32] In recent years, the complex cubic-quintic Ginzburg–Landau equation (CQGLE) has been applied in much nonlinear optical research, such as mode-locked lasers, multimode lasers, short pulse propagation, parametric oscillators, and long-haul optical communication.[33–36] In addition, in order to better reflect the complexity and variability of a dissipative system, it is necessary to choose variable-coefficient CQGLE as the theoretical model. Nozaki and Bekki took effort firstly to defining and using the modified bilinear method to study the exact solutions of the generalized GLE in 1984.[37] So far, some kinds of soliton solutions to the CQGLE have been obtained in the theoretical research.[38–41] However, the discussions on bound states of solitons, especially rectangular soliton pulses, are rare,[42,43] even though they have been studied in the experiment.[44,45] When the energy of the soliton pulse is too high, the single soliton pulse with traditional hyperbolic secant shape usually splits to form the bound state of solitons, which is helpful for improving multi-channel communication and expanding channel capacity.[43] Also, the pulse splitting hinders the formation of soliton pulses with higher energy. The new rectangular solitons just make up for the deficiency, because they break through the bottleneck of soliton area theory and complete the stable output of high energy pulses.[46,47] Therefore, in this Letter, we mainly analyze the dissipative soliton bound states and new rectangular pulses of the variable-coefficient CQGLE, which not only enriches the theoretical system of solitons but also is expected to be helpful for improving the upper limit of the energy output of nonlinear optical devices such as fiber lasers. The variable-coefficient CQGLE can be written as[48] $$\begin{alignat}{1} &i\frac{\partial u}{\partial z}-\frac{B(z)}{2}\frac{\partial^{2}u}{\partial \tau^{2}}+\psi (z)|u|^{2}u-\rho (z)|u|^{4}u\\ ={}&\frac{i}{2}[g(z)-\beta (z)]u+\frac{i}{2}d(z)\frac{\partial^{2}u}{\partial \tau^{2}}-iK(z)|u|^{2}u\\ &-i\nu (z)|u|^{4}u.~~ \tag {1} \end{alignat} $$ Here, $u(z, \tau)$ is the normalized amplitude of optical field, $z$ is the normalized propagation distance, and $\tau$ is the retarded time. The variable coefficient $B(z)$ is related to group velocity dispersion (GVD), $\psi (z)$ and $\rho (z)$ represent the cubic nonlinearities and the quintic nonlinearities, respectively. Accordingly, the first term on the right side of Eq. (1) determines the net gain $\Delta g_{0}(z)= g(z) - \beta (z)$, with $g(z)$ being the peak gain and $\beta (z)$ the linear loss. Then, the coefficient $d(z)$ represents gain dispersion, $\nu (z)$ and $K(z)$ contribute to three-photon absorption and two-photon absorption effects, respectively. In the following, the analytic one-soliton solution to Eq. (1) will be deduced using the modified Hirota method for the first time. Analytic One-Soliton Solution to Eq. (1). On the basis of the modified Hirota method and the general dependent variable transformation, $u(z, \tau)$ can be set as[49] $$ u(z,\tau)=G(z,\tau)/F(z,\tau)^{1/2+i\alpha },~~ \tag {2} $$ here $F(z, \tau)$ is assumed to be a real function, and $G(z, \tau)$ is a complex differentiable function. Then the modified bilinear forms of Eq. (1) can be expressed as $$ \begin{alignat}{1} & iD_{z,\alpha }^{1} G\cdot F-\Big[\frac{B(z)}{2}+\frac{i}{2}d(z)\Big]D_{\tau,\alpha }^{2} G\cdot F\\ ={}&\frac{i}{2}[g(z)-\beta (z)]G\cdot F, \\ &\Big(\frac{1}{2}+i\alpha\Big)\Big(\frac{3}{2}+i\alpha\Big)\Big[\frac{B(z)}{2}+\frac{i}{2}d(z)\Big]D_{\tau }^{2} F\cdot F\\ ={}&2[\rho (z)-i\nu (z)]\vert G\vert^{4}-2[\psi (z)+iK(z)]\vert G\vert^{2}\cdot F,~~ \tag {3} \end{alignat} $$ where $D_{\tau,\alpha }^{2} G\cdot F$ and $D_{z,\alpha }^{1} G\cdot F$ are modified Hirota's bilinear operators, and $D_{\tau }^{2} F\cdot F$ is a Hirota bilinear operator. They are defined as follows:[37,50] $$\begin{align} &D_{z,\alpha }^{n} D_{\tau,\alpha }^{m} G(z,\tau)\cdot F(z,\tau)\\ ={}&\Big({\frac{\partial }{\partial z}-\Big(\frac{1}{2}+i\alpha\Big)\frac{\partial }{\partial z'}}\Big)^{n} \times \Big({\frac{\partial }{\partial \tau }-\Big(\frac{1}{2}+i\alpha\Big)\frac{\partial }{\partial \tau'}}\Big)^{m}\\ &\times G(z,\tau)\cdot F(z',\tau')\vert_{z'=z,\tau'=\tau },~~ \tag {4} \end{align} $$ $$\begin{align} &D_{z}^{n} D_{\tau }^{m} G(z,\tau)\cdot F(z,\tau) =\Big(\frac{\partial }{\partial z}-\frac{\partial }{\partial z'}\Big)^{n}\Big(\frac{\partial }{\partial \tau }-\frac{\partial }{\partial \tau'}\Big)^{m}\\ &\times G(z,\tau)\cdot F(z',\tau')\vert_{z'=z,\tau'=\tau }.~~ \tag {5} \end{align} $$ Further, we can set the form of bright one-soliton solution as $$ \begin{alignat}{1} &G(z,\tau)=\varepsilon G_{1} (z,\tau)+\varepsilon^{3}G_{3} (z,\tau)+\varepsilon^{5}G_{5} (z,\tau)+\cdots,\\ &G_{1} (z,\tau)=\exp (b_{1} (z)+ib_{2} (z)+(a_{1} +ia_{2})\tau\\ &\qquad\qquad~\,+k_{1} +ik_{2}), \\ &F(z,\tau)=1+\varepsilon^{2}F_{2} (z,\tau)+\varepsilon^{4}F_{4} (z,\tau)+\varepsilon^{6}F_{6} (z,\tau)\\ &\qquad\quad~~~\,+\cdots ,\\ &F_{2} (z,\tau)=\sigma (z)\exp ({2b_{1} (z)+2a_{1} \tau+2k_{1}}).~~ \tag {6} \end{alignat} $$ Here, $b_{1}(z)$, $b_{2}(z)$ and $\sigma (z)$ are differentiable functions of $z$ to be determined; $\varepsilon$ is a formal expansion parameter, real constants $a_{1}$ and $a_{2}$ are complex frequencies, real constants $k_{1}$ and $k_{2}$ are initial phases. By substituting Eq. (6) into Eq. (3), the coefficients of different powers of $\varepsilon$ can be extracted, and each coefficient can be divided into two independent equations according to the imaginary part and the real part, and then the expressions of a series of parameters can be solved in turn as follows: $$\begin{align} &b_{1} (z)=\int \frac{1}{2}[g(z)+2a_{1} a_{2} B(z)+(a_{1}^{2} -a_{2}^{2})d(z)\\ &\qquad\quad-\beta (z)]dz,\\ &b_{2} (z)=\int [d(z)a_{1} a_{2} +\frac{1}{2}B[z](a_{2}^{2} -a_{1}^{2})]{\rm d}z,\\ &\varPsi [z]=\frac{1}{2}a_{1}^{2} [(4\alpha^{2}-3)B[z]+8\alpha d(z)]\sigma (z),\\ &K(z)=\frac{1}{2}a_{1}^{2} [(4\alpha^{2}-3)d(z)-8\alpha B(z)]\sigma (z),\\ &\sigma (z)=m\exp \Big[ {\mathop \int\nolimits [d(z)(a_{2} -2\alpha a_{1})^{2}+\beta (z)-g(z)]{\rm d}z} \Big],\\ &\qquad\quad\{m:m\in \Re \cap m\ne 0\},\\ &B(z)=2d(z)(a_{2} /a_{1}-\alpha),\\ &\rho (z)=d(z)a_{1} [(7\alpha -4\alpha^{3})a_{1} +(4\alpha^{2}-3)a_{2} ]\sigma (z)^{2},\\ &\nu (z)=\frac{1}{2}d(z)a_{1} [(3-20\alpha^{2})a_{1} +16\alpha a_{2} ]\sigma (z)^{2},\\ &G_{x} (z,\tau)=0,~x=(3,5,7,\dots);\\ &F_{x} (z,\tau)=0,~x=(4,6,8,\dots).~~ \tag {7} \end{align} $$ Without loss of generality, we set $\varepsilon = 1$. Finally, on the basis of all the above constraints, the one-soliton solution to Eq. (1) can be written as $$\begin{align} u(z,\tau)={}&\Big\{\exp \Big[y(z)+k_{1} +ik_{2} +\tau (a_{1} +ia_{2})\\ &+i\frac{(\alpha a_{1}^{3} -\alpha a_{1} a_{2}^{2} +a_{2}^{3})}{a_{1} }\mathop \int\nolimits d(z)dz\Big]\Big\}\\ &\cdot\Big\{\Big[1+m\exp \Big[\mathop \int\nolimits d(z)[ {(2\alpha a_{1} -2a_{2})^{2}+a_{1}^{2}}]dz\\ &+2\tau a_{1} +2k_{1}\Big]\Big]^{\frac{1}{2}+i\alpha}\Big\}^{-1},~~ \tag {8} \end{align} $$ and $$ y(z)=\frac{1}{2}\mathop \int\nolimits [g(z)+d(z)(a_{1}^{2} -4\alpha a_{1} a_{2} +3a_{2}^{2})-\beta (z)]dz.~~ \tag {9} $$ The free parameters are peak gain $g(z)$, linear loss $\beta (z)$, gain dispersion $d(z)$, modified parameter $\alpha$, frequency parameter $a_{1}$ and $a_{2}$, real constant $m$, and phase parameter $k_{1}$ and $k_{2}$. Results and Discussions—Bound States of Two Solitons. In fiber lasers, optical soliton pulses are formed under the balance of dispersion, nonlinearity and dissipation effect. However, when the effect of dissipation is weak and the balance of dispersion effect and nonlinear effect plays a major role, the dissipative system will be similar to a Hamiltonian system, so the traditional hyperbolic secant solitons can be observed. For the traditional soliton pulse, a single pulse will split to form a soliton bound state when its energy is too high, as shown in Fig. 1. In Fig. 1(a), we set all the coefficients as follows: $a_{1} = 0.0001$, $a_{2} = 3$, $\alpha = 1.46$, $k_{1} = 8$, $k_{2} = -7$, $g(z) = 0.3$, $d(z) = 0.2z$, $m = 60$, $\beta (z) = -0.079+2/(0.32-0.6z)$, and we will keep $\alpha$, $k_{1}$, $k_{2}$ unchanged in all of the following discussions. In this figure, a single soliton pulse with high energy is split into two soliton molecules due to the energy area theory, resulting in the formation of the soliton bound state. Furthermore, we can see that the energy distributions of the two soliton molecules are not equal to each other.
cpl-38-9-094201-fig1.png
Fig. 1. Evolution of soliton bound states. The corresponding parameters are selected to be $a_{1} = 0.0001$, $a_{2} = 3$, $\alpha = 1.46$, $k_{1} = 8$, $k_{2} = -7$, $g(z) = 0.3$, $d(z) = 0.2z$, $m = 60$ with (a) $\beta (z) = -0.079+2 /(0.32-0.6z)$ and (b) $\beta (z) = -0.079+2/(0.12-0.6z)$.
Then we can achieve the soliton bound state with uniform energy distribution by adjusting $\beta (z) = -0.079+2/(0.12-0.6z)$ in Fig. 1(b). Moreover, we observe that the splitting point of the soliton pulse is determined by the point of discontinuity of $\beta (z)$, so we can change the splitting position by controlling the point of discontinuity of $\beta (z)$. The research on the bound states of optical solitons can help to carry out more complex information modulation and coding in optical communication, and may help to advance the multi-channel sharing technology. Soliton Rectangular Pulses. When the dispersion effect is not dominant in fiber lasers, meanwhile the interaction between dissipative and nonlinear effect is more prominent, the shapes of optical soliton pulses will evolve into rectangles as shown in Fig. 2. This is the first time that the rectangular solitons are presented in the variable-coefficient CQGLE. In Fig. 2(a), all coefficients are set as $\alpha = 1.46$, $k_{1} = 8$, $k_{2} = -7$, $a_{1} = 0.6025$, $a_{2} = 3$, $g(z) = 5.138$, $d(z) = 2$, $\beta (z) = 2.06$, $m = 60$, and in this case, the frequency propagation width of the rectangular soliton pulse is about 42, and its amplitude is about 0.13. For the dissipative solitons with a rectangular shape, they are absolutely stable, and can be enhanced with the increase of the net gain in the system without splitting, so as to ensure the output of a high-energy single pulse. In order to further increase the total energy of the soliton pulse, we can make efforts to broaden the propagation width and to increase the amplitude of the soliton. On the one hand, in Fig. 2(b), the net gain $\Delta g_{0}(z)= g(z) - \beta (z)$ of the system is increased from 3.078 in Fig. 2(a) to 10.078, while the values of $a_{1}$, $a_{2}$ and $d(z)$ are replaced with $-1.3$, 2 and 0.3, respectively, to keep the balance among dissipation effect, dispersion effect and nonlinear effect, then the rectangular pulse with nearly doubled frequency range is shown. On the other hand, in Fig. 2(c), the value of $m$ is reduced from 60 to 10, which leads to the enhancement of the nonlinear effect in the dissipative system, and then the amplitude of the rectangular wave increases from 0.14 in Fig. 2(b) to 0.34, that is, the energy of the soliton increases significantly. Our results are beneficial to the preparation of high energy rectangular soliton pulses in the fiber laser. Moreover, due to the advantages of high energy and stability of rectangular pulses, they show the surprising potential in the applications of high-speed and long-haul all-optical networks, optical sensing, optical ranging, all-optical square wave clock, and so on.
cpl-38-9-094201-fig2.png
Fig. 2. Evolution of soliton rectangular pulses. The corresponding parameters are selected to be $\alpha = 1.46$, $k_{1} = 8$, $k_{2} = -7$ with (a) $a_{1} = 0.6025$, $a_{2} = 3$, $g(z) = 5.138$, $d(z) = 2$, $\beta (z) = 2.06$, $m = 60$; (b) $a_{1} = -1.3$, $a_{2} = 2$, $g(z) = 10$, $d(z) = 0.3$, $\beta (z) = -0.078$, $m = 60$; (c) $a_{1} = -1.3$, $a_{2} = 2$, $g(z) = 10$, $d(z) = 0.3$, $\beta (z) = -0.078$, $m = 10$.
In summary, an analytic one-soliton solution to the variable-coefficient CQGLE has been obtained with the modified Hirota method. Then the stable soliton bound states in the weak dissipative system and the rectangular soliton pulses in the strong dissipative system have been observed and studied in detail for the first time. In the weak dissipative system, solitons have appeared as hyperbolic secant pulses, and most of the high energy solitons have split into soliton bound states, as well as the linear loss $\beta (z)$ is the key to the splitting point and energy distribution of soliton molecules. In the strong dissipative system, solitons behave as rectangular pulses without splitting, and increasing the net gain $\Delta g_{0}(z)$ or enhancing the nonlinear effect by reducing $m$ can lead to the increase of the total energy of solitons. Moreover, it is necessary to pay attention to the harmony of dispersion, nonlinear and dissipation effect in the dissipative system by adjusting the relevant parameters. The present results not only are perfect for theoretical systems of solitons but also provide valuable references for applications of optical solitons in optical communication and preparation of ultra-high energy soliton pulses in fiber lasers. References Dark optical solitons: physics and applicationsPhysical mechanisms of the rogue wave phenomenonDark solitary waves in a class of collisionally inhomogeneous Bose-Einstein condensatesThe Peregrine soliton in nonlinear fibre opticsA study on a two-wave mode Kadomtsev-Petviashvili equation: conditions for multiple soliton solutions to existDynamics of ring dark solitons in Bose–Einstein condensates and nonlinear opticsA two-mode modified KdV equation with multiple soliton solutionsSolitons in optical communicationsNonautonomous Solitons in External PotentialsEffect of two photon absorption on nonlinear pulse propagation in gain mediumOptical spatial solitons: historical overview and recent advancesGiant amplification of modes in parity-time symmetric waveguidesExact solutions of the nonlocal Hirota equationsBound-State Soliton Solutions of the Nonlinear Schrödinger Equation and Their Asymmetric DecompositionsInverse scattering solutions and dynamics for a nonlocal nonlinear Gross–Pitaevskii equation with PT-symmetric external potentialsMulti-dark soliton solutions for a coupled AB system in the geophysical flowsSoliton Molecules and Some Hybrid Solutions for the Nonlinear Schrödinger Equation *Multi-soliton solutions for a (2+1)-dimensional variable-coefficient nonlinear Schrödinger equationDark and antidark soliton interactions in the nonlocal nonlinear Schrödinger equation with the self-induced parity-time-symmetric potentialPeriodic, breather and rogue wave solutions for a generalized (3+1)-dimensional variable-coefficient B-type Kadomtsev–Petviashvili equation in fluid dynamicsMixed soliton solutions of the defocusing nonlocal nonlinear Schrödinger equationPeriodic soliton interactions for higher-order nonlinear Schrödinger equation in optical fibersAsymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations with time-dependent potentialDemonstration of soliton transmission over more than 4000 km in fiber with loss periodically compensated by Raman gainExperimental observation of soliton interaction over long fiber paths: discovery of a long-range interactionAccurate Monte-Carlo Tests of the Stochastic Ginzburg-Landau Model with Multiplicative Colored NoiseGinzburg-Landau Equations for ( d+s )-Wave Superconductors in a Non-Fermi LiquidTransitions of stationary to pulsating solutions in the complex cubic-quintic Ginzburg-Landau equation under the influence of nonlinear gain and higher-order effectsDecomposition of Gauge Potential in Ginzburg–Landau TheoryStable transmission of solitons in the complex cubic–quintic Ginzburg–Landau equation with nonlinear gain and higher-order effectsStable soliton propagation in a coupled (2 + 1) dimensional Ginzburg–Landau systemOn the Ginzburg-Landau laser mode-locking model with fifth-order saturable absorber termOptical Bullet Holes: Robust Controllable Localized States of a Nonlinear CavityGroup interactions of dissipative solitons in a laser cavity: the case of 2+1Impact of high-order effects on soliton explosions in the complex cubic-quintic Ginzburg-Landau equationExact Solutions of the Generalized Ginzburg-Landau EquationChirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equationDromion-like structures and periodic wave solutions for variable-coefficients complex cubic–quintic Ginzburg–Landau equation influenced by higher-order effects and nonlinear gainChirped Lambert W-kink solitons of the complex cubic-quintic Ginzburg-Landau equation with intrapulse Raman scatteringInvestigation on breather waves and rogue waves in applied mechanics and physicsSingularities and special soliton solutions of the cubic-quintic complex Ginzburg-Landau equationStable soliton pairs in optical transmission lines and fiber lasersDissipative soliton resonance in an all-normaldispersion erbium-doped fiber laser033 mJ, 1043 W dissipative soliton resonance based on a figure-of-9 double-clad Tm-doped oscillator and an all-fiber MOPA systemSquare-wave pulse with ultra-wide tuning range in a passively mode-locked fiber laserDissipative solitons for mode-locked lasersDispersion-managed soliton fiber laser with random dispersion, multiphoton absorption and gain dispersionAnalytic soliton solutions of cubic-quintic Ginzburg-Landau equation with variable nonlinearity and spectral filtering in fiber lasersDissipative Solitons in a Generalized Coupled Cubic–Quintic Ginzburg–Landau Equations
[1] Kivshar Y S and Luther-Davies B 1998 Phys. Rep. 298 81
[2] Kharif C and Pelinovsky E 2003 Eur. J. Mech. B 22 603
[3] Wang C, Law K J H, Kevrekidis P G, and Porter M A 2013 Phys. Rev. A 87 023621
[4] Kibler B, Fatome J, Finot C, Millot G, Dias F, Genty G, Akhmediev N, and Dudley J M 2010 Nat. Phys. 6 790
[5] Wazwaz A M 2017 Math. Methods Appl. Sci. 40 4128
[6] Kamchatnov A M and Korneev S V 2010 Phys. Lett. A 374 4625
[7] Wazwaz A M 2017 Appl. Math. Lett. 70 1
[8] Haus H A and Wong W S 1996 Rev. Mod. Phys. 68 423
[9] Serkin V N, Hasegawa A, and Belyaeva T L 2007 Phys. Rev. Lett. 98 074102
[10] Roy S and Bhadra S 2008 Commun. Nonlinear Sci. Numer. Simul. 13 2157
[11] Chen Z G, Segev M, and Christodoulides D N 2012 Rep. Prog. Phys. 75 086401
[12] Konotop V V, Shchesnovich V S, and Zezyulin D A 2012 Phys. Lett. A 376 2750
[13] Zuo D W and Zhang G F 2019 Appl. Math. Lett. 93 66
[14] Zhang Y S and He J S 2019 Chin. Phys. Lett. 36 030201
[15] Yu F J 2019 Appl. Math. Lett. 92 108
[16] Xie X Y and Meng G Q 2019 Appl. Math. Lett. 92 201
[17] Wang B, Zhang Z, and Li B 2020 Chin. Phys. Lett. 37 030501
[18] Lan Z Z 2018 Appl. Math. Lett. 86 243
[19] Li M and Xu T 2015 Phys. Rev. E 91 033202
[20] Lan Z Z 2019 Appl. Math. Lett. 94 126
[21] Xu T, Lan S, Li M, Li L L, and Zhang G W 2019 Physica D 390 47
[22] Chen J, Luan Z, Zhou Q, Alzahrani A K, Biswas A, and Liu W J 2020 Nonlinear Dyn. 100 2817
[23] Deng Q and Yao X 2020 J. Math. Phys. 61 041504
[24]Ping T J and Sheng Z 2005 J. Optoelectron. Laser 16 120
[25] Mollenauer L F and Smith K 1988 Opt. Lett. 13 675
[26] Mollenauer L F and Smith K 1989 Opt. Lett. 14 1284
[27] Bao J D 1992 Chin. Phys. Lett. 9 1
[28] Han Q and Zhang L Y 1998 Chin. Phys. Lett. 15 742
[29] Uzunov I M, Georgiev Z D, and Arabadzhiev T N 2018 Phys. Rev. E 97 052215
[30] Li J F, Jiang Y, Sun W M, Wang F, and Zong H S 2010 Chin. Phys. Lett. 27 087403
[31] Yan Y Y and Liu W J 2019 Appl. Math. Lett. 98 171
[32] Wang L L and Liu W J 2020 Chin. Phys. B 29 070502
[33] Moores J D 1993 Opt. Commun. 96 65
[34] Firth W J and Scroggie A J 1996 Phys. Rev. Lett. 76 1623
[35] Grelu P and Akhmediev N 2004 Opt. Express 12 3184
[36] Gurevich S V, Schelte C, and Javaloyes J 2019 Phys. Rev. A 99 61803
[37] Nozaki K and Bekki N 1984 J. Phys. Soc. Jpn. 53 1581
[38] Kalashnikov V L 2009 Phys. Rev. E 80 46606
[39] Yan Y Y, Liu W J, Zhou Q, and Biswas A 2020 Nonlinear Dyn. 99 1313
[40] N, Maan N, Goyal A, Raju T S, and Kumar C N 2020 Phys. Lett. A 384 126675
[41] Yin X, Chen L, Wang J, Zhang X, and Ma G 2021 Alexandria Eng. J. 60 889
[42] Akhmediev N N, Afanasjev V V, and Soto-Crespo J M 1996 Phys. Rev. E 53 1190
[43] Akhmediev N N, Ankiewicz A, and Soto-Crespo J M 1998 J. Opt. Soc. Am. B 15 515
[44] Wu X, Tang D Y, Zhang H, and Zhao L M 2009 Opt. Express 17 5580
[45] Zheng Z, Ouyang D, and Ren X 2019 Photon. Res. 7 513
[46] Zhang X, Gu C, and Chen G 2012 Opt. Lett. 37 1334
[47] Grelu P and Akhmediev N 2012 Nat. Photon. 6 84
[48] Singh P G, Rajib P, Malomed B A, and Soumendu J 2018 J. Opt. 20 105501
[49] Huang L G, Pang L H, Wong P, Li Y Q, and Bai S Y 2016 Ann. Phys. 528 493
[50] Zakeri G A and Yomba E 2013 J. Phys. Soc. Jpn. 82 084002