Nonlinear Schrodinger Equation Research Articles

Accurate Computation of b-Coefficients in Nonlinear Fourier Transform for the Manakov Equation

In recent years, nonlinear Fourier transform (NFT) has been intensely researched as a scheme for communication over the nonlinear optical fiber and characterizing dynamics in nonlinear optical systems. For the accurate computation of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> -coefficients on discrete eigenvalues, the bidirectional algorithm has been proposed where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> -coefficients are calculated at a cutting point within the temporal window rather than two boundaries. The performance of the bidirectional algorithm depends severely upon the selection criterion of the cutting-point, which has been investigated in nonlinear Schrodinger equation and Korteweg–De Vries equation but not yet in Manakov equation accounting for the dual-polarization signals over optical fiber. In this paper, we propose a cutting-point criterion of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> -coefficients for the bidirectional algorithm of Manakov equation and compare it with existing criteria. Numerical results show that, the resulting algorithm has much better accuracy compared with existing criteria, especially for signals containing a large number of eigenvalues with large imaginary parts. The bidirectional algorithm with the proposed criterion is useful for decoding the dual-polarization nonlinear frequency division multiplexing (DP-NFDM) signals for communication and characterization of vector soliton dynamics in Manakov equation described systems.

Analysis of Brownian motion in stochastic Schrödinger wave equation using Sardar sub-equation method

In this paper, the stochastic nonlinear Schrödinger equation (NLSE) forced by multiplicative noise that represents the temporal change of fluctuations is examined in a more extended form. Symmetry reduction (nonclassical symmetry) is used to convert the nonlinear partial differential equation into a nonlinear ordinary differential equation to obtain new optical soliton solutions. The Sardar sub-equation method is suggested to solve this stochastic nonlinear problem. The obtained stochastic solitons illustrate the dispersion of waves in optical fiber transmissions. This approach enables us to study a wide range of solutions with significant physical perspectives, including dark solitons, bright solitons, singular and periodic solitons with their noise term effects involved in Brownian motion based on the Itô sense. The noise term effects on the solitons are presented using 3D and 2D graphs. The results and calculations show the significance, accuracy and efficiency of the method. Various stable and unstable nonlinear stochastic differential equations that are found in mathematics, physics and other applied areas can be solved using this technique.

Study of transition dynamics from breathing single pulse to higher-order pulses and hysteresis in a figure-of-9 fiber laser

This study investigates the nonlinear dynamics and hysteresis phenomena in a figure-of-9 fiber laser. The laser cavity consists of an Erbium doped nonlinear amplifying loop mirror (NALM) containing a Dispersion Compensation Fiber (DCF) segment, coupled to a nonlinear optical loop mirror (NOLM). A lumped-element model is developed for the laser cavity accounting for bi-directional light propagation. The model is based on the modified nonlinear Schrödinger equation (NLSE) with coefficients directly related to the parameters of each cavity segment. Gain saturation is taken into consideration for stable laser operation. The numerical results show that self-starting passive mode-locking and self-structuration of soliton lattices occur in the cavity of the laser. The soliton structuring is based on the concept of dissipative solitons, which requires a balance between dispersion-nonlinearity and gain-loss. The gain-loss parameters influence both the temporal pulse interactions and the NALM’s transmission, leading to a large variety of single and multiple breathing dissipative soliton dynamics. In that context, riveting studies of the transition dynamics from breathing single pulses to high-order multiple breathing dissipative solitons and hysteresis with respect to unsaturated amplifier gain in a figure-of-9 all-fiber laser are explored.

Ground states for the NLS equation with combined local nonlinearities on noncompact metric graphs

We investigate the existence of ground states for the nonlinear Schrödinger equation with combined local L2-critical and subcritical nonlinearities on noncompact metric graphs. The interplay of the combined local nonlinearities creates some new phenomena. Precisely, we show that the existence (or nonexistence) of ground states mainly depends on the topological and metric properties of the noncompact metric graphs. Our results rely on the Concentration-compactness principle and an application of Gagliardo-Nirenberg inequalities.

Study of a nonlinear Schrodinger equation with truncated M proportional derivative

In this work, we introduce a novel truncated M proportional (T-MP) derivative and consider a T-MP perturbed derivative nonlinear Schrodinger equation (PDNLSE). The PDNLSE is a nonlinear model which arises in nano optical fibers (photonic nanowires). While the DNLSE exhibits self-steepening (SS), the PDNLSE exhibits self-phase modulation (SPM) and Raman scattering (RS) effects, Here, the exact solutions of the PDENLSE are derived, here, by implementing the unified method. These solutions are displayed in graphs. It is found that a sufficient condition for a hyper-chaotic solution to hold is that an elliptic function solution exists. Non elliptic solutions may be, also, hyper-chaotic. It is worth mentioning that hyper chaotic may occur in economic and financial mathematics. Nonchaotic solutions exhibit many geometric structures, chirped solitons, and rhombus-shaped and M-shaped solitons. It is found that modulation instability triggers when the coefficient of the third-order dispersion exceeds a critical value. Further, the global bifurcation is investigated via phase portrait by constructing the Hamiltonian function.

On the solitonic wave structures and stability analysis of the stochastic nonlinear Schrödinger equation with the impact of multiplicative noise

The nonlinear Schrödinger equation (NLSE) is a fundamental and significant physical model that plays a crucial role in demonstrating the dynamics of optical solitons in the theory optical fibers. The propagation of solitons in nonlinear optical fibers has garnered substantial interest due to its broad spectrum of applications, particularly in ultrafast signal routing systems and short light pulses for communications. This article focuses on investigating the impact of multiplicative noise on the solutions of this governing model. To achieve this, three distinct methods, namely the enhanced modified extended tanh expansion (METE) method, the generalized Kudryashov (GK) approach and the extended modified auxiliary equation mapping (EMAEM) method are employed to obtain a new variety of stochastic solutions. The perception of how noise intensity impacts the transmission of waves is a fundamental and critical issue in various fields. In the context of this study, an equation describes an exact wave pattern under the condition that s (representing noise intensity) equals zero. In such cases, the observed pattern remains undisturbed and matches the expected theoretical wave pattern. However, as the strength of noise s increases, the observed pattern begins to deviate from the ideal wave pattern and exhibits signs of damage or distortion. Furthermore, the stability analysis for the stochastic NLSE is performed, providing valuable insights into the behavior solutions as the strength of multiplicative noise increases. The validation of computational results through stability analysis enhances the confidence in the findings of the study. All the mathematical calculations, as well as graphical representations, have been meticulously achieved to present a comprehensive and detailed understanding of the impact of multiplicative on soliton propagation in nonlinear optical fibers.

(2+1)-dimensional modulated nonlinear ion-acoustic soliton packets in magnetized electronegative plasma

The dynamics of propagation of (2+1)-dimensional ion-acoustic wave (IAW) packets in magnetized electronegative plasma made of negative ions with electrons, both in Boltzmann distribution and cold mobile positive ions is studied. For this purpose, the standard reductive perturbation method is used to derive the (2+1)-dimensional nonlinear Schrödinger (NLS) equation, which governs the modulation of positive IAW packets. The coefficients of the obtained NLS equation do not only depend on the system parameters specially the electronegative parameters such as the negative ion concentration ratio and the electron-to-negative ion temperature ratio (α and σn, respectively), but also on the magnetic field (ωp). The modulational instability is studied to analyze the areas of the emergence of waves. In order to describe the main characteristics of IAWs, the variational method is performed involving trial function solution of NLS equation and the exact (input profile) and numerical (output profile) solutions are also discussed.

Soliton Interactions with Different Dispersion Curve Functions in Heterogeneous Systems

In practical optical communication systems, there are some factors that can affect transmission quality of optical solitons. The constant coefficient nonlinear Schrödinger (NLS) equation has been unable to meet the actual research needs. We need to use the variable coefficient NLS equation to simulate an actual system, so as to explore its potential application value. Based on the variable coefficient NLS equation, six dispersion decreasing fibers (DDFs) with different dispersion curve functions are used as transmission media to study generation and interaction of two solitons in an optical communication system. The two soliton interaction phenomena, such as the bound state solitons, are theoretically obtained. Moreover, the output characteristics of bound state solitons in different DDFs are discussed, which enriches the transmission phenomenon of two solitons in the optical communication system. This study has great application value in fields such as optical information processing devices, condensed matter physics, and plasma, and provides an indispensable theoretical basis for development of new optical devices.

A self-consistent magnetoelectric coupling model for GaN-based /Terfenol-D composites

A magnetoelectric composites is designed and investigated, which combines GaN-based semiconductor and magnetostrictive Terfenol-D. In this work, we study the polarization charge and strain of the multilayer composites under an applied magnetic field and prestress. In order to quantitatively describe the response of composites under different loading environments, a nonlinear magneto-electric self-consistent coupling model is established based on nonlinear constitutive equation, Schrodinger equation, and Poisson equation. The validity of this model is well-proven by comparing experimental data and numerical results under different magnetic fields. The results demonstrate that the giant magnetostriction effect of Terfenol-D offered large tensile strain to decrease the polarization charge and strain in GaN-based semiconductor layer. These conclusions will be of great help to the basic research of piezomagnetic composite materials as well as guidance for the future device design.

Dynamical evolutions of optical smooth positons in variable coefficient nonlinear Schrödinger equation with external potentials

We construct the optical smooth positon solutions for the generalized variable coefficient nonlinear Schrödinger equation with longitudinally varying dispersion, nonlinearity, linear and harmonic potentials. By employing the similarity transformation technique, we derive this solution along with the homogeneous smooth positon solutions of the basic NLS equation and the integrable requirement. The dynamical properties of the constructed second- and third-order positon solutions are investigated considering the two physical scenarios. In the first situation, we consider a longitudinally variable linear potential while keeping the dispersion parameter constant. In the second scenario, we take into account three different types of modulated dispersion parameters, namely quadratic, exponential, and linear, to analyze the various features in the intensity profile of inhomogeneous optical smooth positons. In both situations, we observe a wide range of significant nonlinear phenomena in the intensity profiles of positons. These phenomena include amplification, deformation, compression, prolongation, suppression, and distortion. Our findings could make a significant contribution to intriguing nonlinear optics experiments focused on investigating optical positons.

Large deviations principle for the cubic NLS equation

AbstractIn this paper, we present a probabilistic study of rare phenomena of the cubic nonlinear Schrödinger equation on the torus in a weakly nonlinear setting. This equation has been used as a model to numerically study the formation of rogue waves in deep sea. Our results are twofold: first, we introduce a notion of criticality and prove a Large Deviations Principle (LDP) for the subcritical and critical cases. Second, we study the most likely initial conditions that lead to the formation of a rogue wave, from a theoretical and numerical point of view. Finally, we propose several open questions for future research.

Hybrid Mode-Locked Fiber Ring Laser Using Graphene Saturable Absorbers to Generate 20 and 50-GHz Pulse Trains

Optical pulses at high repetition rates are generated using rational harmonic mode locking and saturable absorber made of graphene nanoparticles in a fiber laser. The pulse generation from the fiber laser is modeled by solving the Generalized Nonlinear Schrodinger Equation. The computation involved varying the various saturable absorption parameters, such as linear and nonlinear absorption coefficients. Experimentally stable pulse trains at 20 GHz and 50 GHz are generated with a pulse width of ∼ 2.7 ps. This result agrees with the simulation.

Equivalent transformation and integrability of the nonlinear Schrödinger equations with time-dependent coefficients

The nonlinear Schrödinger (NLS) types of equations play a key role in quantum mechanics, Quantum communication and physical applications. However, how to deal with explicit solutions and other properties of the NLS equations, especially for the variable-coefficient NLS (vc-NLS) types of equations is a difficult problem. In this paper, we construct the form-preserving equivalent transformations (ETs) to transform the vc-NLS systems into constant-coefficient NLS (cc-NLS) systems, and the form-preserving ETs are given explicitly. Then, based on the equivalent transformation method, we deal with the integrability of the NLS equations, and the Lax pairs (LPs) are provided as verification of the integrability.

Transport of Gaussian measures with exponential cut-off for Hamiltonian PDEs

We show that introducing an exponential cut-off on a suitable Sobolev norm facilitates the proof of quasi-invariance of Gaussian measures with respect to Hamiltonian PDE flows and allows us to establish the exact Jacobi formula for the density. We exploit this idea in two different contexts, namely the periodic fractional Benjamin–Bona–Mahony (BBM) equation with dispersion β > 1 and the periodic one-dimensional quintic defocussing nonlinear Schrödinger equation (NLS). For the BBM equation we study the transport of the cut-off Gaussian measures on fractional Sobolev spaces, while for the NLS equation we study the measures based on the modified energies introduced by Planchon–Visciglia and the third author. Moreover, for the BBM equation we also show almost sure global well-posedness for data in {C^alpha }(mathbb{T}) for arbitrarily small α > 0 and invariance of the Gaussian measure associated with the {H^{beta /2}}(mathbb{T}) norm.

Transmission of hyper-multi-level eigenvalue-modulated signal using arbitrary optical multi-eigenvalue

Optical eigenvalues of the eigenvalue equation associated with the nonlinear Schrödinger equation (NLSE) are invariant during propagation in nonlinear dispersive fiber. Optical eigenvalue communication is expected to be a technology for overcoming the Kerr nonlinear limit. Several modulation methods based on optical eigenvalue have been proposed. In this paper, we demonstrate an arbitrary optical eigenvalue modulation scheme by numerical simulations and experiments. Based on the proposed scheme, the generation and transmission of a 65 536-ary eigenvalue-modulated signal is demonstrated by numerical simulation. Moreover, we experimentally demonstrate modulation, transmission, and demodulation of a 16-ary eigenvalue-modulated signal. The 16-ary signal represented by combinations of four eigenvalues can be successfully transmitted through a 100-km standard single-mode fiber line.

Physical wave propagation and dynamics of the Ivancevic option pricing model

In this study, the Ivancevic option pricing model (IOPM), a nonlinear wave alternative to the traditional Black–Scholes option pricing model, is investigated. The Ivancevic option pricing model is formalized by adaptive nonlinear Schrodinger equations that characterize the option-pricing wave function in terms of stock price and time. This model exhibits the characteristic regulated Brownian behavior observed in financial markets. We use the unified auxiliary equation method, which is one of the most powerful methods to explore analytical solutions of the Ivancevic option pricing model. Via this method, we obtain several new exact solutions such as singular, periodic, bright, dark, hyperbolic, trigonometric, exponential and Jacobi elliptic functions solutions. In addition, we simulate 3D surface 2D graphs and counter plots graphics for some obtained solutions by giving specific values for the involved parameters. To further analyze the obtained solutions, we conduct simulations and generate visual representations in the form of 3D surface plots, 2D graphs, and counter plots. By assigning specific values to the parameters involved in the model, we are able to visualize and examine the dynamics of the solutions in a more tangible and intuitive manner. These graphical representations provide valuable insights into the behavior and patterns exhibited by the solutions, enhancing our understanding of the model’s implications and potential applications Furthermore, we present the option price wave functions of the dependent variable in 2D graphs under the effect of the time variable t.

Continuum limits for rational, breather solutions, and gauge equivalent model of the generalized discrete nonlinear Schrödinger equation

In this paper, we focus on an integrable higher‐order discrete nonlinear Schrödinger (dNLS) equation, which yields the Lakshmanan–Porsezian–Daniel (LPD) equation in the continuum limit. The breather solutions of the higher‐order dNLS are obtained for the first time. We show that the rational solutions and breather solutions of the higher‐order dNLS equation yield the counterparts of the integrable fourth‐order NLS equation under proper continuous limits. Most importantly, we also explore the gauge equivalent equation of the higher‐order dNLS equation from the point of view of the continuous limit theory. We make integrable discretization for the linear spectrum problem of the integrable generalized Heisenberg ferromagnetic equation that is just the gauge equivalent structure of the LPD equation. Compatibility condition of the discrete linear spectrum problem leads to a discrete integrable generalized Heisenberg Ferromagnetic equation. It is revealed that the discrete linear spectrum problem of the discrete integrable generalized ferromagnetic equation converges to the corresponding spectrum problem of the integrable generalized Feisenberg spin model in the continuum limit.In addition, with the help of the method of linear combinations for the conserved fluxes and densities, the conservation laws of the integrable higher‐order dNLS equation converge to the homologous results of the LPD equation.

Soliton solutions of fractional extended nonlinear Schrödinger equation arising in plasma physics and nonlinear optical fiber

In this research, we study traveling wave solutions to the fractional extended nonlinear SchrÖdinger equation (NLSE), and the effects of the third-order dispersion parameter. This equation is used to simulate the propagation of femtosecond, plasma physic and in nonlinear optical fiber. To accomplish this goal, we use the extended simple equation approach and the improved F-expansion method to secure a variety of distinct solutions in the form of dark, singular, periodic, rational, and exponential waves. Also, the stability of the outcomes is effectively examined. Several graphs have been sketched under appropriate parametric values to reinforce some reported findings. Computational work along with a graphical demonstration confirms the exactness of the proposed methods. The issue has not previously been investigated by taking into account the impact of the third order dispersion parameter. The main objective of this study is to obtain the different kinds of traveling wave solutions of fractional extended NLSE which are absent in the literature which justify the novelty of this study. We believe that these novel solutions hold a prominent place in the fields of nonlinear sciences and optical engineering because these solutions will enables a through understanding of the development and dynamic nature of such models. The obtained results indicate the reliability, efficiency, and capability of the implemented technique to determine wide-spectral stable traveling wave solutions to nonlinear equations emerging in various branches of scientific, technological, and engineering domains.

Localized wave solutions to the vector nonlinear Schrödinger equation with nonzero backgrounds

The vector nonlinear Schrödinger (NLS) system is studied through the generalized Darboux transformation (DT). We begin with the nonzero seed solution in the exponential form of evolution variable and explore the eigenfunction of the ‐component NLS equation. With the formulae of the eigenfunction, localized wave solutions of the two‐component and three‐component NLS equations are studied. By choosing different free parameters, some special solutions and the dynamic evolutions are presented, including breather–rogue wave, breather fission, breather fusion, and breather–soliton.

Propagation of the ordinary and extraordinary modulated optical pulses in a nonlinear Kerr-type birefringent optical waveguide: Analytical description

The purpose of this work is to describe theoretically the behavior of modulated pulses in nonlinear birefringent optical waveguides and particularly in the optical fibers, by taking into account the anisotropy of the medium as well as the orientation of the optical axis with respect to the propagation direction of an incoming pulse. Firstly, the multiple times scale method is used to describe the behavior of an optical pulse in the waveguide whose optical axis is perpendicular to the direction of propagation of the pulse. It appears that in the nonlinear regime, the optical pulse splits into two different modes in propagation namely an ordinary mode and an extraordinary mode which propagate with different group velocities. These modes are described by the different nonlinear Schrodinger (NLS) equations involving nonlinear Kerr-like responses. Next, we turn attention to the waveguide whose the optical axis is parallel to the direction of propagation of an incoming pulse and found that the two modes continue to exist, and are also described by different NLS equations. In addition, we show that for some parameter regime, the two modes become identical and described by the same NLS equation.