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Output spectra vs input pump power on u and v axis. (a,b) numerical simulation of the CNLSEs (1) with loss; (c,d) experiments.
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The Manakov model is the simplest multicomponent model of nonlinear wave theory: It describes elementary stable soliton propagation and multisoliton solutions, and it applies to nonlinear optics, hydrodynamics, and Bose-Einstein condensates. It is also of fundamental interest as an asymptotic model in the context of the widely used wavelength-divis...
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... that some narrow FWM sidebands cannot be completely suppressed before injection into the telecom fiber, thus leading to their subsequent amplification upon propagation. Figure 3 compares the numerical output spectra emerging from the two orthogonal polarization states of the fiber (left column) with the experimental spectra, as a function of the input power P u and P v of the two pumps, which are linearly polarized along orthogonal u and v directions of the fiber, respectively. Note that the orientation of these polarization directions is completely arbitrary, since the fiber is nominally circular; hence it does not exhibit two principal axes of polarization. ...
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... of these polarization directions is completely arbitrary, since the fiber is nominally circular; hence it does not exhibit two principal axes of polarization. Here the frequency spacing between the two pumps was set to 200 GHz, corresponding to P o = 3.3 W and G sat = 62 dB/km. We consider the passband PMI regime. As it can be seen from Fig. 3, for pump powers above 0.5 W, a broadband PMI spectrum develops as a mirror (and orthogonally polarized) image of each pump. Figure 3 reveals the excellent agreement between the numerical solutions of the CNLSEs (1) with added loss and the experimental results. Moreover, Fig. 3 shows the emergence of secondary outer sidebands, as ...
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... it can be seen from Fig. 3, for pump powers above 0.5 W, a broadband PMI spectrum develops as a mirror (and orthogonally polarized) image of each pump. Figure 3 reveals the excellent agreement between the numerical solutions of the CNLSEs (1) with added loss and the experimental results. Moreover, Fig. 3 shows the emergence of secondary outer sidebands, as predicted by the linear stability analysis at the highest powers as P = 2P u = 2P v approaches P o . ...
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... 62 dB/km. We consider the passband PMI regime. As it can be seen from Fig. 3, for pump powers above 0.5 W, a broadband PMI spectrum develops as a mirror (and orthogonally polarized) image of each pump. Figure 3 reveals the excellent agreement between the numerical solutions of the CNLSEs (1) with added loss and the experimental results. Moreover, Fig. 3 shows the emergence of secondary outer sidebands, as predicted by the linear stability analysis at the highest powers as P = 2P u = 2P v approaches P o . Note that the growth of these secondary sidebands is reinforced by the cascaded FWM between each pump and the parallel primary PMI sideband. We can also observe from Fig. 3 that the ...
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... results. Moreover, Fig. 3 shows the emergence of secondary outer sidebands, as predicted by the linear stability analysis at the highest powers as P = 2P u = 2P v approaches P o . Note that the growth of these secondary sidebands is reinforced by the cascaded FWM between each pump and the parallel primary PMI sideband. We can also observe from Fig. 3 that the gravity center of the primary sideband approaches the pump as the pump power increases, in good qualitative agreement with the analytical predictions of the linear stability ...
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... As an important branch of nonlinear science, soliton is widely applied due to the property of maintaining unchanged waveform and velocity before and after collision, which is of great interest to scholars [1][2][3]. Particularly, it has been confirmed through experiments in the optical and electronic fields that soliton controls information propogation effectively [4]. Thus, optical soliton is employed widely for its characteristics of carrying information over long distances with less distortion [5,6]. ...
In this paper, the mixed 4-coupled nonlinear Schrödinger equations with different nonlinear signs are studied to derive a new type of soliton solutions called the superposition soliton solutions. By using the Hirota method, we obtain the exact one-bright-three-superposition N-soliton solutions analytically. Notably, this kind of soliton solutions have not been researched in prior literature. Under certain conditions, the general mixed (bright-dark) soliton solutions can be obtained from our results such as all bright soliton solutions. In addition, the propagation characteristics, including elastic collision, time periodicity and soliton reaction, are displayed through graphic simulation. On this basis, the influence of various parameters on the phase, direction, and amplitude of soliton propogation is concluded. Finally, the asymptotic behaviors of 2, 3-soliton solutions are analyzed in detail.
... The Manakov system has accommodated the development of new models to represent complex wave propagations, such as CNLS equations of three or multiple components (Kanna and Lakshmanan 2001), Manakov model with variable coefficients, varying potential and nonlinearities (Zhong et al. 2015;Su et al. 2013;Cheng et al. 2014), modified Manakov equations (Tsoy and Akhmediev 2006), coupled optical fiber system (Li and Guan 2021), two-component Gross-Pitaevskii equations and others. The Manakov system has significant applications in biology (Scott 1984), finance (Yan 2011), fluid dynamics (Dhar and Das 1991), Bose-Einstein condensates (Busch and Anglin 2001), nonlinear fiber optics (Frisquet et al. 2015) etc. , Gerdjikov and Todorov (2019), Mumtaz et al. (2012), Radhakrishnan et al. (1999), Özışık et al. (2022). ...
In nonlinear optical telecommunication networks and optical switching devices, the study of optical solitons is critical. In recent years, coupled nonlinear Schrödinger equations have been studied regarding the optical solitons and their collisions. When the coupled nonlinear Schrödinger equations are of Manakov type, the optical solitons collide with each other elastically and after collision their polarization may change depending on the polarization of incoming optical solitons. In order to develop and improve innovative optical devices, enhance the stability of optical communication networks, and minimize fiber losses, it is imperative to establish an analytical approach capable of generating a diverse range of optical solitons. The goal of this manuscript is the utilization of a specific integration scheme to produce a diverse range of optical solitons for the Manakov model, with the aim of reducing both experimental costs and time. In this study, the extended sinh-Gordon equation expansion method and the two variable G′/G,1/G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( G'/G, 1/G \right)$$\end{document}-expansion method are employed to enable a comparison of the solutions and demonstrate the originality of this research. For the considered expansion methods, optical soliton solutions such as dark-dark soliton, bright-bright soliton, combined dark-bright soliton, multi soliton and periodic solitary waves are achieved. Moreover, the graphical demonstration of these solitons is made in order to better understand the obtained results.
... In the literature, there are many studies on models that include the term CD. Up to now, an extensive class of models regarding this matter in literature has been developed, such as Schrödinger-Hirota model [3,4], Radhakrishnan-Kundu-Lakshmanan model [5][6][7][8][9], Manakov system [10][11][12][13][14], Sasa-Satsuma model [15][16][17][18][19], Kundu-Eckhaus equation [20][21][22][23][24], and many more. Recently, some models have been derived by reckoning without the term chromatic dispersion and adding higher-order nonlinear terms from the literature. ...
In this study, one of our main subjects is the examination of optical solitons of the nonlinear Schrödinger equation having cubic-quintic-septic-nonic nonlinearities via the modified F-expansion method. The other subject is also the analysis of the impacts of some parameters in the model on the soliton shape, which is examined for the first time in this study. According to the modified F-expansion method, we select the suitable transformation to gain the nonlinear ordinary differential equation for the nonlinear Schrödinger equation having cubic-quintic-septic-nonic nonlinearities in the first stage. Then, we get a system consisting of linear equations in polynomial form with the aid of the modified F-expansion method. Various solution sets consisting of the parameters of the nonlinear Schrödinger equation having cubic-quintic-septic-nonic nonlinearities are achieved. Inserting the selected sets and transformations into the serial form of the presented method and utilizing the solutions of the auxiliary equation in the presented method, the optical soliton solutions of the model are derived. Furthermore, varied optical soliton solutions, such as anti-kink, singular, and bright, are achieved, and 3D and 2D projections of the generated soliton solutions have been illustrated. The impact of some parameters on each soliton behavior has also been examined. It is found that these parameters have a significant impact on the soliton structure.
... Recently, optical dark-dark RWs have been observed experimentally [36,37] in two orthogonally polarized optical fibers [48]. We set the parameter settings to be identical to Ref. [36] to discuss the possibilities of experimentally exciting other types of vector RWs in the two-component defocusing case. ...
... Referring to the experiments [36,37,48], the optical fiber can be a reverse-TrueWave fiber with a chromatic dispersion of −14 ps nm −1 km −1 (equivalent to β 2 = 18 ps 2 /km), a nonlinear coefficient γ = 2.4 W −1 km −1 , and an attenuation of 0.25 dB/km at the central wavelength λ 0 = 1554.7 nm. P 0 = 9π 2 2 β 2 8γ denotes the power for which the low cutoff frequency vanishes. ...
We systematically investigate the spatial-temporal patterns of rogue waves in N-component coupled defocusing nonlinear Schrödinger equations where N≥2. The fundamental rogue-wave solutions are given in a unified form for both focusing and defocusing cases. We establish the quantitative correspondence between modulation instability and rogue-wave patterns, which develops the previously reported inequality relation into an equation correspondence. As an example, we demonstrate phase diagrams for rogue-wave patterns in a two-component coupled system, based on the complete classification of their spatial-temporal structures. The phase diagrams enable us to predict various rogue-wave patterns, such as the ones with a four-petaled structure in both components. These results are meaningful for controlling the rogue-wave excitations in two orthogonal polarization optical fibers.
... In particular, this happens when several wave components are involved in the dynamics [16][17][18]. The MI of the coupled evolution equations has been studied in several publications [19][20][21][22][23][24]. However, the role of additional MI bands in the nonlinear stage of MI evolution remains largely unexplored. ...
We study higher-order modulation instability phenomena in the frame of Manakov equations. Evolution that starts with a single pair of sidebands expands over several higher harmonics. The choice of initial pair of sidebands influences the structure of unstable frequency components and changes drastically the wave evolution leading, in some cases, to jumps across spectral components within the discrete spectrum. This complex dynamics includes several growth-decay cycles of evolution. We show this using numerical simulations of the MI process and confirm the results using the exact multi-Akhmediev breather solutions. Detailed explanation of the observed phenomena are given.
... The applicability of Eqs. (1) with N = 2 in physics has been verified experimentally in optics [50][51][52][53] and for description of multicomponent BECs [54,55]. This task becomes significantly more difficult when the number of components in Eqs. ...
We present exact multi-parameter families of soliton solutions for two- and three-component Manakov equations in the \emph{defocusing} regime. Existence diagrams for such solutions in the space of parameters are presented. Fundamental soliton solutions exist only in finite areas on the plane of parameters. Within these areas, the solutions demonstrate rich spatio-temporal dynamics. The complexity increases in the case of 3-component solutions. The fundamental solutions are dark solitons with complex oscillating patterns in the individual wave components. At the boundaries of existence, the solutions are transformed into plain (non-oscillating) vector dark solitons. The superposition of two dark solitons in the solution adds more frequencies in the patterns of oscillating dynamics. These solutions admit degeneracy when the eigenvalues of fundamental solitons in the superposition coincide.
... F I G U R E 1 Vector breather of type I, see Equation (43), which can be obtained from scalar breather solution using the transformation (40). The solution parameters are defined in (47). ...
... Similar to type I, the general breathers of type II and III are localized and move on the condensate background. At the same time, the structure of type II and III solutions fundamentally differs from type I and cannot be retrieved by a solution transformation, similar to Equation (40). One can say that the general breathers of type II and III represent a nontrivial vector counterpart of the scalar NLSE breather. ...
... One can see that the numerator (97) is exactly canceled at any and , when 12 = ℎ 13 , 22 = ℎ 23 , where ℎ is an arbitrary constant. The latter happens only when 0,1 = 0, 0,2 = 0, so that ℎ = 2 ∕ 1 , that is, when both breathers are of type I and the vector two-breather solution is the trivial generalization of the scalar one, see transformation (40). ...
We study theoretically the nonlinear interactions of vector breathers propagating on an unstable wavefield background. As a model, we use the two‐component extension of the one‐dimensional focusing nonlinear Schrödinger equation—the Manakov system. With the dressing method, we generate the multibreather solutions to the Manakov model. As shown previously in [D. Kraus, G. Biondini, and G. Kovačič, Nonlinearity 28(9), 3101, (2015)], the class of vector breathers is presented by three fundamental types I, II, and III. Their interactions produce a broad family of the two‐component (polarized) nonlinear wave patterns. First, we demonstrate that the type I and the types II and III correspond to two different branches of the dispersion law of the Manakov system in the presence of the unstable background. Then, we investigate the key interaction scenarios, including collisions of standing and moving breathers and resonance breather transformations. Analysis of the two‐breather solution allows us to derive general formulas describing phase and space shifts acquired by breathers in mutual collisions. The found expressions enable us to describe the asymptotic states of the breather interactions and interpret the resonance fusion and decay of breathers as a limiting case of infinite space shift in the case of merging breather eigenvalues. Finally, we demonstrate that only type I breathers participate in the development of modulation instability from small‐amplitude perturbations withing the superregular scenario, while the breathers of types II and III, belonging to the stable branch of the dispersion law, are not involved in this process.
... Recently, the optical dark-dark RWs have been observed experimentally [35,36] in two orthogonally polarized optical fibers [42]. We set the parameter settings to be identical to Ref. [35] to discuss the possibilities of experimentally exciting other types of vector RWs in two-component defocusing case. ...
We systematically investigate rogue wave's spatial-temporal pattern in $N$ $(N\geq2)$-component coupled defocusing nonlinear Schr\"{o}dinger equations. The fundamental rogue wave solutions are given in a unified form for both focusing and defocusing cases. We establish the quantitative correspondence between modulation instability and rogue wave patterns, which develops the previously reported inequality relation into an equation correspondence. As an example, we demonstrate phase diagrams for rogue wave patterns in a two-component coupled system, based on the complete classification of their spatial-temporal structures. The phase diagrams enable us to predict various rogue wave patterns, such as the ones with a four-petaled structure in both components. These results are meaningful for controlling the rogue wave excitations in two orthogonal polarization optical fibers.
... Since then, the system has been becoming a fundamental model in nonlinear science and technology for its wide applications from optical, electrical, magnetic and fluid dynamics to Bose-Einstein condensates and finance [4,5]. As we know, optical experiments have confirmed the effectiveness of information transmission through the solitons governed by this system [6][7][8]. In fiber communication, it is shown experimentally that impairments induced by dispersion and Kerr nonlinearity can be compensated digitally for polarization-division multiplexed wavelength-division multiplexing transmission via applying the Manakov system [9]. ...
By the means of reconstructing the Darboux transformation to the Manakov system, we obtain the new explicit solutions composed of a rogue wave and a breather for the system. Via controlling the parameter involved in the solutions, we reveal a novel ‘firewall’ effect during the rogue wave and breather interactions. There are three cases of inelastic collisions and one type of semi-elastic collision between the rogue wave and breather. Particularly, either of the rogue wave or breather cannot cross over another during the collision.
... where the nonlinear coefficients 1 = ±1 and 2 = ±1. These equations govern many physical processes such as the interaction of two incoherent light beams in crystals [28,29,65], transmission of light in randomly birefringent optical fibers [27,66,67], and evolution of two-component Bose-Einstein condensates [68,69]. Rogue waves in the Manakov system are rational solutions that satisfy the following boundary conditions, ...
... where s j is as defined in Theorem 4 and numerically given in Eq. (67). The error of this lower-order rogue wave approximation is O(|a m | −1 ). ...
... Internal parameters in these predicted lower (N 1Q , N 2Q )-th order rogue waves in the center region are all zero, due to our choices of internal parameters in the original rogue waves and the s j values shown in Eq. (67). Plotting these (N 1Q , N 2Q )-th order rogue waves from Theorem 4, we get the predicted center-region solutions in the second to fourth rows of Fig. 10. ...
We show that new types of rogue wave patterns exist in integrable systems, and these rogue patterns are described by root structures of Okamoto polynomial hierarchies. These rogue patterns arise when the $\tau$ functions of rogue wave solutions are determinants of Schur polynomials with index jumps of three, and an internal free parameter in these rogue waves gets large. We demonstrate these new rogue patterns in the Manakov system and the three-wave resonant interaction system. For each system, we derive asymptotic predictions of its rogue patterns under a large internal parameter through Okamoto polynomial hierarchies. Unlike the previously reported rogue patterns associated with the Yablonskii-Vorob'ev hierarchy, a new feature in the present rogue patterns is that, the mapping from the root structure of Okamoto-hierarchy polynomials to the shape of the rogue pattern is linear only to the leading order, but becomes nonlinear to the next order. As a consequence, the current rogue patterns are often deformed, sometimes strongly deformed, from Okamoto root structures, unless the underlying free parameter is very large. Our analytical predictions of rogue patterns are compared to true solutions, and excellent agreement is observed, even when rogue patterns are strongly deformed from Okamoto root structures.
