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Optical Solitons in Fibers
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... They have also shown analytically that the bound state Nth order soliton is breaking, i.e., experiences fission, under sufficient high-order linear and/or nonlinear perturbation in the fiber. According to the ISM, each fundamental soliton after fission carries an energy given by [1,13,17] ...
... and where fundamental solitons are expressed as [17] ...
... The observed ECE enhancement with respect to the ISM prediction is explained from a transfer of energy in between solitons following the fission process, known as interpulse Raman scattering [21,22], and in contrast with intrapulse Raman scattering. In a bound state, a highorder soliton experiences a periodic evolution due to a mutual interaction between GVD and self-phase modulation (SPM) [17,20]. High-order linear and/or nonlinear perturbations break down the bound state, which triggers a fission process, and the MS breaks into fundamental soliton components. ...
We formulate the energy conversion efficiency from a high-order soliton to fundamental solitons by including the influence of interpulse Raman scattering in the fission process. The proposed analytical formula agrees closely with numerical results of the generalized nonlinear Schrodinger equation as well as to experimental results, while the resulting formulation significantly alters the energy conversion efficiency predicted by the Raman-independent inverse scattering method. We also calculate the energy conversion efficiency in materials of different Raman gain profiles such as silica, ZBLAN and chalcogenide glasses (As2S3 and As2Se3). It is predicted that ZBLAN glass leads to the largest energy conversion efficiency of all four materials. The energy conversion efficiency is a notion of utmost practical interest for the design of wavelength converters and supercontinuum generation systems based on the dynamics of soliton self-frequency shift.
... They have also shown analytically that the bound state Nth-order soliton is breaking, i.e., experiences fission, under sufficient high-order linear and/or nonlinear perturbation in the fiber. According to the ISM, each fundamental soliton after fission carries an energy given by [1,13,17] ...
... and where fundamental solitons are expressed as [17] q k (τ, ...
... The observed ECE enhancement with respect to the ISM prediction is explained from a transfer of energy in between solitons following the fission process, known as interpulse Raman scattering [21,22], and in contrast to intrapulse Raman scattering. In a bound state, a high-order soliton experiences a periodic evolution due to a mutual interaction between GVD and self-phase modulation (SPM) [17,20]. High-order linear and/or nonlinear perturbations break down the bound state, which triggers a fission process, and the MS breaks into fundamental soliton components. ...
We formulate the energy conversion efficiency from a high-order soliton to fundamental solitons by including the influence of interpulse Raman scattering in the fission process. The proposed analytical formula agrees closely with numerical results of the generalized nonlinear Schrödinger equation as well as to experimental results, while the resulting formulation significantly alters the energy conversion efficiency predicted by the Raman-independent inverse scattering method. We also calculate the energy conversion efficiency in materials of different Raman gain profiles such as silica, ZBLAN, and chalcogenide glasses ( ${{\rm As}_2}{{\rm S}_3}$ A s 2 S 3 and ${{\rm As}_2}{{\rm Se}_3}$ A s 2 S e 3 ). It is predicted that ZBLAN glass leads to the largest energy conversion efficiency of all four materials. Energy conversion efficiency is a notion of utmost practical interest for the design of wavelength converters and supercontinuum generation systems based on the dynamics of a soliton self-frequency shift.
... The idea to use solitons for data transmission in optical fiber lines arose for the first time in [5]. After this work, the nonlinear Schrödinger equation and its modifications were extremely intensively studied in relation to fiber telecommunication systems [6,7,8,9]. Later, the idea was put forward to use multisoliton pulses in fiber-optic data transmission lines, when information is modulated and restored in the so-called nonlinear Fourier domain [10,11,12]. Despite the fact that NLSE is an integrable system [13], its numerous studies were carried out by numerical methods. ...
... Formulas for the expansion of Ω(t) up to the 8th order in τ are given in [40]. In this case, 18 nested commutators are added, which additionally contain Q (6) and Q (5) . To obtain a consistent 8th order finite-difference scheme, at least a 7-point stencil must be used. ...
... We compared the forementioned exponential 6th order schemes with the CF [6] 4 scheme [33,37]. This is a commutator-free quasi-Magnus (CFQM) exponential integrator with complex coefficients. ...
Based on the generalized Cayley transform, a family of conservative one-step schemes of the sixth order of accuracy for the Zakharov-Shabat system is constructed. The exponential integrator is a special case. Schemes based on rational approximation allow the use of fast algorithms to solve the initial problem for a large number of values of the spectral parameter.
... As it is well known, dispersion and dissipation are extremely important for soliton pulse propagation in nonlinear media. These two processes are the main cause for the distortion and losses of the signal (Agrawal, 2001;Hasegawa, & Matsumoto, 2003;Ablowitz, Prinari, & Trubatch, 2004), and have been studied since the end of | 1960's, although it was not until 1980's that people began to use amplifiers to compensate those losses (Hasegawa, & Matsumoto, 2003). In the amplification process, the silica doping of fibers is commonly used (Ablowitz, Prinari, & Trubatch, 2004). ...
... As it is well known, dispersion and dissipation are extremely important for soliton pulse propagation in nonlinear media. These two processes are the main cause for the distortion and losses of the signal (Agrawal, 2001;Hasegawa, & Matsumoto, 2003;Ablowitz, Prinari, & Trubatch, 2004), and have been studied since the end of | 1960's, although it was not until 1980's that people began to use amplifiers to compensate those losses (Hasegawa, & Matsumoto, 2003). In the amplification process, the silica doping of fibers is commonly used (Ablowitz, Prinari, & Trubatch, 2004). ...
... Theoretically, this kind of propagation is mainly described by the nonlinear Schrödinger equation (NLS), but other equations can be used, such as the Sine-Gordon, Korteweg-de Vries, and Ginzburg-Landau equations that can also describe this kind of soliton propagation (Agrawal, 2001;Hasegawa, & Matsumoto, 2003;Ablowitz, Prinari, & Trubatch, 2004). ...
We find and discuss the non-autonomous soliton solutions in the case of variable nonlinearity and dispersion implied by the Ginzburg-Landau equation with variable coefficients. In this work we obtain non-autonomous Ginzburg-Landau solitons from the standard autonomous Ginzburg-Landau soliton solutions using a simplified version of the He-Li mapping. We find soliton pulses of both arbitrary and fixed amplitudes in terms of a function constrained by a single condition involving the nonlinearity and the dispersion
of the medium. This is important because it can be used as a tool for the parametric manipulation of these non-autonomous solitons.
... First, derivation of an approximate solution for the soliton based on a variational formulation for (3.2). This variational approximation uses a trial function for ψ S depending on a small set of parameters, like frequency and duration, for which a set of ODEs is derived, [43]. It will be found that the direct variational approximation for the given equation does not capture the soliton dynamics -it misses to predict any change in soliton peak power. ...
... The soliton equation (3.2) is reformulated as a perturbation equation for the standard NLS equation, following a well known procedure described, e.g., in [43]. ...
... SPT is a widely used method to determine propagation behavior of localized solutions under the influence of small perturbations. Various approaches have been developed: direct perturbation theory [88], variation of conserved quantities [41], variation of the scattering data of inverse scattering transform [48], and Lagrange formulation of perturbation theory [43]. ...
Das Thema dieser Arbeit ist eine mögliche Steuerung eines optischen Solitons in nichtlinearen optischen Fasern. Es gelang, die interessierenden Solitonparameter wie Intensität, Dauer und Zeitverschiebung durch die Wechselwirkung mit einer dispersiven Welle geringer Intensität kontrollierbar zu modifizieren. Es wird eine neue analytische Theorie vorgestellt für die Wechselwirkung zwischen Solitonen und dispersiven Wellen, die auf der Kreuzphasenmodulation in nichtlinearen Fasern beruht. Das vorgestellte Modell kombiniert quantenmechnische Streutheorie und eine Erweiterung der Störungstheorie für Solitonen aus der nichtlinearen Optik. Damit wurden folgende neue Ergebnisse erzielt: (1) Die Entwicklung aller Solitonparameter wird korrekt vorhergesagt. Insbesondere wird die mögliche Verstärkung der Solitonamplitude erfolgreich bestimmt. (2) Passende Intervalle der Kontrollparameter, die eine effektive Solitonmanipulation garantieren, können quantitativ bestimmt werden. (3) Der Raman-Effekt wurde in die Modellbeschreibung eingebunden. Die klassische Abschätzung der Eigenfrequenzverschiebung des Solitons durch den Raman-Effekt wurde verbessert und erweitert durch eine neue Relation für den einhergehenden Amplitudenverlust. Weiterhin wurden solche Kontrollpulse bestimmt, die dieser Schwächung des Solitons entgegenwirken. Im Unterschied zu früheren Versuchen liefert die hier entwickelte Modellbeschreibung die passenden Parameterbereiche für eine stabile Auslöschung des Raman-Effektes. (4) Obwohl die Wechselwirkung selbst auf der Kreuzphasenmodulation basiert, spielt der ”self-steepening“- Effekt, der die Bildung von optischen Schocks beschreibt, eine entscheidende Rolle für eine effiziente Veränderung der Solitonparameter.
... Maxwell's equations can be found in [5] or [6],but is not included here. Instead, the physical processes behind the NLSE are described. ...
... The dimensionless quantities used here are < = z/zL, s = r/tO, and For example, a fiber made of silica glass (Si02) has a group velocity dispersion of k" = -10ps2/km at a wavelength of 1.5 pm. The constant n2 is known to be approximately 1.22 x 10-22m2/V2 and no is about 1.4 [5]. For a pulse with width TO = 5ps, a silica fiber has to x 2.8ps, Po x 3.6 x 10f2V2/m2, and z~ x 807m . ...
... Two years later, in 1967, Gardner et al.[4] used the inverse-scattering method to solve the KdV equation. This method is described in some detail in[5]. In the inverse-scattering solution, solitons are described as bound states of a Schriidinger operator. ...
... Using ultrashort pulses and optical fibers, strong nonlinear optical effects can be induced. So far, many phenomena have been discovered with ultrashort pulses and optical fibers, such as the soliton effect [2], the soliton self-frequency shift (SSFS) [3], supercontinuum generation [4], temporal and spectral compression [5,6], and pulse trapping [7,8]. These nonlinear phenomena have been applied to several fields, such as light source technology, optical communication, optical frequency combs, optical metrology, and biomedical imaging [9][10][11]. ...
... It is well known that a soliton pulse in a fiber preserves its shape and shows periodical evolution when propagating along the fiber [1,2]. In a mode-locked fiber laser operating in the soliton regime, a soliton pulse accompanies the so-called Kelly sidebands, which are caused by the periodical interaction with dispersive waves inside the cavity [12]. ...
Using nonlinear optical effects, we can control the optical spectra in various ways. Here, we discovered a novel phenomenon in ultrafast nonlinear fiber optics, namely, periodical spectral peaking. A narrow spectral dip on a pulse turns into a sharp, intense peak through a nonlinear effect. This phenomenon shows periodical evolution behavior in the soliton regime. If we use absorption spectra with constant frequency separation, like molecular gas absorption spectra, we can generate almost uniform frequency spanning sharp spectra with a linewidth on the order of a few picometers and sub-THz frequency separation. The generated spectral peak almost replicates the characteristics of the absorption spectra. Fundamental characteristics and physical mechanisms were investigated both numerically and experimentally. This phenomenon provides us with a novel approach to control the optical spectra and opens up new aspects and applications of nonlinear fiber optics.
... The inverse scattering [1], variation, and perturbation methods [2] could obtain the analytical solutions under some special conditions. These included the inverse scattering method for classical solitons [3], the dam-break approximation for the non-return-to-zero pulses with the extremely small chromatic dispersion [4], and the perturbation theory for the multidimensional NLSE in the field of molecular physics [5]. ...
... The corresponding MI gain g MI in the side bands of ω 0 (the frequency of signal) is given by Figure 2 shows a comparison of the gain spectra between Eq. (11) and [6] for the case Pz ðÞ hi = P 0 z ðÞ hi ¼ 1. The maximum frequency modulation index caused by dispersion corresponds to 1 2 β 2 ω 2 dz ¼ π [22,23], and the maximum value of the sideband is ω c ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4γ Pz ðÞ hi = β 2 jj p , so the choice of dz satisfies 1 2 β 2 ω 2 dz ¼ π, which makes Eq. (11) have the same frequency regime as [26]. In Figure 2, the curves are different but have the same maximum value of g MI .I n practice, researchers generally utilize the maximum value of g MI to estimate the amplified noises and SNR [3]. ...
... As previously noted, the required balance between dispersion and nonlinearity shown in figure 1.1 can be maintained on average in a lossy fibre, if the pulse is periodically amplified. Such results are vitally linked to the NLS equation, which describes propagation in the fibre between amplifiers [63,171]. ...
... Reviews of this method can be found in references [~) 8,88,111,110] and examples relating to lumped nonlinear transmission lines in references [38,104,133,151,180]. A MMS derivation for propagation in optical fibre can be found in refereI;lces [63,64,171] Boyd [25] has said that these methods are the Justification for Inverse Scattering the sense that so many physical systems are reducible to either the KdV or NLS equations by these techniques. Without them, the KdV and NLS equations would be mathematical curios. ...
... Nonlinear Schrödinger equations are a kind of the most important mathematical models in physics with applications in many contexts [1][2][3][4], such as plasma physics, nonlinear optics and bimolecular dynamics. Recently, a growing interest is focused on the numerical solution for the coupled nonlinear Schrödinger (CNLS) equations. ...
... The larger β, more new waves will be produced. -For all β, new waves begin to be created at time interval [2,3]. That is, they maintains one wave as t < 2. -The conservation quantities are always keep constants for all β and every time. ...
In the manuscript, we present several numerical schemes to approximate the coupled nonlinear Schrödinger equations. Three of them are high-order compact and conservative, and the other two are noncompact but conservative. After some numerical analysis, we can find that the schemes are uniquely solvable and convergent. All of them are conservative and stable. By calculating the complexity, we can find that the compact schemes have the same computational cost with the noncompact ones. Numerical illustrations support our analysis. They verify that compact schemes are more efficient than noncompact ones from computation cost and accuracy. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2015
... When viewing the matrix IDNLS equation as a finite difference approximation of the matrix NLS equation, it has the same applications as the matrix NLS equation, namely electromagnetic wave propagation in nonlinear media [7,8], surface waves on sufficiently deep waters [7], and signal propagation in optical fibers [9,10]. Apart from that, the matrix IDNLS equation has applications to the dynamics of a discrete curve on an ultraspherical surface [11], the dynamics of triangulations of surfaces [12], and Hamiltonian flows [13,14]. ...
... It appears [4,5] that the spectrum of the discrete matrix Zakharov-Shabat system (4) is invariant under the sign inversion z → −z and that, as a result, the Marchenko kernels F (n; τ ) andF (n; τ ) vanish if n is an even integer. Thus further constraints on the triplets (A, B, C) and (Ā,B,C) are required to represent the Marchenko kernels in the form (9). In fact, the matrix triplets have to be decomposed as in (29) below in terms of matrix triplets (A, B, C) and (Ā,B,C), where A andĀ have half the matrix orders that A andĀ have. ...
In this article we derive explicit solutions of the matrix integrable discrete nonlinear Schrödinger equation under a quasiscalarity condition by using the inverse scattering transform and the Marchenko method. The Marchenko equation is solved by separation of variables, where the Marchenko kernel is represented in the form CA −(n+j+1) e iτ (A−A −1) 2 B, (A, B, C) being a matrix triplet where A has only eigenvalues of modulus larger than one. The class of solutions obtained contains the N -soliton and breather solutions as special cases. Unitarity properties of the scattering matrix are derived.
... The NLSEs have widespread applications in describing the propagation properties of optical solitons in nonlinear optics. Researchers have extensively studied NLSEs over a considerable period, focusing on their role in determining the dynamic behavior of light pulses [22][23][24][25][26][27]. Furthermore, NLSEs are of significant importance in various scientific disciplines, including optical fibers, hydrodynamics, biology, elastic media, quantum mechanics, magneto-static rotating waves, and optics [28][29][30][31][32][33]. ...
The motivation of this work is to attain optical solitons of the generalized third-order nonlinear Schrödinger equation (NLSE) and the integrable (2+1)-dimensional coupled nonlinear Schrödinger equation (CNLSE). These models have various applications, including ultra-short pulses in optical fibers, making them noteworthy in applied sciences and mathematical physics. To extract the optical solitons of these models, the unified solver method (USM) is applied. The USM is a mathematically modified version of the Riccati equation and its related solutions. A unique advantage of this method is the elimination of the need for a balancing constant, simplifying the analytical process and enhancing the utility of these soliton solutions in practical applications. The obtained solutions include dark, singular, and periodic singular soliton solutions, enriching the understanding of optical solitons. In addition to the analytical solutions, the article also includes graphical representations to visually depict the obtained soliton solutions. These visualizations enhance the understanding and provide a clear picture of how the solitons behave in the given system.
... Ultrashort pulses and optical fibers can be used to induce significant nonlinear effects that can be exploited in a wide variety of optical control techniques [1]. So far, many effects have been discovered, for example, the soliton effect, soliton self-frequency shifting (SSFS), and supercontinuum generation [1][2][3]. They have been used for light source technology, all-optical signal processing, and wavelength conversion and have been applied to biomedical imaging, optical frequency combs, spectroscopy, and optical communication [4]. ...
Nonlinear fiber effects are useful for controlling optical spectra in a wide variety of ways. Here, we report the demonstration of freely controllable, intense spectral peaking using a high-resolution spectral filter with a liquid-crystal spatial light modulator and nonlinear fibers. A large enhancement of spectral peak components by more than a factor of 10 was achieved by employing phase modulation. Multiple spectral peaks with an extremely high signal-to-background ratio (SBR) of up to 30 dB were generated simultaneously in a wide wavelength range. It was shown that part of the energy from the whole pulse spectrum was concentrated at the filtering part and constructed the intense spectral peaks. This technique is very useful for highly sensitive spectroscopic applications and comb mode selection.
... The nonlinear Schrödinger (NLS) equation, as one of nonlinear partial differential equations, has been studied in such fields as nonlinear optics, plasma physics and condensed matter physics due to its potential mathematical properties and physical applications [1][2][3][4]. It has been found that the NLS equation can describe the dynamic of nonlinear localized wave usually associated with soliton, breather and rogue wave. ...
In this paper, we investigate the coupled modified nonlinear Schrödinger equation. Breather solutions are constructed through the traditional Darboux transformation with the nonzero plane-wave solutions. To obtain the higher-order localized wave solution, N -fold generalized Darboux transformation is given. Under the condition that the characteristic equation admits a double-root, we present the expression of first-order interactional solution. Then we graphically analyze the dynamics of breather and rogue wave. Due to the simultaneous existence of nonlinear and self-steepening terms in the equation, there presents different profile in two components for the breathers.
... The nonlinear Schrödinger (NLS) equation, as one of nonlinear partial differential equations, has been studied in such fields as nonlinear optics, plasma physics and condensed matter physics due to its potential mathematical properties and physical applications [1][2][3][4]. It has been found that the NLS equation can describe the dynamic of nonlinear localized wave usually associated with soliton, breather and rogue wave. ...
In this paper, we investigate the coupled modified nonlinear Schödinger equation. Through the traditional Darboux transformation, we construct the first-order breather solution which can exhibit Akhmediev breather and general breather. To obtain the higher-order localized wave solution, N-fold generalized Darboux transformation is given. Under the condition that the characteristic equation admits a double-root, we present the expression of first-order interactional solution. Then we graphically analyze the dynamics of breather and rogue wave. Due to the simultaneous existence of nonlinear and derivative terms in the equation, there presents different profile in two components for the breathers.
... OLITIONS formation in optical fibers is a meaningful nonlinear wave phenomenon that has been observed and extensively investigated in diverse laser systems [1][2][3], owning to its fundamental importance and potential applications in optical signal processing and optical communications. It is well recognized that the dynamics of the soliton formation can be described by the nonlinear Schrödinger equation (NLSE), governing optical pulse propagation in ideal single-mode fibers (SMFs). ...
We report on the first experimental observation of passively harmonic mode-locking (HML) in a bidirectional domain-wall dark-soliton fiber laser. The domain-wall solitons (DWSs) are characterized, in which, the strong coupling between the two directions takes great responsibility for the generation. By appropriately altering the polarization state in the cavity, the stable fundamental domain-wall solitons and HML states could be obtained and switched with increasing the pump power. In our experiment, the repetition rate can be tunable from the fundamental mode locking of 2.96 MHz up to eighth-order HML of 23.68 MHz when the pump power is increased from 120 to 542.5mW. Improtantly, this HML operation of dark-pulse DWSs confirms that the DWSs are the intrinsic feature of the mode- locked lasers, as well as the conventional bright solitons. This work could provide a new perspective to the understanding of the formation and transformation of dark solitons, which may also find great potential applications in fiber sensing and future optical communication systems.
... Dispersion managed (DM) optical soliton is a proven and sought after constituent of optical communication, all-optical data processing and fiber laser (Kurtzke 1993;Suzuki et al. 1996;Sugahara et al. 1998;Hasegawa and Matsumoto 2003;Mak et al. 2005;Biswas and Wheeler 2010;Mishra and Hong 2011;Turitsyn et al. 2012). It occurs in a nonlinear system (e.g., an optical fiber) with periodic dispersion variation. ...
We present the generation and interaction dynamics of dispersion managed soliton in a dissipative cubic-quintic nonlinear optical fiber coupled with frequency-selective feedback. Analytically, perturbative variational method is used to obtain evolution equations for different soliton parameters that are subsequently solved to study the propagation characteristics of the dispersion managed dissipative solitons (DMDS). The DMDSs show breather like characteristics and are found to be robust against certain level of initial noise. Both the standard Gaussian and more generic super-Gaussian pulse profile are shown to yield stable DMDS. Interaction of two DMDSs of same phase may lead to a bound state. By varying the temporal separation and phase difference between the DMDSs the bound state can be switched between low to high speed regime.
... It has been shown by [4] that E l (r, θ , ξ , τ; ε) can be expanded in terms of ε: ...
This study presents the nonlinearity and dispersion effects involved in the propagation of optical solitons which can be understood by using a numerical routine to solve the Generalized Nonlinear Paraxial equation. A sequence of code has been developed in Mathematica, to explore in depth several features of the optical soliton's for mation and propagation. These numerical routines were implemented through the use with Mathematica and the results give a very clear idea of this interesting and important practical phenomenon.
... Nonlinear phenomena, which have been found in optics, plasma physics, condensed matter physics and other fields, can be described by the nonlinear evolution equations (NLEEs) [1][2][3][4][5]. Solitons, as the balance between the nonlinear and dispersion terms, are the nonlinear excitations represented by the solutions of the NLEEs [6,7]. ...
Under investigation in this paper is a fourth-order variable-coefficient nonlinear Schrödinger equation, which can describe an inhomogeneous one-dimensional anisotropic Heisenberg ferromagnetic spin chain or alpha helical protein. With the aid of the Hirota method and symbolic computation, bilinear forms, dark one- and two-soliton solutions are obtained. Influences of the variable coefficients on the dark one and two solitons are graphically shown and discussed. Amplitude and shape of the dark one soliton keep invariant during the propagation when the variable coefficients are chosen as the constants. With the variable coefficients being the functions, amplitude of the dark soliton keeps unchanged during the propagation, but direction of the soliton curves. Head-on and overtaking collisions between the dark two solitons are displayed with the variable coefficients chosen as the constants, and it is shown that the shapes of the two solitons do not change during the collision. When we choose the variable coefficients as the functions, directions of the two solitons change and elastic collisions occur between the two solitons.
... The generalized nonlinear Schrödinger equation (NLSE) is a generic model that is very important in NL optics, where it describes the full spatiotemporal optical solitons or light bullets (Akhmediev and Ankiewicz 1997;Kivshar and Agrawal 2003;Hasegawa and Matsumoto 2003;Malomed 2006). For this equation, along with several other related equations such as the Klein-Gordon (KG) equation and the Korteweg de-Vries (KdV) equation, there is a large interest in finding novel exact solutions (Drazin and Johnson 1989). ...
We generalize the Jacobi elliptic function expansion method used to solve the (3 + 1)-dimensional nonlinear Schrödinger equation for the case of an arbitrary inverse of an elliptic integral. Among the obtained solutions are functions based on the Weierstrass elliptic function and the inverses of Carlson’s elliptic integrals.
... However, such a derivation is cumbersome and we refer to [27,28] for details. Here for simplicity we will derive it by means of a Taylor expansion of the dispersion relation [29,30], assuming that the waves are quasi-monochromatic. We start with the electromagnetic waves by expanding the k(ω, |A| 2 ) around ω = ω 0 and |A| 2 = 0, that is ...
... Solitons are solutions to a conservative equation. To use them in applications such as mode-locked lasers and optical telecommunications[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34], loss must be compensated. This is naturally accomplished by laser gain, or by amplifiers placed throughout the line. ...
Despite the abundance and importance of three-dimensional systems, relatively
little progress has been made on spatiotemporal nonlinear optical waves
compared to time-only or space-only systems. Here we study radiation emitted by
three-dimensionally evolving nonlinear optical waves in multimode fiber.
Spatiotemporal oscillations of solitons in the fiber generate multimode
dispersive wave sidebands over an ultrabroadband spectral range. This work
suggests routes to multipurpose sources of coherent electromagnetic waves, with
unprecedented wavelength coverage.
... Let us say some words about the mathematical model and classification of regimen of SRSS. Major details can be found in well devoted books Agrawal (Agrawal, 1995) and Hasegawa (Hasegawa, 1989), and papers (Serkin, et.al. 2003, Karasik & Chunaev, 2007. ...
A comprehensive resume on the soliton solutions obtained by studying the Raman Effect on nonlinear propagation of optical pulses is presented. Additionally, we show the powerful method for studying the complex nonlinear differential equation that describes the Raman waves, by means of the mechanical analogy method.
... Equation (1.3) with vanishing boundary values, where q n is a rectangular matrix, has been studied in [5,12,13,17,18]. As a discretization of the nonlinear Schrödinger (NLS) equation (1.1), the IDNLS equation (1.3) has important applications to electromagnetic wave propagation in nonlinear media [33], surface waves on deep waters [33], and signal propagation in optical fibers [19,20]. On its own behalf, the IDNLS equation has applications to Heisenberg spin chains [22,27], self-trapping on a dimer [23], anharmonic lattices [29], the dynamics of a discrete curve on an ultraspherical surface [15], the dynamics of triangulations of surfaces [21], and Hamiltonian flows [24,28]. ...
In this article we develop the direct and inverse scattering theory of the Ablowitz-Ladik system with potentials having limits of equal positive modulus at infinity. In particular, we introduce fundamental eigensolutions, Jost solutions, and scattering coefficients, and study their properties.We also discuss the discrete eigenvalues and the corresponding norming constants. We then go on to derive the left Marchenko equations whose solutions solve the inverse scattering problem. We specify the time evolution of the scattering data to solve the initial-value problem of the corresponding integrable discrete nonlinear Schrödinger equation. The one-soliton solution is also discussed.
... Recently, soliton theory, one of typical topics in nonlinear science, has been widely applied in optics of nonlinear media [1], photonics [2], plasmas [3], mean-field theory of Bose–Einstein condensates [4], condensed matter physics [5], and many other fields. ...
Analytical solutions in terms of rational-like functions are derived for the (2 + 1)-dimensional nonlinear Schrodinger equation with time-varying coefficients using the similarity transformation and direct ansatz. By applying specific time-modulated nonlinearities, dispersions, and potentials, the dynamics of the solutions can be controlled. As a result, abundant wave structures are exhibited through chosen three types of elementary functions.
... The formation and propagation of solitary waves being able to propagate for a long distances maintaining their shape is one of the most interesting phenomena in physics of nonlinear waves. This is possible owing to the balance between the cubic nonlinearity and frequency dispersion (see, for example, [1] [2] [3] [4] [5] [6] [7] [8] [9]). Such propagation regimes can be described within the framework of slowly varying envelope approximation (SVE) as solutions of nonlinear Schrödinger equation, when the polarization of the propagating light is assumed to be constant, and when the propagating pulse envelope contains a big number of light field oscillations . ...
... This permits very strong confinement, for example to enhance nonlinear optical effects. Optical nonlinearities can be exploited to obtain a number of useful phenomena: optical solitons [48,69,177,194], which are pulses that don't spread out as they propagate; conversion of light at one wavelength to another wavelength [4, 27, 78, 132]; generation of white light (many colors) from a single input wavelength (an effect called supercontinuum generation) [33,40,103,110]. ...
This thesis focuses on achieving new understanding of the principles and phenomena involved in the interaction of light with a variety of complicated material systems, including biomaterials and nanostructured materials. We will show that bone piezoelectricity may be a source of intense blast-induced electric fields in the brain, with magnitudes and timescales comparable to fields with known neurological effects, and may play a role in blast-induced traumatic brain injury. We will also shed new light on the localization of photons in a variety of complex microstructured waveguides. We will reveal the principles behind the design of single-polarization waveguides, including design strategies that did not seem to have been considered previously. Finally, we designed a 3D photonic crystal slab structure to exhibit negative-index behavior at visible wavelengths, which was fabricated and experimentally demonstrated by our collaborators to show negative refraction with, to our knowledge, the lowest loss at visible wavelengths to date.
... Nowadays, it continues to be one of the major attractive subjects deserving deep investigations . In the wake of possible generation of very short light waves in laboratory, the nonlinear Schrödinger equation , being used to describe weakly nonlinear wave packets [1] [2] under slowly varying envelope approximation, become lest accurate. Accordingly, Schäfer and Wayne [3] derived an evolution equation expressed as [4] u xt ¼ u þ u 3 ...
This article studies the propagation of the short pulse optical model governed by higher order nonlinear Schrödinger equation (HNLSE) with non-Kerr nonlinearity. The model is used to describe the propagation of ultrashort photons in highly nonlinear media. Upon establishing the general framework, we discuss the statics and dynamics of HNLSE by employing an extended modified auxiliary equation mapping (EMAEM) architectonic to obtain some new solitary wave solutions like bright dromion (soliton), domain wall, singular, periodic, doubly periodic, trigonometric, rational and hyperbolic solutions etc. Obtained optical soliton solutions are analysed graphically to represent features like as width, amplitude, and structure of solitons. We will also discuss our governing model for M-shaped, Homoclinic breathers, multiwave, kink cross rational, and interaction of M-shaped with 1 and 2 kink solutions with the help of various ansatz transformations.
This paper presents an investigation of soliton solutions for the perturbed Fokas-Lenells (pFL) equation, which has a vital role in optics, using Sardar sub-equation method. The equation models the propagation of ultrashort light pulses in optical fibers. Using appropriate wave transformation, the pFL equation is reduced to a nonlinear ordinary differential equation (NLODE). The solutions of this NLODE equation are assumed to be in the suggested form by the Sardar sub-equation method. Hence, an algebraic equation system is obtained by substituting the trial solution and their necessary derivatives into the NLODE. After finding the unknowns in the system, the soliton solutions of the perturbed Fokas-Lenells equation are extracted. The method produces various kinds of solitons such as dark, periodic, singular periodic, combined bright-dark. To show physical representations of the solitons, 2D, 3D and contour plots of the solutions are demonstrated via computer algebraic systems. It is expected that derived solutions may be useful for future works in various fields of science, especially optics and so, it may contribute to the optic fiber industry.
The integrating factor technique is widely used to solve numerically (in particular) the Schrödinger equation in the context of spectral methods. Here, we present an improvement of this method exploiting the freedom provided by the gauge condition of the potential. Optimal gauge conditions are derived considering the equation and the temporal numerical resolution with an adaptive embedded scheme of arbitrary order. We illustrate this approach with the nonlinear Schrödinger (NLS) and with the Schrödinger–Newton (SN) equations. We show that this optimization increases significantly the overall computational speed, sometimes by a factor five or more. This gain is crucial for long time simulations, as, with larger time steps, less computations are performed and the overall accumulation of round-off errors is reduced.
In this paper, the optical solitons with M-truncated derivative and other solutions to the coupled nonlinear Schrodinger equation (NLSE) are obtained. The coupled NLSE describes the pulse propagation through a two-mode optical fiber. Complex amplitude ansatz and polynomial expansion methods are employed to construct such novel solutions. We successfully construct dark, mixed dark-bright, singular optical solitons and other solutions to the considered nonlinear model. Dark soliton describes the solitary waves with lowerintensity than the background, bright soliton describes the solitary waves whose peak intensity is larger thanthe background and the singular soliton solutions is a solitary wave with discontinuous derivatives; examples ofsuch solitary waves include compactions, which have finite (compact) support, and peakons, whose peaks have adiscontinuous first derivative. Numerical simulations of the solutions are used to analyze some of the propagation properties of pulses and their collision dynamics. Fusion along quasi-periodic waves is found to be exhibited. Under nonlinear refractive elements, the graded or phase index waveguide is seen to change the strength of the soliton. In addition, the properties are shown with figures for these solutions. These findings have been usefully extended to the transmission of long-wave and high-power telecommunication systems. In addition, through the figures presented, the effect of the fractional order \(\alpha\) is reflected.
We study the modulational stability of periodic travelling wave solutions to equations of nonlinear Schrödinger type. In particular, we prove that the characteristics of the quasi‐linear system of equations resulting from a slow modulation approximation satisfy the same equation, up to a change in variables, as the normal form of the linearized spectrum crossing the origin. This normal form is taken from Stability of Travelling wave solutions of Nonlinear Dispersive equations of NLS type, where Leisman et al. compute the spectrum of the linearized operator near the origin via an analysis of Jordan chains. We derive the modulation equations using Whitham's formal modulation theory, in particular the variational principle applied to an averaged Lagrangian. We use the genericity conditions assumed in the rigorous theory of Leismen et al. to direct the homogenization of the modulation equations. As a result of the agreement between the equation for the characteristics and the normal form from the linear theory, we show that the hyperbolicity of the Whitham system is a necessary condition for modulational stability of the underlying wave.
This survey focuses on the following problem: it is necessary, observing the behaviour of the object, automatically figure out how to improve (optimize) the quality of his functioning and to identify constraints to the improvement of this quality. In other words, build the objective function (or set of objective functions in multiobjective case) and constraints - i.e. the mathematical model of optimization - by mean machine learning. We present the main developed to date methods and algorithms that enable the automatic construction of mathematical models of planning and management objects by the use of arrays of precedents. The construction of empirical optimization models by reliable case information allows us to obtain an objective control model that reflects real-world processes. This is their main advantage compared to the traditional, subjective approach to the construction of control models. Relevant to the task a set of mathematical methods and information technologies called ``Extraction optimization models from data'', ``BOMD: Building Optimization Models from Data'', ``Building Models from Data'', ``The LION Way: Learning plus Intelligent Optimization'', ``Data-Driven Optimization''. The incompleteness of information and uncertainty are understood in different ways. Significantly different are the problem settings - deterministic, stochastic, parametric, mixed. Therefore, the consideration of a wider range of tasks leads to a variety of (primarily statistical) and other formulations of the problem and interpretations of uncertainty and incompleteness of initial information. The survey contains the following sections: Empirical synthetic of pseudoBoolean models; Empirical linear models with real variables; Empirical neural network optimization models; Iterative models; Models, including statistical statements; Problems, associated with the lack of the training set of points not belonging to the region of feasible solutions.
We report on the first experimental observation of vector dark solitons in an erbium-doped single mode fiber ring laser. We show experimentally that two orthogonal linearly polarized dark solitons could couple together incoherently through the cross polarization phase modulation and form a bound state of dark solitons. In particular, the formed vector dark solitons are stable even under existence of high frequency periodic background intensity modulation. Numerical simulations have well confirmed the existence of vector dark solitons in single mode fiber lasers and their stability under periodic background intensity modulation.
Two-dimensional Rossby solitary waves propagating in a line have attracted much attention in the past decade, whereas there is few research on three-dimensional Rossby solitary waves. But as is well known, three-dimensional Rossby solitary waves are more suitable for real ocean and atmosphere conditions. In this paper, using multiscale and perturbation expansion method, a new Zakharov-Kuznetsov (ZK)-Burgers equation is derived to describe three-dimensional Rossby solitary waves that propagate in a plane. By analyzing the equation we obtain the conservation laws of three-dimensional Rossby solitary waves. Based on the sine-cosine method, we give the classical solitary wave solutions of the ZK equation; on the other hand, by the Hirota method we also obtain the rational solutions, which are similar to the solutions of the Benjamin-Ono (BO) equation, the solutions of which can describe the algebraic solitary waves. The rational solutions of the ZK equations are worth of attention. Finally, with the help of the classical solitary wave solutions, similar to the fiber soliton communication, we discuss the dissipation and chirp effect of three-dimensional Rossby solitary waves.
A Darboux-Bäcklund transformation is used to obtain a positon type solution of the nonlinear equations describing the propagation of coupled nonlinear optical pulses. This form of the positon solution is then compared with that obtained by the special limiting procedure applied to a two-soliton solution. It is observed that though the algebraic form of the two solutions is different yet both of these have singularities and the position of the singularities remains on the similar curve in the (x, t) plane. We also depict the form of these solutions graphically. Finally, it may be added that the method of Darboux-Bäcklund transformation is convenient for generating more than one-positon solution.
We consider the cubic-quintic nonlinear Schrödinger equation describing the propagation of femtosecond pulses in nonlinear optical fibers. Through analysis of different phase portraits, we predict the existence of solitary and periodic wave solutions of the previous equation. Numerical solutions confirm such existence. In addition, by using a special rational exponential ansatz, we find analytic solitary and periodic wave solutions. These interesting solutions may be useful to explain dynamical properties of optical traveling wave solutions. © 2015, Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg.
In this study, we consider a natural integrable generalization of the defocusing cubic nonlinear Schrödinger equation to two dimensions and we classify the admissible boundary conditions. In particular, we determine whether the classical physical observables are conserved: mass, momentum, and Hamiltonian. We find that this is the case when a certain integral (the mass constraint) vanishes. The vanishing of the mass constraint, and thus the existence of conserved quantities, is contingent on the boundary conditions adopted. In particular, under decaying boundary conditions, the Hamiltonian is not necessarily conserved.
The exact chirped soliton-like and quasi-periodic wave solutions of the (3+1)-dimensional generalized nonlinear Schrödinger equation including linear and nonlinear gain (loss) with variable coefficients are obtained detailedly in this paper. The form and the behaviour of solutions are strongly affected by the modulation of both the dispersion coefficient and the nonlinearity coefficient. In addition, self-similar soliton-like waves precisely piloted from our obtained solutions by tailoring the dispersion and linear gain (loss).
Modern computer algebra systems are revolutionizing the teaching and learning of mathematically intensive subjects in science and engineering, enabling students to explore increasingly complex and computationally intensive models that provide analytic solutions, animated numerical solutions, and complex two-and three-dimensional graphic displays. This self-contained text benefits from a spiral structure that regularly revisits the general topics of graphics, symbolic computation, and numerical simulation with increasing intricacy at each turn. The text is built around a large number of computer algebra worksheets or "recipes" that have been designed using MAPLE to provide tools for problem solving and to stimulate critical thinking. No prior knowledge of MAPLE is assumed. All relevant commands are introduced on a need-to-know basis and are indexed for easy reference. Each recipe is associated with a scientific model or method and an interesting or amusing story designed to both entertain and enhance concept comprehension and retention. All recipes are included on the CD-ROM enclosed with the book. Aimed at third- and fourth-year undergraduates in science and engineering, the text contains numerous examples in disciplines that will challenge students progressing in mathematics, physics, engineering, game theory, and physical chemistry. Computer Algebra Recipes: An Advanced Guide to Mathematical Modeling can serve as an effective computational science text, with a set of problems following each section of recipes to enable readers to apply and confirm their understanding. The book may also be used as a reference, for self-study, or as the basis of an online course. © 2007 Springer Science+Business Media, LLC. All rights reserved.
We consider a one-dimensional modified complex Ginzburg-Landau equation, which governs
the dynamics of matter waves propagating in a discrete bi-inductance nonlinear
transmission line containing a finite number of cells. Employing an extended Jacobi
elliptic functions expansion method, we present new exact analytical solutions which
describe the propagation of periodic and solitary waves in the considered network.
Drawing examplesfrom mathematics, physics, chemistry, biology, engineering, economics, medicine, politics, and sports, this book illustrates how nonlinear dynamics plays a vital role in our world. Examples cover a wide range from the spread and possible control of communicable diseases, to the lack of predictability in long-range weather forecasting, to competition between political groups and nations. After an introductorychapter that explores what it means to be nonlinear, the book covers the mathematical conceptssuch as limit cycles, fractals, chaos, bifurcations, and solitons, that will be applied throughout the book. Numerous computer simulations and exercises allow students to explore topics in greater depth using the Maple computer algebra system. The mathematical level of the text assumes prior exposure to ordinary differential equations and familiarity with the wave and diffusion equations.No prior knowledge of Maple is assumed, and all Maple examples are included on a CD. The book may be used at the undergraduate or graduate level to prepare science and engineering students for problems in the "real world", or for self-study by practicing scientists and engineers.
We consider the stability of soliton-like pulses propagating in nonlinear optical fibers with periodically spaced phase-sensitive amplifiers, a situation where the averaged pulse evolution is governed by a fourth-order nonlinear diffusion equation similar to the Kuramoto-Sivashinsky equation or Swift-Hohenberg equations. A bifurcation and stability analysis of this averaged equation is carried out, and in the limit of small amplifier spacing, a steady-state pulse solution is shown to be asymptotically stable. Furthermore, both a saddle-node bifurcation and a subcritical bifurcation from the zero solution are found. Analytical results are confirmed using the bifurcation software package AUTO. The analysis provides evidence for the existence of stable pulse solutions for a wide range of parameter values, including those corresponding to physically realizable soliton communications systems.
A numerical study of two-dimensional Schrödinger equation is carried out using the improved complex variable element-free Galerkin (ICVEFG) method. The ICVEFG method involves employment of the improved complex variable moving least-squares (ICVMLS) in the element-free Galerkin (EFG) procedure for numerical approximation. The ICVMLS is used to construct trial functions for the two-dimensional Schrödinger equation in the form of one-dimensional basis function that effectively reduces the number of unknown coefficients. In this study, the applicability of the ICVEFG method is examined through a number of numerical example problems. Convergence studies are carried out for these example problems by varying the number of nodes to ascertain convergent results are achieved as the number of nodes increases. The stability and accuracy of the ICVEFG method are validated by comparing the computed results with the exact solutions.
Lately, there is much interest in the use of the spin of carriers in semiconductor quantum well (QW) structures together with their charge to realize novel concepts like spintronics. The necessary conditions to develop spintronic devices are high spin polarizations in QWs and a large spin-splitting of subbands in k-space. The latter is important for the ability to control spins with an external electric field by the Rashba effect. Significant progress has been achieved recently in generating large spin polarizations, in demonstrating the Rashba splitting and also in using the splitting for manipulating the spins. At the same time as these conditions are fulfilled and spins are polarized in-plane of QW, it has been shown that the spin polarization itself drives a current resulting in the spin galvanic effect [1,2]. The spin-galvanic effect is due to asymmetric spin-flip scattering of spin polarized carriers and it is determined by the process of spin relaxation. In some optical experiments, where circularly polarized radiation is used to orient spins, the photocurrent may represent a sum of spin-galvanic and circular photogalvanic effects effects.2,3 Both effects provide methods to determine spin relaxation times and the relative strength of the Rashba/Dresselhaus spin-splitting in semiconductor quantum wells.²
The inverse spin-galvanic effect⁴ has also been detected demonstrating that electric current in non-magnetic but gyrotropic QWs results in a non-equilibrium spin orientation. Just recently a first direct experimental proof of this effect was obtained in semiconductor QWs5,6 as well as in strained bulk material.⁷ Microscopically the effect is a consequence of spin-orbit coupling which lifts the spin-egeneracy in k-space of charge carriers together with spin dependent relaxation.
Note from Publisher: This article contains the abstract only.
Based on the bifurcation and the idea that the solitary waves and shock waves of partial differential equations correspond respectively to the homoclinic and heteroclinic trajectories of nonlinear ordinary differential equations satisfied by the travelling waves, different conditions for the existence of solitary waves of a perturbed sine-Gordon equation are obtained. All of the corresponding approximate solitary wave solutions are given by integrating the derived approximate equations directly.
Transmission of data over fiber optic communication lines in the form of solitons appears to be a promising way to increase the speed of transmission. Such a mode of data transmission requires proper matching of light source and light guide parameters to ensure self-compression of anomalously scattering wave packets and formation of steady one dimensional solitons. Theoretically, propagation of a soliton through a real optical fiber can be described a cubic Schroedinger equation. This equation can be solved as an inverse problem of scattering with the aid of perturbation theory, which insures exact integration of the corresponding system of Marchenko equations. The solution indicates the requirements for formation and propagation of solitons. An optical data transmission system built experimentally on this basis for design and performance analysis includes a source of ultrashort light pulses, namely a parametric light generator without resonator, a more than 20 percent efficient coupling, a single mode quartz fiber 600 m long with a core diameter of 7 microns and a loss factor not exceeding 1.5 dB/km, and a device for recording solitons within the 1.06 to 1.60 micron range of crystal wavelengths.
The operation of directional couplers in the soliton mode in optical communication systems is considered. Results are presented on soliton evolution in cases of strong and weak coupling between the optical fibers.
A set of equations describing pulse propagation in multimode optical fibers in the presence of an intensity-dependent refractive index is derived by taking advantage of the coupled-mode theory usually employed for describing the influence of fiber imperfections on linear propagation. This approach takes into account in a natural way the role of the waveguide structure in terms of the propagation constants and the spatial configurations of the propagating modes and can be applied to the most general refractive-index distribution. The conditions under which soliton propagation and longitudinal self-confinement can be achieved are examined.
An analysis is made of the problems encountered in controlled nonlinear modification and stabilization of picosecond pulses of the fundamental and higher-order TE and TM modes traveling in fiber-optic waveguides. The sensitivity of such processes to initial amplitude-phase characteristics of signals, waveguide geometry, and propagation mode is the reason for the great diversity of the energy and spatial tuning scales. The advantages and shortcomings of soliton and nonsoliton mode signals in waveguides are considered. A review is given of near-threshold regimes for the stabilization of nonsoliton pulses and of some of their parameters over finite distances in graded-index waveguides. Data-handling capabilities of nonlinear communication channels, governed by the transmission distance and channel geometry, are discussed.
The asymptotic dispersive broadening of soliton pulses in optical fibers in the presence of weak losses has recently, following extensive numerical calculations, been shown to be less than the linear dispersion. We derive an explicit analytical approximation for the asymptotic dispersion that is in good agreement with numerical results. We also consider the case of strongly damped frequency-chirped pulses, where again the action of the nonlinearity is found to affect the long-term asymptotic properties of the pulses.
Solitons in optical fibers: light pulse with surprising properties
- F M Mitschke
Optical soliton propagation in ideal monomode fiber
- T Weisi