Computational and approximate solutions of complex nonlinear Fokas–Lenells equation arising in optical fiber
TL;DR: In this paper, the authors used the generalized Khater (GK) method and the trigonometric quintic B-spline (TQBS) scheme to study the calculations and approximate solutions of complex nonlinear Fokas-Lenells (FL) equations.
Abstract: This manuscript uses the generalized Khater (GK) method and the trigonometric quintic B-spline (TQBS) scheme to study the calculations and approximate solutions of complex nonlinear Fokas–Lenells (FL) equations. This model describes the propagation of short pulses in optical fibers. Many novel computing solutions have been obtained. The absolute, real, and imaginary values of some solutions are plotted in two three-dimensional and density graphs to explain the dynamic behavior of short pulses in the fiber. The use of constructed analytical solutions to evaluate initial and boundary conditions allows the application of numerical solutions to study the accuracy of our novel computational techniques. The performance of both methods demonstrates the ability, effectiveness, and ability to apply them to different forms of nonlinear evolution equations to check the accuracy of analytical and numerical solutions.
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TL;DR: In this article, the evolution dynamics of closed-form solutions for a new integrable nonlinear fifth-order equation with spatial and temporal dispersion which describes shallow water waves moving in two directions was investigated.
Abstract: This paper investigates the evolution dynamics of closed-form solutions for a new integrable nonlinear fifth-order equation with spatial and temporal dispersion which describes shallow water waves moving in two directions. Some novel computational soliton solutions are obtained in the form of exponential rational functions, trigonometric and hyperbolic functions, and complex-soliton solutions. Some dynamical wave structures of soliton solutions are achieved in evolutionary dynamical structures of multi-wave solitons, double-solitons, triple-solitons, multiple solitons, breather-type solitons, Lump-type solitons, singular solitons, and Kink-wave solitons using the generalized exponential rational function (GERF) technique. All newly established solutions are verified by back substituting into the considered fifth-order nonlinear evolution equation using computerized symbolic computational work via Wolfram Mathematica. These newly formed results demonstrate that the considered fifth-order equation theoretically possesses very rich computational wave structures of closed-form solutions, which are also useful in obtaining a better understanding of the internal mechanism of other complex nonlinear physical models arising in the field of plasma physics and nonlinear sciences. The physical characteristics of some constructed solutions are also graphically displayed via three-dimensional plots by selecting the best appropriate constant parameter values to easily understand the complex physical phenomena of the nonlinear equations. Eventually, the results validate the effectiveness and trustworthiness of the used technique.
42 citations
TL;DR: The trigonometric quintic B-spline scheme was used in this paper to solve the ZK nonlinear dimensional equation and explain the relati cation of the relatio cation.
Abstract: The trigonometric quintic B-spline scheme is used in this research paper to research Zakharov’s (ZK) nonlinear dimensional equation’s numerical solution. The ZK model’s solutions explain the relati...
28 citations
TL;DR: A variety of evolution equations have been developed from the Gilson-Pickering (GP) model, including the Fornberg-Whitham (FW) equation, the Rosenau-Hyman (RH) equation and the Fuchssteiner-Fokas-Camassa-Holm (FFCH) equation as discussed by the authors .
Abstract: In this study, we use cutting-edge analytical and numerical approaches to the Gilson–Pickering (GP) problem in order to get precise soliton solutions. This model explains wave propagation in plasma physics and crystal lattice theory. A variety of evolution equations have been developed from the GP model, including the Fornberg–Whitham (FW) equation, the Rosenau–Hyman (RH) equation, and the Fuchssteiner-Fokas-Camassa–Holm (FFCH) equation, to name a few. The GP model has been studied using these evolution equations. To investigate the characterizations of new waves, crystal lattice theory and plasma physics use the Khater II, and He’s variational iteration approaches. Many alternative responses may be achieved by utilizing various formulae; each of these solutions is shown by a distinct graph. The validity of such methods and solutions may be demonstrated by assessing how well the relevant techniques and solutions match up. The results of this study suggest that the technique is preferred for successfully resolving nonlinear equations that emerge in mathematical physics.
26 citations
TL;DR: In this paper , Cattaneo-Christov heat flux law exhibits hyperbolic equation which follows the causality principle and make the problem more compatible to real world applications.
Abstract: In view of increaing significance of non-isothermal flow of non-Newtonian fluids over exponential surfaces in numerous industrial and technological procedures such as film condensation, extrusion of plastic sheets, crystal growth, cooling process of metallic sheets, design of chemical processing equipment and various heat exchangers, and glass and polymer industries current disquisition is addressed. For comprehensive examination Williamson model expressing the attributes of shear thickening and thinning liquids is taken under consideration. The physical aspects of magnetic field applied in transverse direction to flow is also accounted. Heat transfer aspects are incorporated and analyzed by employing Cattaneo-Christov heat flux model. Mathematical formulation of problem is conceded in the form of PDE’s by implementing boudary layer approach and later on converted into ODE’s with the assistance of transformation procedure. The resulting equations are solved numerically using shooting and Runge–Kutta methods. Impact of involved parameters on flow distributions is displayed through graphs. From the analysis it is inferred that Cattaneo Christov heat flux law exhibits hyperbolic equation which follows the causality principle and make the problem more compatible to real world applications. It is also deduced that magnetic field suppresses the velocity field and associated boundary layer region. Decrease in temperature profile and heat transfer coefficient is found against inciting magnitude of thermal relaxation parameter. Substantial decrease in velocity is found against increasing magnitude of Williamson fluid parameter and magnetic field parameter whereas skin friction coefficient increments. Confirmation about present findings is executed by making comparison with existing literature.
22 citations
TL;DR: In this article , the analytical and numerical solutions' structure of the combined mKdV equation and KDV equation were analyzed using the Khater II (Khat. II) method and three accurate B-spline numerical schemes.
Abstract: This paper analyzes the analytical and numerical solutions’ structure of the combined mKdV equation and KdV equation (mKdV+KdV equation) using the Khater II (Khat. II) method and three accurate B-spline numerical schemes. ExCBS, SBS and TQBS numerical schemes are the numerical systems used. The handled model describes many distinct phenomena such as wave propagation of bounded particles with a harmonic force in a one-dimensional nonlinear lattice, propagation of ion-acoustic waves of small amplitude without Landau damping in plasma physics, and propagation of thermal pulse through a single sodium fluoride crystal in solid physics. Numerous examples show the relationship between quick and slow soliton, which generates phase shift. This phase shift is shown in a contour map to show the modest and colossal energy density along the path of fast and slow colliding solitons. Calculating the difference between analytical and numerical solutions shows whether they match spline-connected and distribution graphs.
22 citations
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Journal Article•
TL;DR: This work puts forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time by approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks.
Abstract: We put forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time. Specifically, we approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks. The first network acts as a prior on the unknown solution and essentially enables us to avoid numerical differentiations which are inherently ill-conditioned and unstable. The second network represents the nonlinear dynamics and helps us distill the mechanisms that govern the evolution of a given spatiotemporal data-set. We test the effectiveness of our approach for several benchmark problems spanning a number of scientific domains and demonstrate how the proposed framework can help us accurately learn the underlying dynamics and forecast future states of the system. In particular, we study the Burgers', Korteweg-de Vries (KdV), Kuramoto-Sivashinsky, nonlinear Schrodinger, and Navier-Stokes equations.
395 citations
TL;DR: The underlying physical principles of metasurface optical elements are introduced and, drawing on various works in the literature, how their constituent nanostructures can be designed with a highly customizable effective index of refraction that incorporates both phase and dispersion engineering are discussed.
Abstract: Control over the dispersion of the refractive index is essential to the performance of most modern optical systems. These range from laboratory microscopes to optical fibres and even consumer products, such as photography cameras. Conventional methods of engineering optical dispersion are based on altering material composition, but this process is time-consuming and difficult, and the resulting optical performance is often limited to a certain bandwidth. Recent advances in nanofabrication have led to high-quality metasurfaces with the potential to perform at a level comparable to their state-of-the-art refractive counterparts. In this Review, we introduce the underlying physical principles of metasurface optical elements (with a focus on metalenses) and, drawing on various works in the literature, discuss how their constituent nanostructures can be designed with a highly customizable effective index of refraction that incorporates both phase and dispersion engineering. These metasurfaces can serve as an essential component for achromatic optics with unprecedented levels of performance across a broad bandwidth or provide highly customized, engineered chromatic behaviour in instruments such as miniature aberration-corrected spectrometers. We identify some key areas in which these achromatic or dispersion-engineered metasurface optical elements could be useful and highlight some future challenges, as well as promising ways to overcome them. Flat metasurface optics provides an emerging platform for combining semiconductor foundry methods of manufacturing and assembling with nanophotonics to produce high-end and multifunctional optical elements. This Review highlights the design of metasurfaces, recent advances in the field and initial promising applications.
366 citations
TL;DR: This paper aims to elevate the notion of joint harmonic analysis to a full-fledged framework denoted as time-vertex signal processing, that links together the time-domain signal processing techniques with the new tools of graph signal processing.
Abstract: An emerging way to deal with high-dimensional noneuclidean data is to assume that the underlying structure can be captured by a graph. Recently, ideas have begun to emerge related to the analysis of time-varying graph signals. This paper aims to elevate the notion of joint harmonic analysis to a full-fledged framework denoted as time-vertex signal processing, that links together the time-domain signal processing techniques with the new tools of graph signal processing. This entails three main contributions: a) We provide a formal motivation for harmonic time-vertex analysis as an analysis tool for the state evolution of simple partial differential equations on graphs; b) we improve the accuracy of joint filtering operators by up-to two orders of magnitude; c) using our joint filters, we construct time-vertex dictionaries analyzing the different scales and the local time-frequency content of a signal. The utility of our tools is illustrated in numerous applications and datasets, such as dynamic mesh denoising and classification, still-video inpainting, and source localization in seismic events. Our results suggest that joint analysis of time-vertex signals can bring benefits to regression and learning.
161 citations
TL;DR: In this article, the existence and properties of rogue-wave solutions in different nonlinear wave evolution models that are commonly used in optics and hydrodynamics were studied, including the Fokas-Lenells equation, the defocusing vector nonlinear Schrodinger equation, and the long-wave--short-wave resonance equation.
Abstract: We study the existence and properties of rogue-wave solutions in different nonlinear wave evolution models that are commonly used in optics and hydrodynamics. In particular, we consider the Fokas-Lenells equation, the defocusing vector nonlinear Schr\"odinger equation, and the long-wave--short-wave resonance equation. We show that rogue-wave solutions in all of these models exist in the subset of parameters where modulation instability is present if and only if the unstable sideband spectrum also contains cw or zero-frequency perturbations as a limiting case (baseband instability). We numerically confirm that rogue waves may only be excited from a weakly perturbed cw whenever the baseband instability is present. Conversely, modulation instability leads to nonlinear periodic oscillations.
130 citations
Proceedings Article•
21 Jun 2015TL;DR: Extreme wave events, also referred to as rogue waves, are mostly known as oceanic phenomena responsible for a large number of maritime disasters as mentioned in this paper, have height and steepness much greater than expected from the sea average state, and not only appear in oceans, but also in the atmosphere, in optics, in plasmas, in superfluids, in Bose-Einstein condensates and as capillary waves.
Abstract: Extreme wave events, also referred to as rogue waves, are mostly known as oceanic phenomena responsible for a large number of maritime disasters. These waves have height and steepness much greater than expected from the sea average state [1]: not only appear in oceans, but also in the atmosphere, in optics, in plasmas, in superfluids, in Bose-Einstein condensates and as capillary waves. The common features and differences among freak wave manifestations in their different contexts is a subject of intense discussion [2].
112 citations