Generalized Nonlinear Schrodinger Equation Research Articles

On examining the predictive capabilities of two variants of the PINN in validating localized wave solutions in the generalized nonlinear Schrödinger equation

We introduce a novel neural network structure called strongly constrained theory-guided neural network (SCTgNN), to investigate the behaviour of the localized solutions of the generalized nonlinear Schrödinger (NLS) equation. This equation comprises four physically significant nonlinear evolution equations, namely, the NLS, Hirota, Lakshmanan–Porsezian–Daniel and fifth-order NLS equations. The generalized NLS equation demonstrates nonlinear effects up to quintic order, indicating rich and complex dynamics in various fields of physics. By combining concepts from the physics-informed neural network and theory-guided neural network (TgNN) models, the SCTgNN aims to enhance our understanding of complex phenomena, particularly within nonlinear systems that defy conventional patterns. To begin, we employ the TgNN method to predict the behaviour of localized waves, including solitons, rogue waves and breathers, within the generalized NLS equation. We then use the SCTgNN to predict the aforementioned localized solutions and calculate the mean square errors in both the SCTgNN and TgNN in predicting these three localized solutions. Our findings reveal that both models excel in understanding complex behaviour and provide predictions across a wide variety of situations.

Non-Kerr Law Medium: Optical Solitons and their Perturbation

This investigation focuses on the perturbation analysis of optical solitons within a medium in accordance with the non-Kerr law. In the field of partial differential equations, generalized nonlinear Schrodinger equation (GNLSE) is an integrable nonlinear equation. In the non-Kerr law non-linear medium, GNLSE is utilized for doing an analytical analysis of soliton perturbations as well as the soliton itself. In case of non-Kerr law instance, the application of quasi-stationarity results in a soliton that is very close to being approximated. Several edge scenarios of nonlinearity that vary from the Kerr law remain the primary focus of the current study. Although it was found that a disturbance of the nonlinear damping kind remains present, equations can be solved to find solutions. Consequently, GNLSE cannot be integrated because of the presence of higher-order dispersion.

Major Role of Multiscale Entropy Evolution in Complex Systems and Data Science.

Complex systems are prevalent in various disciplines encompassing the natural and social sciences, such as physics, biology, economics, and sociology. Leveraging data science techniques, particularly those rooted in artificial intelligence and machine learning, offers a promising avenue for comprehending the intricacies of complex systems without necessitating detailed knowledge of underlying dynamics. In this paper, we demonstrate that multiscale entropy (MSE) is pivotal in describing the steady state of complex systems. Introducing the multiscale entropy dynamics (MED) methodology, we provide a framework for dissecting system dynamics and uncovering the driving forces behind their evolution. Our investigation reveals that the MED methodology facilitates the expression of complex system dynamics through a Generalized Nonlinear Schrödinger Equation (GNSE) that thus demonstrates its potential applicability across diverse complex systems. By elucidating the entropic underpinnings of complexity, our study paves the way for a deeper understanding of dynamic phenomena. It offers insights into the behavior of complex systems across various domains.

Frequency downshifting in decaying wavetrains on the ocean surface covered by ice floes

We study analytically and numerically a frequency downshifting due to power-type frequency-dependent decay of surface waves in the ocean covered by ice floes. The downshifting is obtained both within the linear model and within the nonlinear Schrödinger (NLS) equation augmented by viscous terms for the initial condition in the form of an NLS envelope soliton. It is shown that the frequency-dependent dissipation produces a more substantial downshifting when the spectrum is relatively wide. As a result, the nonlinear adiabatic scenario of wavetrain evolution provides a downshifting remarkably smaller in magnitude than in the linear regime. Meanwhile, interactions between nonlinear wavegroups lead to spectral broadening and, thus, result in fast substantial frequency downshifts. Analytic estimates are obtained for an arbitrary power n of the dependence of a dissipation rate on frequency ∼ωn. The developed theory is validated by the numerical modeling of the generalized NLS equation with dissipative terms. Estimates of frequency downshift are given for oceanic waves of realistic parameters.

Butterfly-shaped and dromion-like optical similaritons in an asymmetric twin-core fiber amplifier

Butterfly-shaped and dromion-like optical waves in a tapered graded-index waveguide (GRIN) with an external source are reported for the first time, to our knowledge. More pertinently, we obtain these waves both analytically and numerically in a generalized nonlinear Schrödinger equation (GNLSE), which describes self-similar wave propagation in GRIN with variable group-velocity dispersion (GVD), nonlinearity, gain, and source. The proposed GNLSE appertains to the study of similariton propagation through asymmetric twin-core fiber amplifiers. Dromion-like structures, which have generally been investigated in the (2+1) or higher dimensional systems, are reported in the (1+1) dimensional GNLSE with an external source. Herein, we introduce the concept of soliton management when the variable group-velocity dispersion and Kerr nonlinearity functions are suggested. For example, when the GVD parameter is perturbed, we observe the emergence of vibration of dromion-like structures. Then the dromion-like structure is transformed into oscillation by the modulation instability of the modified coefficient of the Gaussian GVD function, exhibiting interference based on two dromion-like structures. Additionally, the phenomenon of unbreakable PT symmetry of these nonlinear waves has been demonstrated for three explicit examples.

Generalized and multi-oscillation solitons in the nonlinear Schrödinger equation with quartic dispersion.

We study different types of solitons of a generalized nonlinear Schrödinger equation (GNLSE) that models optical pulses traveling down an optical waveguide with quadratic as well as quartic dispersion. A traveling-wave ansatz transforms this partial differential equation into a fourth-order nonlinear ordinary differential equation (ODE) that is Hamiltonian and has two reversible symmetries. Homoclinic orbits of the ODE that connect the origin to itself represent solitons of the GNLSE, and this allows one to study the existence and organization of solitons with advanced numerical tools for the detection and continuation of connecting orbits. In this paper, we establish the existence of new types of connecting orbits, namely, PtoP connections from one periodic orbit to another. As we show, these global objects provide a general mechanism that generates additional families of two types of solitons in the GNLSE. First, we find generalized solitons with oscillating tails whose amplitude does not decay but reaches a nonzero limit. Second, PtoP connections in the zero energy level can be combined with EtoP connections from the origin to a selected periodic orbit to create multi-oscillation solitons; their characterizing property is to feature several episodes of different oscillations in between decaying tails. As is the case for solitons that were known previously, generalized solitons and multi-oscillation solitons are shown to be an integral part of the phenomenon of truncated homoclinic snaking.

Soliton solution, breather solution and rational wave solution for a generalized nonlinear Schrödinger equation with Darboux transformation

In this paper, the exact solutions of generalized nonlinear Schrödinger (GNLS) equation are obtained by using Darboux transformation(DT). We derive some expressions of the 1-solitons, 2-solitons and n-soliton solutions of the GNLS equation via constructing special Lax pairs. And we choose different seed solutions and solve the GNLS equation to obtain the soliton solutions, breather solutions and rational wave solutions. Based on these obtained solutions, we consider the elastic interactions and dynamics between two solitons.

A study of breather lump wave, rogue wave, periodic cross kink wave, multiwave, M-shaped rational and their interactions for generalized nonlinear Schrödinger equation

In this paper, we examine generalized nonlinear Schrödinger equation (GNLSE) for lump, breather lump wave, rogue wave, periodic cross kink wave, lump with one kink, lump with two kink, interaction between lump periodic and kink wave, periodic cross lump wave, periodic wave, and multiwave. Ansatz transformations will be used to study homoclinic breather, kink cross rational, M-shaped rational, periodic cross rational, M-shaped rational with one kink, M-shaped rational with two kink, M-shaped interaction with rogue and kink and M-shaped interaction with periodic and kink. We will also study the stability of the obtained solutions. Additionally, we present 3D, 2D, contour, and stream plot illustrations for our graphs.

A study of breather lump wave, rogue wave, periodic cross kink wave, multi-wave, M-shaped rational and their interactions for generalized nonlinear Schrödinger equation

In this paper, we examine generalized nonlinear Schrödinger equation (GNLSE) for lump, breather lump wave, rogue wave, periodic cross kink wave, lump with one kink, lump with two-kink, interaction between lump periodic and kink wave, periodic cross lump wave, periodic wave and multi-wave. Ansatz transformations will be used to study homoclinic breather, kink cross rational, M-shaped rational, periodic cross rational, M-shaped rational with one kink, M-shaped rational with two-kink, M-shaped interaction with rogue and kink and M-shaped interaction with periodic and kink. We will also study the stability of the obtained solutions. Additionally, we present three-dimensional, two-dimensional, contour, and stream plot illustrations for our graphs.

Hybrid accurate simulations for constructing some novel analytical and numerical solutions of three-order GNLS equation

This study presents analytical and numerical solutions of a simplified third-order generalized nonlinear Schrödinger equation (GNLSE) to demonstrate how ultrashort pulses behave in optical fiber and quantum fields. The investigated model can be used as a wave model to illustrate the wave aspect of the matter. It is called a quantum-mechanical state function because it might show how atoms and transistors move and act physically. Four analytical and numerical schemes are used to construct an accurate novel solution. Khater II (Kha II) and novel Kudryashov (NKud) methods are present in the employed analytical scheme. In contrast, the exponential cubic-B-spline and trigonometric-quantic-B-spline schemes represent the simulated numerical techniques. Many novel solitary wave solutions are constructed and formulated in some distinct forms and represented through density, three-, and two-dimensional graphs. The built analytical solutions accuracy is investigated by deriving the requested boundary and initial conditions for implementing the suggested numerical schemes that show the matching between both solutions (analytical and numerical). This matching between solutions proves the accuracy of the obtained solutions. Additionally, to guarantee the applicability of our solutions, we investigate their stability by using the Hamiltonian systems properties. Finally, the novelty of our study and its scientific contributions are illuminated by comparing our results with recently published ones.

On the dynamics of soliton waves in a generalized nonlinear Schrödinger equation

The main objective of the present paper is to investigate the dynamics of soliton waves in a generalized nonlinear Schrödinger (GNLS) equation. To achieve such a goal, the real and imaginary portions of the GNLS equation are first extracted using a complex wave transformation. Solitons and Jacobi elliptic structures of the governing model, describing the propagation of femtosecond pulses in nonlinear optical fibers, are then constructed through applying the modified Jacobi elliptic expansion method (MJEEM). In the end, by employing a series of 2 and 3D-dimensional numerical representations, it is exposed that the width of bright and dark solitons respectively decreases and increases, while the amplitude of both waves decreases with the increase of nonlinear parameters.

High coherent supercontinuum generation in nitrobenzene liquid-core photonic crystal fiber with elliptical air-hole inner ring

A nitrobenzene liquid-core photonic crystal fiber (NLC-PCF) with eight elliptical air-holes in the innermost ring of the cladding is proposed in this work, and this fiber structure can contribute to flexibly adjusting dispersion and constraining the fiber core. The appreciable dispersion is realized by adjusting the structural parameters of NLC-PCF, which is characterized by three zero-dispersion wavelengths (ZDWs), flat dispersion with the fluctuation of fewer than 40 ps nm−1 km−1, and high nonlinearity as high as 5500 W−1 km−1. The propagation of femtosecond pulse and supercontinuum generation (SCG) in NLC-PCF is studied numerically when the pump wavelength is located in the normal and abnormal dispersion region near different ZDW through solving the generalized nonlinear Schrödinger equation (GNLSE) by the split-step Fourier method. The numerical results show that a highly coherent supercontinuum (SC) spanning from 1.3 to 2.8 µm is obtained when the pump pulse with the center wavelength of 1810 nm, the peak power of 1000 W, and pulse width of 50 fs propagated in the 5 cm long NLC-PCF. This research can find applications in the fields of novel liquid-core PCF design and ultrashort pulse propagation.

Generation of High Power Ultrashort Pulses in Tapered Yb-Doped PCF Through Self-Similar Compression

We present a numerical approach for analysing simultaneous amplification and compression of a chirped hyperbolic secant soliton using a tapered Yb-doped photonic crystal fiber (PCF) based self-similar compressor. Theoretical studies reveal the conservation of input soliton during the amplification process due to self-similar property, which in turn helps for high quality compression of amplified soliton. In order to satisfy the self-similar condition, we note that the Yb-doped fiber should exhibit a decreasing group velocity dispersion (GVD). Such a constraint is met using a tapered PCF in our work, which is modeled using a fully vectorial effective index method (FVEIM). Influence of spatial and spectral dependent gain, saturation energy, amplified spontaneous emission (ASE), dispersion and nonlinearity on pulse propagation are analyzed using the combined solution of dynamic rate equations (DRE) and the generalized nonlinear Schrödinger equation (GNLSE). Numerical outcomes upon solving the aforementioned equations show that a 1 ps pulse is compressed down to a pulsewidth of 55.4 fs with a peak power of 1.03 kW in a total fiber length of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L=2.5$ </tex-math></inline-formula> m and taper ratio of 2.34. Overall quality of the compression scheme is evidenced by a relatively high compression factor (F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>c</i></sub> ) = 18.12, and low percentage of pedestal energy ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$PE$ </tex-math></inline-formula> ) = 10.95 %

Coupled-generalized nonlinear Schrödinger equations solved by adaptive step-size methods in interaction picture

We extend two adaptive step-size methods for solving two-dimensional or multi-dimensional generalized nonlinear Schrödinger equation (GNLSE): one is the conservation quantity error adaptive step-control method (RK4IP-CQE), and the other is the local error adaptive step-control method (RK4IP-LEM). The methods are developed in the vector form of fourth-order Runge–Kutta iterative scheme in the interaction picture by converting a vector equation in frequency domain. By simulating the supercontinuum generated from the high birefringence photonic crystal fiber, the calculation accuracies and the efficiencies of the two adaptive step-size methods are discussed. The simulation results show that the two methods have the same global average error, while RK4IP-LEM spends more time than RK4IP-CQE. The decrease of huge calculation time is due to the differences in the convergences of the relative photon number error and the approximated local error between these two adaptive step-size algorithms.

Nth-order smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger equation

In this paper, we investigate smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger (GNLS) equation which contains higher-order nonlinear effects. With the help of generalized Darboux transformation (GDT) method, we construct Nth-order smooth positon solutions of GNLS equation. We study the effect of higher-order nonlinear terms on these solutions. Our investigations show that the positon solutions are highly compressed by higher-order nonlinear effects. The direction of positons also get changed. We also derive Nth-order breather-positon (B-P) solution with the help of GDT. We show that these B-Ps are well compressed by the effect of higher-order nonlinear terms, but the period of B-P solution is not affected as in the breather solution case.

Feed-forward neural network as nonlinear dynamics integrator for supercontinuum generation.

The nonlinear propagation of ultrashort pulses in optical fibers depends sensitively on the input pulse and fiber parameters. As a result, the optimization of propagation for specific applications generally requires time-consuming simulations based on the sequential integration of the generalized nonlinear Schrödinger equation (GNLSE). Here, we train a feed-forward neural network to learn the differential propagation dynamics of the GNLSE, allowing emulation of direct numerical integration of fiber propagation, and particularly the highly complex case of supercontinuum generation. Comparison with a recurrent neural network shows that the feed-forward approach yields faster training and computation, and reduced memory requirements. The approach is generic and can be extended to other physical systems.

Dispersive waves and radiation trapping in optical fibers with a zero-nonlinearity wavelength

The formation of dispersive waves and radiation trapping are two most relevant phenomena in supercontinuum generation. In spite the fact that the nonlinear coefficient of many waveguides depends strongly on wavelength and even changes sign, the influence of a frequency-dependent nonlinearity on these phenomena have seldom been studied. The goal of this work is to fill this gap in the literature by thoroughly studying the behavior of dispersive waves and radiation trapping in the presence of a zero-nonlinearity wavelength (ZNW). Although the generalized nonlinear Schrödinger equation (GNLSE) is widely used to model optical pulse propagation in waveguides, it has been shown that the GNLSE may lead to unphysical results in the case of frequency-dependent nonlinearities. For this reason, we perform our analysis by resorting to the newly introduced photon-conserving generalized nonlinear Schrödinger equation (pcGNLSE). Further, we discuss results obtained with both equations and put in evidence the inadequacy of the GNLSE to appropriately account for radiation trapping.

Design of Broadband Flat Optical Frequency Comb Based on Cascaded Sign-Alternated Dispersion Tellurite Microstructure Fiber.

We designed a tellurite microstructure fiber (TMF) and proposed a broadband optical frequency comb generation scheme that was based on electro-optical modulation and cascaded sign-alternated dispersion TMF (CSAD-TMF). In addition, the influence of different nonlinear effects, the ultrashort pulse evolution in the CSAD-TMF with the anomalous dispersion (AD) zones and the normal dispersion (ND) zones were analyzed based on the generalized nonlinear Schrodinger equations (GNLSE) modelling. According to the simulations, when the input seed comb had a repetition rate of 20 GHz and had an input pulse peak power of 30 W, the generation scheme could generate optical frequency combs with a 6 dB spectral bandwidth spanning over 170 nm centered at 1550 nm. Furthermore, the generated combs showed good coherence in performance over the whole 6 dB spectral bandwidth. The highly coherent optical frequency combs can be used as high-repetition-rate, multi-wavelength light sources for various integrated microwave photonics and ultrafast optical signal processing applications.

Analytic combined bright-dark, bright and dark solitons solutions of generalized nonlinear Schrödinger equation using extended Sinh-Gordon equation expansion method

In this paper, we find analytic solutions of generalized nonlinear Schrödinger equation (GNLSE), the underlying equation for studying pulse propagation through optical waveguides, by implementing the extended Sinh-Gordon equation expansion method (ShGEEM). Here, the terms corresponding to second and third order dispersion, waveguide’s loss, Kerr Effect, self-steepening and Raman Effect are included in GNLSE. The obtained analytic solutions represent bright, dark and combined bright-dark solitons. The constraint conditions for the existence of valid solutions are given. Also, the new solutions are graphically shown for pulse propagation along a photonic crystal fiber.

Design of a novel star type photonic crystal fiber for mid-infrared supercontinuum generation

In this work, we have designed two simple photonic crystal fibers (PCFs) with the highly nonlinear chalcogenide material of Ge11.5As24Se64.5 for broadband mid-infrared supercontinuum generation. The hexagonal lattice of air holes in a star type arrangement has been considered in the designs, where only two different air hole sizes and a constant pitch are employed. The study shows that the diameter of the air hole has significant influence on both the dispersion and the nonlinearity parameters of the PCFs. The numerical solutions of the generalized nonlinear Schrödinger equation (GNLSE) for a secant hyperbolic input pulse at different pumping wavelengths at mid-infrared region are investigated by varying input pulse power and fiber lengths. About 3000 nm wide supercontinuum can be generated at pumping wavelength of 2400 nm and input pulse power of 3000W with the designs. Thus, the designs of the PCF can be employed for generating supercontinuum to be useful for medical imaging, bio-sensing, spectroscopy, wave division multiplexing (WDM) systems, and optical communication systems.