Nonlinear Propagation Research Articles

DISSIPATION AND DISPERSION IN FRACTIONAL NONLINEAR WAVE DYNAMICS: ANALYTICAL AND NUMERICAL EXPLORATIONS

The fractional nonlinear generalized [Formula: see text]-dimensional Chaffee–Infante (FGCI) equation is a nonlinear evolution equation that governs the propagation of waves in dispersive media, exhibiting both dissipative and dispersive effects. It finds applications in various fields such as fluid mechanics, plasma physics, and nonlinear optics, where the interplay between nonlinearity, dispersion, and dissipation plays a crucial role in determining the dynamics of the system. This study aims to obtain analytical and numerical solutions for the FGCI equation, which is a generalization of the Chaffee–Infante equation. The analytical approach employed is the Khater II (Khat II) method, an efficient technique for constructing approximate analytical solutions to nonlinear fractional differential equations. Additionally, He’s variational iteration (HVI) method is implemented as a numerical scheme to validate the accuracy of the analytical solutions obtained from the Khat II method. The results obtained through these methods demonstrate the effectiveness of the proposed techniques in solving the FGCI equation and provide valuable insights into the dynamics of wave propagation phenomena governed by the equation. The significance of this study lies in its contribution to the understanding of nonlinear wave propagation phenomena and the development of analytical and numerical techniques for solving fractional differential equations. The novelty of the research lies in the application of the Khat II method to the FGCI equation, which has not been previously explored, and the validation of the analytical solutions through the HVI method. The study is conducted in the field of applied mathematics, with a focus on nonlinear fractional differential equations and their applications in various physical systems.

Dynamics of pulse propagation with solitary waves in monomode optical fibers with nonlinear Fokas system

In this study, a unified auxiliary equation method, which is one of the powerful methods for exploring nonlinear model solutions, is used in the Fokas system, with complex functions representing nonlinear pulse propagation in monomode optical fibers. As a result, we get some solutions, including dark–bright, singular, periodic, bright–dark, Jacobi elliptic functions, trigonometric, hyperbolic and exponential ones. In addition, we use a computer program to generate 3D, 2D and counterplot graphics from the obtained solutions by assigning specific values to the involved parameters. While discussing, the graphs for various values of an arbitrary constant are examined. These findings constitute an important step in understanding how solitary waves are generated in nonlinear media. Since the studied model is used in many domains, including Bose–Einstein condensates and plasma physics, these results improve our theoretical knowledge and open up new avenues for potential real-world applications and the development of cutting-edge technologies.

An iteration method to study nonlinear wave propagation for a non-Green elastic 1D bar

Abstract The problem of propagation of nonlinear waves in a 1D bar is studied, wherein the linearized strain tensor is considered as a function of the Cauchy stress tensor. Specifically, two constitutive equations for non-Green elastic solids are investigated, introducing a novel numerical iterative method capable of obtaining approximate solutions of one nonlinear constitutive equation for rock, and one constitutive equation that shows a strain-limiting behaviour. The numerical results are compared with exact solutions for the case of a linearized elastic solid.

Laser-accelerated MeV-scale collimated electron bunch from a near-critical plasma of a liquid jet target

Generation of a collimated electron bunch with energy of a few MeV is demonstrated experimentally during propagation of 1 TW 10 Hz femtosecond laser radiation through a near-critical plasma formed from a micrometer-scale liquid jet (ethanol) target by ablation and boring with an intense nanosecond pulse. Hydrodynamic and particle-in-cell simulations reveal the evolution of the plasma cloud and help to identify the acceleration mechanism, which is related to self-modulated laser Wakefield acceleration during nonlinear propagation of laser radiation through plasma. The measured bunch divergence is at the level of 0.04 rad with high shot-to-shot stability. The total charge of the particles with energy above 1.6 MeV was estimated at ∼15 pC. The simplicity and robustness of the target design allows for enhanced pulse repetition rate with suppressed debris formation.

A hybrid variational method for beam propagation and interaction in a graded-index nonlinear waveguide

The hybrid variational method can be used to simplify multidimensional numerical simulations. We examine the applicability of this method to study the nonlinear propagation of a single beam and the interference of two spatial soliton beams in a graded-index optical waveguide governed by a two-dimensional nonlinear Schrodinger equation. We classified three distinct regimes of the dynamics that emerge during single Gaussian beam propagation, corresponding to beam decay, breather formation, and self-focusing. When two spatial soliton beams were set up to interfere in the waveguide, we identified the boundary where the nonlinear interaction during the interference led to excitation or self-focusing. These quite involved dynamics were used to test the predictive power of our variational models by comparing them with direct numerical simulations, obtaining good agreement.

Improving PID Controller Performance in Nonlinear Oscillatory Automatic Generation Control Systems Using a Multi-objective Marine Predator Algorithm with Enhanced Diversity

Power systems are pivotal in providing sustainable energy across various sectors. However, optimizing their performance to meet modern demands remains a significant challenge. This paper introduces an innovative strategy to improve the optimization of PID controllers within nonlinear oscillatory Automatic Generation Control (AGC) systems, essential for the stability of power systems. Our approach aims to reduce the integrated time squared error, the integrated time absolute error, and the rate of change in deviation, facilitating faster convergence, diminished overshoot, and decreased oscillations. By incorporating the spiral model from the Whale Optimization Algorithm (WOA) into the Multi-Objective Marine Predator Algorithm (MOMPA), our method effectively broadens the diversity of solution sets and finely tunes the balance between exploration and exploitation strategies. Furthermore, the QQSMOMPA framework integrates quasi-oppositional learning and Q-learning to overcome local optima, thereby generating optimal Pareto solutions. When applied to nonlinear AGC systems featuring governor dead zones, the PID controllers optimized by QQSMOMPA not only achieve 14%\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\%$$\\end{document} reduction in the frequency settling time but also exhibit robustness against uncertainties in load disturbance inputs.

Traveling wave solution and qualitative behavior of fractional stochastic Kraenkel–Manna–Merle equation in ferromagnetic materials

The main purpose of this article is to investigate the qualitative behavior and traveling wave solutions of the fractional stochastic Kraenkel–Manna–Merle equations, which is commonly used to simulate the zero conductivity nonlinear propagation behavior of short waves in saturated ferromagnetic materials. Firstly, fractional stochastic Kraenkel–Manna–Merle equations are transformed into ordinary differential equations by using the traveling wave transformation. Secondly, the phase portraits, sensitivity analysis, and Poincaré sections of the two-dimensional dynamic system and its perturbation system of ordinary differential equations are drawn. Finally, the traveling wave solutions of fractional stochastic Kraenkel–Manna–Merle equations are obtained based on the analysis theory of planar dynamical system. Moreover, the obtained three-dimensional graphs of random solutions, two-dimensional graphs of random solutions, and three-dimensional graphs of deterministic solutions are drawn.

On the maximization of entropy in the process of thermalization of highly multimode nonlinear beams.

We present a direct experimental confirmation of the maximization of entropy which accompanies the thermalization of a highly multimode light beam, upon its nonlinear propagation in standard graded-index (GRIN) optical fibers.

Shape unrestricted topological corner state based on Kekulé modulation and enhanced nonlinear harmonic generation

Abstract Topological corner states have been extensively utilized as a nanocavity to increase nonlinear harmonic generation due to their high Q-factor and robustness. However, the previous topological corner states based nanocavities and nonlinear harmonic generation have to comply with particular spatial symmetries of underlying lattices, hindering their practical application. In this work, we design a photonic nanocavity based on shape unrestricted topological corner state by applying Kekulé modulation to a honeycomb photonic crystal. The boundaries of such shape unrestricted topological corner state are liberated from running along specific lattice directions, thus topological corner states with arbitrary shapes and high Q-factor are excited. We demonstrate enhancement of second (SHG) and third harmonic generation (THG) from the topological corner states, which are also not influenced by the geometry shape of corner. The liberation from the shape restriction of corner state and nonlinear harmonic generation are robust to lattice defects. We believe that the shape unrestricted topological corner state may also find a way to improve other nonlinear optical progress, providing great flexibility for the development of photonic integrated devices.

Tunable third harmonic generation based on high-Q polarization-controlled hybrid phase-change metasurface

Abstract Dielectric metasurfaces have made significant advancements in the past decade for enhancing light–matter interaction at the nanoscale. Particularly, bound states in the continuum (BICs) based on dielectric metasurfaces have been employed to enhance nonlinear harmonic generation. However, conventional nonlinear metasurfaces are typically fixed in their operating wavelength after fabrication. In this work, we numerically demonstrate tunable third harmonic generation (THG) by integrating a dielectric metasurface with the phase-change material Ge2Sb2Te5 (GST). The hybrid phase-change metasurface can support two BICs with different electromagnetic origins, which are transformed into two high-Q quasi-BICs through the introduction of structural asymmetry. The two quasi-BICs are selectively excited by controlling the polarization of incident light, and their wavelengths are tunable due to the phase transition of GST. Notably, the efficiency of THG is significantly enhanced at the fundamental wavelengths corresponding to the two quasi-BICs, and the operating wavelength for THG enhancement can be dynamically tuned through the GST phase transition. Furthermore, the wavelength of THG enhancement can be further tuned by manipulating the polarization of pump light. Additionally, a high-Q analog of electromagnetically induced transparency is numerically achieved through the interaction between a low-Q Mie resonance and a quasi-BIC mode, which also improves the THG efficiency. The high-Q polarization-controlled hybrid phase-change metasurface holds promise for applications in dynamically tunable nonlinear optical devices.

Noncontact hardness assessment of carbon steel with nonlinear Rayleigh waves induced by grating laser

Hardness is a crucial mechanical property to characterize the metal performance in service, and its nondestructive assessment is undoubtedly a challenging work. The objective of this paper is to realize a noncontact assessment of the heat-treated medium carbon steel by using grating laser-induced nonlinear Rayleigh waves. The generation and suppression mechanism of the instrument nonlinearity introduced by a grating laser source is explained theoretically. A different nonlinear parameter is adopted and its relevant feasibility is discussed. Ten heat-treated 45 steel samples underwent NDE using grating laser ultrasonics and the destructive testing using a Rockwell hardness tester in a successive manner. A logarithmic function is utilized to fit the correlation between the Rockwell A hardness and nonlinear parameter with a goodness of fit of 0.9102. The reasonableness of experimental fitting outcomes is discussed in the following, which confirms the potential application prospect.

Physics-informed neural networks for fully non-linear free surface wave propagation

This study proposed fully nonlinear free surface physics-informed neural networks (FNFS-PINNs), an advancement within the framework of PINNs, to tackle wave propagation in fully nonlinear potential flows with the free surface. Utilizing the nonlinear fitting capabilities of neural networks, FNFS-PINNs offer an approach to addressing the complexities of modeling nonlinear free surface flows, broadening the scope for applying PINNs to various wave propagation scenarios. The improved quasi-σ coordinate transformation and dimensionless formulation of the basic equations are adopted to transform the time-dependent computational domain into the stationary one and align variable scale changes across different dimensions, respectively. These innovations, alongside a specialized network structure and a two-stage optimization process, enhance the mathematical formulation of nonlinear water waves and solvability of the model. FNFS-PINNs are evaluated through three scenarios: solitary wave propagation featuring nonlinearity, regular wave propagation under high dispersion, and an inverse problem of nonlinear free surface flow focusing on the back-calculation of an initial state from its later state. These tests demonstrate the capability of FNFS-PINNs to compute the propagation of solitary and regular waves in the vertical two-dimensional scenarios. While focusing on two-dimensional wave propagation, this study lays the groundwork for extending FNFS-PINNs to other free surface flows and highlights their potential in solving inverse problems.

Robust LFC design using adaptive neuro‐fuzzy inference‐aided optimal fractional‐order PIDA control for perturbed power systems with solar and wind power sources

AbstractMaintaining stability in modern power systems is challenging due to complex structures, rising power demand, and load disturbances. The integration of renewable energy sources further threatens stability by causing imbalances between generation and demand. Conventional load frequency stabilization methods fall short in such scenarios. This paper proposes an optimal fractional‐order proportional‐integral‐derivative‐acceleration (FOPIDA) controller, enhanced by a robust adaptive neuro‐fuzzy inference system (ANFIS), to improve load frequency control and reliability in power systems with wind and solar generators. First, the dynamical model of a multi‐area interconnected power system, including a thermal power plant, wind turbine, and solar photovoltaic generators, is developed. A decentralized ANFIS‐FOPIDA controller is then designed for load frequency control objectives. The gains of this controller are optimized using the whale optimization algorithm (WOA), focusing on frequency deviation and tie‐line power exchange. Simulations on a New England IEEE 10‐generator 39‐bus power system demonstrate the approach's effectiveness under various disturbances, including random load‐generation disturbances and nonlinear generation behaviors. Comparisons with other strategies, such as fractional order (FO) beetle swarm optimization algorithm (FOBSOA)‐FOPIDA, WOA‐PIDA, and WOA‐ANFIS‐PIDA, and recent control approaches highlight the superior performance of the WOA‐ANFIS‐FOPIDA method in enhancing power system stability.

Local plasticity-induced nonlinear surface wave scattering in thick-walled structures

Thick-walled structures (TWSs) are frequently utilized as primary load-bearing structures. The early detection of damage is crucial to ensure the operational safety and integrity of these structures. Nonlinear elastic-wave-based techniques provide a potential solution to the problem, which heavily relies on the good understanding of nonlinear wave generation and propagation characteristics. The task, however, is technically challenging due to the complex wave modes existing in a TWS. Targeting a meticulously manufactured local plasticized damage region on the surface of a TWS, which can be regarded as the precursor of incipient damage, this study investigates the scattering features of nonlinear elastic waves resulting from the interaction between the fundamental quasi-surface wave and the local plasticity. Finite element simulations and experiments show that, upon interacting with the nonlinear material zone, both nonlinear quasi-surface waves and shear bulk waves can be generated, with the latter scattered towards the opposite surface of the TWS. The scattered nonlinear shear waves exhibit a strong and predictable directivity with high nonlinear energy content, which is conducive to the detection and localization of the incipient surface damage of the TWS.

Nonlinear propagation of chirped laser pulses through a dispersive and turbulent atmosphere

The evolution of ultrashort laser pulses in dispersive, turbulent, nonlinear, and dissipative media is discussed in connection with nonlinear self-focusing collapse and the onset of laser filamentation. In quiescent air, a laser pulse propagating with a peak power greater than a critical power for self-focusing will undergo a catastrophic, transverse collapse until the intensity is large enough for photoionization. At this point, self-focusing is arrested and balanced by plasma refraction, forming a laser filament. By applying an appropriate chirp, the dispersive properties of the medium can be used to enhance this process and control its onset, and to counter dissipative effects such as molecular absorption and atmospheric scattering. This paper presents an analysis of the effect of atmospheric turbulence on the propagation of nonlinear pulses with dispersion compensation (chirp). The analytical results are compared with wave optics simulations and found to be in reasonable agreement as long as the pulse maintains a near-Gaussian spatiotemporal profile.

Extended finite element method with cell-based smoothing for modeling frictional contact crack-induced acoustic nonlinearity involving distorted mesh

This study proposes a 2D cell-based smoothed extended finite element method (CS-XFEM) for accurate and efficient simulation of nonlinear ultrasonic wave propagation in solid structures, specifically addressing the effects of frictional contact in cracks. Traditional mesh discretization methods for cracks often suffer from mesh distortion and computational instability owing to their high aspect ratios. To overcome this, CS-XFEM integrates a cell-based smoothing technique into XFEM to model the frictional contact of a crack. A comprehensive numerical example demonstrates the advantages of CS-XFEM. The results show that CS-XFEM exhibits a higher convergence rate and enables a larger critical time increment than XFEM. Specifically, the critical time increment of CS-XFEM was found to be twice that of XFEM, leading to a 50% reduction in the total computational time. These findings confirm that CS-XFEM is an efficient, accurate, and robust numerical method for studying the acoustic nonlinearity induced by crack-induced frictional contact.

Optimized technique and dynamical behaviors of fractional Lax and Caudrey–Dodd–Gibbon models modelized by the Caputo fractional derivative

The presented paper aims to investigate, examine, and analyze the nonlinear time-fractional evolution partial differential equations (TFNE-PDEs) in the sense of Caputo essential in numerous nonlinear wave propagation phenomena. To achieve this, the Laplace-residual power series method (L-RPSM) is used to construct approximate Laplace-residual power series solutions (AL-RPSSs) for fifth-order Korteweg–de Vries PDEs (KdV-PDEs). The L-RPSM provides analytical solutions of the dynamic wavefunction of several equations including time-fractional Lax PDE (TFL-PDE), and time-fractional Caudrey–Dodd–Gibbon PDEs (TFCDG-PDEs). The theoretical and numerical consequences of the models are discussed, and error’s analysis of the method are discussed. The outcomes are introduced in 2D and 3D figures, and dynamic behaviors of parameters are discussed for distinct values of α. Moreover, the accuracy of this method is demonstrated by comparing it with results obtained using other methods. The results show that the suggested iterative approach is a suitable tool with computational efficiency for long-wavelength solutions of nonlinear time-fractional PDEs in various phenomena.

Large‐Scale Bottom‐Up Fabricated 3D Nonlinear Photonic Crystals

Nonlinear optical effects are used to generate coherent light at wavelengths difficult to reach with lasers. Materials periodically poled or nanostructured in the nonlinear susceptibility in three spatial directions are called 3D nonlinear photonic crystals (NPhCs). They enable enhanced nonlinear optical conversion efficiencies, emission control, and simultaneous generation of nonlinear wavelengths. The chemical inertness of efficient second‐order nonlinear materials () prohibits their nanofabrication until 2018. The current methods are restricted to top‐down laser‐based techniques limiting the periodicity along the z‐axis to . The first bottom‐up fabricated 3D NPhC is demonstrated in sol–gel‐derived barium titanate by soft‐nanoimprint lithography: a woodpile with eight layers and periodicities of (‐plane) and (z‐plane). The surface areas exceed , which is two orders of magnitude larger than the state‐of‐the‐art. This study is expected to initiate bottom‐up fabrication of 3D NPhCs with a supremely strong and versatile nonlinear response.

Influences of damping, perturbation and variable coefficient on an extended nonlinear Gardner model

Describing the nonlinear wave propagation in fluids has posed a challenging task. The nonlinear Gardner equation stands as a crucial model in fluid dynamics. An extended Gardner model incorporates variable coefficients, damping, and perturbation terms, offering a more comprehensive approach to understanding complex wave dynamics in fluids. In our study, we employ the symbolic computation tool, specifically the Darboux transformation, to derive new analytic solutions up to the Nth order for this extended Gardner equation. By assigning the spectrum parameters in these solutions into real values, we are able to obtain line-like solitons. Moreover, when N⩾2 and pairs of complex conjugate values are assigned to two of the spectrum parameters, breathers emerge. This enables a comprehensive analysis of the dynamics involving solitons, breathers, and their hybrid forms, shedding light on their interactions. Furthermore, our work introduces several noteworthy insights by investigating the influences of three key factors: the damping, perturbation, and variable coefficient. Through a few of specific examples, we can confirm the following facts: (i) The damping factor plays a crucial role in rendering solitons and breathers unstable, causing them to undergo exponential changes in both amplitudes and directions during propagation. (ii) The perturbation function creates non-zero background waves along the time axis, which interact with solitons and breathers. (iii) The variable coefficient can introduce disturbances to solitons and breathers. Our findings disclose that this equation exhibits a wealth of intricate propagation patterns. Various factors can exert varying degrees of influence on solitons and their dynamics. This observation just underscores the complexity inherent in the real world and should be useful in applications.

Bifurcation analysis and soliton solutions to the doubly dispersive equation in elastic inhomogeneous Murnaghan's rod.

The doubly dispersive (DD) equation finds extensive utility across scientific and engineering domains. It stands as a significant nonlinear physical model elucidating nonlinear wave propagation within the elastic inhomogeneous Murnaghan's rod (EIMR). With this in mind, we have focused on the integration of the DD model and the modified Khater (MK) method. Through the wave transformation, this model is effectively converted into an ordinary differential equation. In this paper, the goal of our work is to explore new wave solutions to the DD model by using the MK scheme. These solutions provide extremely helpful insights into the operation of the system. The three-dimensional (3D) plot and two-dimensional (2D) combined plot via the impacts of the parameters are provided for various parameters in this manuscript. We also discussed the dynamical properties of the model, which are accomplished through the bifurcation analysis, and also found the Hamiltonian function. This research makes a substantial contribution to the area by increasing our understanding of wave solutions in the DD, introducing novel investigation tools, and carrying out an in-depth investigation of the bifurcation and stability aspects of the system. As a direct result of this research, novel openings have been uncovered for further investigation and application in the various disciplines of science and engineering.