Nonlinear Wave Propagation Research Articles

On the Large-x Asymptotic of the Classical Solutions to the Non-Linear Benjamin Equation in Fractional Sobolev Spaces

In this work, we study the large-x asymptotic of classical solutions to the non-linear Benjamin equation modeling propagation of small amplitude internal waves in a two fluid system. In our analysis, we extend known Hs-well-posedness results to the case of the variable-weight Sobolev spaces. The spaces provide a direct control over the asymptotics of classical solutions and their weak derivatives, and permit us to compute the bulk large-x asymptotic of classical solutions explicitly in terms of input data. The asymptotic formula provides a precise description of the qualitative behaviour of classical solutions in weighted spaces and yields a number of weighted persistence and continuation results automatically.

Dynamic of bifurcation, chaotic structure and multi soliton of fractional nonlinear Schrödinger equation arise in plasma physics

In this study, we examine the third-order fractional nonlinear Schrödinger equation (FNLSE) in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+1)$$\\end{document}-dimensional, by employing the analytical methodology of the new extended direct algebraic method (NEDAM) alongside optical soliton solutions. In order to better understand high-order nonlinear wave behaviors in such systems, the researched model captures the physical and mathematical properties of nonlinear dispersive waves, with applications in plasma physics and optics. With the aid of above mentioned approach, we rigorously assess the novel optical soliton solutions in the form of dark, bright–dark, dark–bright, periodic, singular, rational, mixed trigonometric and hyperbolic forms. Additionally, stability assessments using conserved quantities, such as Hamiltonian property, and consistency checks were used to validate the solutions. The dynamic structure of the governing model is further examined using chaos, bifurcation, and sensitivity analysis. With the appropriate parameter values, 2D, 3D, and contour plots can all be utilized to graphically show the data. This work advances our knowledge of nonlinear wave propagation in Bose–Einstein condensates, ultrafast fibre optics, and plasma physics, among other areas with higher-order chromatic effects.

Simulating overtaking interactions of dust-acoustic N-shock waves in oppositely charged dusty plasmas with Q-distributed electrons

Simulating overtaking interactions (OIs) of dust-acoustic N-shock waves (DANSWs) in dust plasmas with opposite polarities and q-distributed electrons and Boltzmann distributed ions are examined in this work. Reductive perturbation theory is used to obtain a Burgers’ equation (BE), and dust-acoustic N-shock wave solutions are derived using the Cole-Hopf transformation and exponential function. Numerical simulations show that the evolution and OIs of DANSWs and their electric fields are modified by the effects of negative/positive dust fluid kinematic viscosity, the ratio of negative dust to positive dust mass, and the nonextensive q-distribution of electrons. This work may help explain the properties of dust-acoustic shock waves in cometary tails and Jupiter's magnetosphere that support the propagation of nonlinear waves.

Derivation of Sawada-Kotera and Kaup-Kupershmidt Equations KdV Flow Equations from Modified Nonlinear Schrödinger Equation (MNLS)ns from Modfied Nonlinear Schrödinger Equation (MNLS)

Mathematical models of problems that arise in almost every branch of science are nonlinear evolution equations (NLEE). As a result, nonlinear evolution equations have served as a language for formulating many engineering and scientific problems. For this reason, many different and effective techniques have been developed regarding nonlinear evolution equations and solution methods. The main reason for this situation is that nonlinear evolution equations involve the problem of nonlinear wave propagation. In this study, (1 + 1) dimensional fifth-order nonlinear Korteweg-de Vries (fKdV) type equations were obtained by applying the multi-scale method known as the perturbation method for the modified nonlinear Schrödinger (MNLS) equation. Thus, we showed the relationship between KdV equations and MNLS type equations.

Solitonic Analysis of the Newly Introduced Three-Dimensional Nonlinear Dynamical Equations in Fluid Mediums

The emergence of higher-dimensional evolution equations in dissimilar scientific arenas has been on the rise recently with a vast concentration in optical fiber communications, shallow water waves, plasma physics, and fluid dynamics. Therefore, the present study deploys certain improved analytical methods to perform a solitonic analysis of the newly introduced three-dimensional nonlinear dynamical equations (all within the current year, 2024), which comprise the new (3 + 1) Kairat-II nonlinear equation, the latest (3 + 1) Kairat-X nonlinear equation, the new (3 + 1) Boussinesq type nonlinear equation, and the new (3 + 1) generalized nonlinear Korteweg–de Vries equation. Certainly, a solitonic analysis, or rather, the admittance of diverse solitonic solutions by these new models of interest, will greatly augment the findings at hand, which mainly deliberate on the satisfaction of the Painleve integrability property and the existence of solitonic structures using the classical Hirota method. Lastly, this study is relevant to contemporary research in many nonlinear scientific fields, like hyper-elasticity, material science, optical fibers, optics, and propagation of waves in nonlinear media, thereby unearthing several concealed features.

Nonlinear Ion Acoustic Waves in Multicomponent Plasmas with Nonthermal Electrons-positron and Bipolar Ions

Abstract This paper studied the propagating characteristics of (2+1) dimensional nonlinear ion acoustic waves in a multicomponent plasma with nonthermal electrons, positrons and bipolar ions. The dispersion relations are initially explored by using the small amplitude wave's dispersion relation. Then, the Sagdeev potential method is employed to study large amplitude ion acoustic waves. The analysis involves examining the system's phase diagram, Sagdeev potential function, and solitary wave solutions through numerical solution of an analytical process in order to investigate the propagation properties of nonlinear ion acoustic waves under various parameters. It is found that the propagation of nonlinear ion acoustic waves is subject to the influence of various physical parameters, including the ratio of number densities between the unperturbed positrons, electrons to positive ions, nonthermal parameters, the masses ratio of positive ions to negative ions, and the charge numbers ratio of negative ions to positive ions, the ratio of the electrons’ temperature to positrons’ temperature. In addition, the multicomponent plasma system has a compressive solitary waves with amplitude greater than zero or a rarefactive solitary waves with amplitude less than zero, in the meantime, compressive and rarefactive ion acoustic wave characteristics depend on the charge numbers ratio of negative ions to positive ions.

Exploring the interplay of dispersion, self-steepening, and self-frequency shift in nonlinear wave propagation

Exploring the interplay of dispersion, self-steepening, and self-frequency shift in nonlinear wave propagation

On acoustic solitary waves in a multispecies degenerate relativistic magnetized plasma using physics informed neural networks

In this paper, we investigate the nonlinear electrostatic wave propagation in a two-dimensional magnetized plasma. The plasma consists of electron and positron components with relativistic degeneracy and stationary ions for neutralizing its background. Using the basic equations for this type of plasma in combination with the reductive perturbation method, we derived the Zakharov–Kuznetsov equation using the Lorentz transformation stretching method (LT). For the first time, we compared the results of the Galilean transformation stretching method (GT) and the LT method to investigate the effect of plasma parameters, such as the relativistic degeneracy parameter of electron particles (re0), the density ratio of ion to electrons (δ), and the normalized electron cyclotron (Ωe), on the amplitude and width of the wave solutions. The plasma parameters used in this research are representative of compact astrophysical objects. Numerical results showed that the amplitude of wave solutions obtained by the LT method is smaller than the GT method, but the width is greater. We provide a physical explanation for these differences. Furthermore, we present a physics-informed neural network (PINN) approach to directly recover the intrinsic nonlinear dynamics from spatiotemporal data. The PINN model uses a deep neural network constrained by the governing equations to learn the optimal parameters, with the aim of enhancing the predictive capabilities of the system. The results of this study provide valuable insight into the propagation of nonlinear waves in white dwarfs, where relativistic effects are significant. These findings could substantially advance the development of emerging machine learning applications in astrophysics.

Generation of nonlinear gravity waves based on the harmonic polynomial cell method incorporated with a mass source

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections.

Non-Linear Plasma Wave Dynamics: Investigating Chaos in Dynamical Systems

This work addresses the significant issue of plasma waves interacting with non-linear dynamical systems in both perturbed and unperturbed states, as modeled by the generalized Whitham–Broer–Kaup–Boussinesq–Kupershmidt (WBK-BK) Equations. We investigate analytical solutions and the subsequent emergence of chaos within these systems. Initially, we apply advanced mathematical techniques, including the transform method and the G′G2 method. These methods allow us to derive new precise solutions and enhance our understanding of the non-linear processes dominating plasma wave dynamics. Through a systematic analysis, we identify the conditions under which the system transitions from orderly patterns to chaotic behavior. This investigation provides valuable insights into the fundamental mechanisms of non-linear wave propagation in plasmas. Our results highlight the dynamic interplay between non-linearity and variation, leading to chaos, which may be useful in predicting and potentially controlling similar phenomena in practical applications.

Spectral integrated neural networks (SINNs) for solving forward and inverse dynamic problems

This study introduces an innovative neural network framework named spectral integrated neural networks (SINNs) to address both forward and inverse dynamic problems in three-dimensional space. In the SINNs, the spectral integration technique is utilized for temporal discretization, followed by the application of a fully connected neural network to solve the resulting partial differential equations in the spatial domain. Furthermore, the polynomial basis functions are employed to expand the unknown function, with the goal of improving the performance of SINNs in tackling inverse problems. The performance of the developed framework is evaluated through several dynamic benchmark examples encompassing linear and nonlinear heat conduction problems, linear and nonlinear wave propagation problems, inverse problem of heat conduction, and long-time heat conduction problem. The numerical results demonstrate that the SINNs can effectively and accurately solve forward and inverse problems involving heat conduction and wave propagation. Additionally, the SINNs provide precise and stable solutions for dynamic problems with extended time durations. Compared to commonly used physics-informed neural networks, the SINNs exhibit superior performance with enhanced convergence speed, computational accuracy, and efficiency.

Diverse soliton wave profile analysis in ion-acoustic wave through an improved analytical approach

In engineering and applied sciences, several physical phenomena can be more precisely characterized by employing nonlinear fractional partial differential equations. The primary goal of this research is to examine the traveling wave solution in closed form for the nonlinear acoustic wave propagation model known as the time fractional simplified modified Camassa–Holm equation, which is used to explain the unidirectional propagation of shallow-water waves with non-hydrostatic pressure and explains the dispersion properties of numerous phenomena like fluid flow, control theory, liquid drop patterning in plasma, acoustics, fusion, and fission processes, etc. The utmost potential approach, namely the new auxiliary equation technique, is applied for analyzing the time nonlinear fractional simplified modified Camassa-Holm equation in the logic of the newest established truncated M-fractional derivative. The fractional partial differential equations have been transformed to the ordinary differential equation using the complex wave transformation in the sense of truncated M-fractional derivative. A variety of soliton solutions, including anti-kink, single soliton, anti-bell, bell, kink, multiple soliton, double soliton, singular-kink, compacton shape, periodic shape, and so many, are displayed in the diagram of 3D and contour plots by taking into account a number of various parameters. It is essential to point out that all derived outcomes are directly compared to the original solutions to certify their exactness. Results show that the used scheme is capable, simple, and straightforward and can be useful to a variety of complex phenomena. The acquired results are unique for the model equation and could be applied to the analysis of several nonlinear study fields.

On the study of double dispersive equation in the Murnaghan’s rod: Dynamics of diversity wave structures

This article secures the various wave structures of the fractional double dispersive equation, a significant nonlinear equation that describes the propagation of nonlinear waves within the elastic, uniform, and inhomogeneous Murnaghan’s rod. The model under discussion has a wide range of applications in science and engineering. Two recently developed analytical techniques known as the improved generalized Riccati equation mapping method and the multivariate generalized exponential rational integral function method have been applied to the proposed equation for the first time. A variety of solutions have been revealed such that dark, singular, bright-dark, bright, complex, and combined solitons. Furthermore, we include a diverse array of plots that illustrate the physical interpretation of the obtained solutions in relation to a number of significant parameters, thereby highlighting the impact of fractional derivatives. Within the context of the proposed model, these visualizations give a clear understanding of the behavior and characteristics of the solutions. This study’s results have the potential to enhance comprehension of the nonlinear dynamic characteristics exhibited by the specified system and validate the efficacy of the implemented techniques. The achieved results significantly enhance our understanding of nonlinear science and the nonlinear wave fields associated with more complex nonlinear models.

Modelling of non-linear elastic constitutive relationship and numerical simulation of rocks based on the Preisach–Mayergoyz space model

SUMMARY The existence of pores, cracks and cleavage in rocks results in significant non-linear elastic phenomena. One important non-linear elastic characteristic is the deviation of the stress–strain curve from the linear path predicted by Hooke's law. To provide a more accurate description of the non-linear elastic characteristics of rocks and to characterize the propagation of non-linear elastic waves, we introduce the Preisach–Mayergoyz space model. This model effectively captures the non-linear mesoscopic elasticity of rocks, allowing us to observe the stress–strain and modulus–stress relationships under different stress protocols. Additionally, we analyse the discrete memory characteristics of rocks subjected to cyclic loading. Based on the Preisach–Mayergoyz space model, we develop a new non-linear elastic constitutive relationship in the form of an exponential function. The new constitutive relationship is validated through copropagating acousto-elastic testing, and the experimental result is highly consistent with the data predicted by the theoretical non-linear elastic constitutive relationship. By combining the new non-linear elastic constitutive relationship with the strain–displacement formula and the differential equation of motion, we derive the non-linear elastic wave equation. We numerically solve the non-linear elastic wave equation with the finite difference method and observe two important deformations during the propagation of non-linear elastic waves: amplitude attenuation and dispersion. We also observe wave front discontinuities and uneven energy distribution in the 2D wavefield snapshot, which are different from those of linear elastic waves. We qualitatively explain these special manifestations of non-linear elastic wave propagation.

Nonlinear dust ion acoustic solitary waves propagation in a magnetized plasma with Tsallis electron distribution

Abstract We have done a theoretical investigation into the propagation of nonlinear dust ion acoustic solitary waves and their soliton properties in a three-component magnetized collisionless plasma consisting of inertial positively charged ions, noninertial electrons following a Tsallis q–distribution, and stationary negatively charged dust grains. We consider a uniform external magnetic field along the z–direction, and the wave propagation occurs obliquely to the magnetic field direction. It is observed that the two types of wave modes namely slow and fast modes, are appears in the linear analysis. By employing the reductive perturbation method, the Korteweg-de Vries (KdV) and modified KdV equations are determined to describe the small amplitude dust ion acoustic soliton. The dependence of several physical plasma parameters including nonextensive q–parameter, magnetic field strength etc. of our plasma on the propagating dust ion acoustic solitary waves potential are numerically examined. This study shows the simultaneous existence of compressive and rarefactive solitons due to the variation of first order nonlinearity coefficient represented by the KdV equation and it is found that there is a critical point for the plasma parameters where the amplitude of the soliton of KdV equation become diverges. The mKdV equation with second order nonlinearity coefficient is derived from there and observed the solitons only with finite amplitude. The present study could be helpful for understanding the nonlinear travelling waves propagating in laboratory and space plasma environments.

Role of Cairns–Gurevich ion distribution on nonlinear wave propagation in negatively charged dusty plasma

The study of dust-acoustic (DA) nonlinear electrostatic waves in dusty plasma is crucial for understanding plasma behavior in both astrophysical and laboratory environments. Dusty plasmas, characterized by the presence of negatively charged dust particles, exhibit complex dynamics influenced by various factors, including nonthermal electron and ion populations following Cairns–Gurevich distributions. This study addresses the problem of understanding wave propagation under these specific conditions by employing a set of fluid hydrodynamic equations for dust fluid, alongside appropriate electron and Cairns–Gurevich ion distributions, to model the dynamics and propagation of DA nonlinear electrostatic waves under specific plasma conditions with negatively charged dust particles. We derived a nonlinear Zakharov–Kuznetsov (ZK) equation to model the evolution of these waves and further transformed it into the nonlinear Schrödinger (NLS) equation to analyze rogue wave formation and identify unstable and stable zones within the plasma. We focused our analysis on the effects of key parameters, such as the nonthermal index, magnetic field strength, negative dust temperature ratio, polarization force, and density ratio, on the behavior of dust-acoustic waves (DAWs). The results demonstrated that soliton waves, explosive waves, and kink waves exhibit distinct behaviors depending on these parameters and the spatial context of Earth's magnetosphere. These findings provide new insights compared to previous studies into the conditions under which these waveforms emerge and evolve, especially in magnetized environments where earlier models were less effective at accurately characterizing or predicting wave behavior. By applying two different analytical methods, a direct integration approach and a generalized Kudryashov method, we expanded our understanding of nonlinear wave phenomena in dusty plasmas. Our results not only advance theoretical knowledge but also have practical implications for interpreting space plasma observations and guiding experimental design in plasma physics research. This study thus contributes to a more comprehensive understanding of plasma dynamics, with potential applications to astrophysical observation and laboratory plasma experimentation.

Soliton solutions of cubic quintic septimal nonlinear Schrödinger wave equation with conformable derivative by two distinct algorithms

This paper investigates the cubic-quintic-septimal nonlinear Schrödinger wave equation with a conformable derivative, which governs the evolution of light beams in a weak nonlocal medium. The analysis utilizes the Kudryashov method and the enhanced modified tanh expansion method. By utilizing these analytical integration schemes, various optical wave solutions are derived within the present conformable model. The paper demonstrates the significance of these optical soliton solutions by illustrating different soliton solutions, including kink-type, bell-shaped, singular, dark, and wave soliton solutions, depicted via contour, three-dimensional, and two-dimensional representations. Moreover, it is crucial to emphasize the importance of analyzing the cubic-quintic-septimal nonlinear Schrödinger wave equation, which finds utility across a spectrum of fields including optics, quantum mechanics, and the study of nonlinear wave propagation. Moving forward, these approaches hold promise for investigating diverse sets of differential equations within multiple domains of applied sciences. This governing equation also has numerous applications in nonlinear optics, such as describing the propagation of laser beams through materials with nonlinear optical properties. The inclusion of these nonlinearities illustrates the interaction and behavior of light beams in weakly non-local media.

Periodic Solutions of Wave Propagation in a Strongly Nonlinear Monatomic Chain and Their Novel Stability and Bifurcation Analyses

Abstract A modified incremental harmonic balance (IHB) method is used to determine periodic solutions of wave propagation in discrete, strongly nonlinear, periodic structures, and solutions are found to be in a two-dimensional hyperplane. A novel method based on the Hill’s method is developed to analyze stability and bifurcations of periodic solutions. A simplified model of wave propagation in a strongly nonlinear monatomic chain is examined in detail. The study reveals the amplitude-dependent property of nonlinear wave propagation in the structure and relationships among the frequency, the amplitude, the propagation constant, and the nonlinear stiffness. Numerous bifurcations are identified for the strongly nonlinear chain. Attenuation zones for wave propagation that are determined using an analysis of results from the modified IHB method and directly using the modified IHB method are in excellent agreement. Two frequency formulae for weakly and strongly nonlinear monatomic chains are obtained by a fitting method for results from the modified IHB method, and the one for a weakly nonlinear monatomic chain is consistent with the result from a perturbation method in the literature.

Obliquely Propagating Self-Gravitational Shock Waves in Non-Relativistic Degenerate Quantum Plasmas

A rigorous theoretical investigation has been carried out on the propagation of non-linear self-gravitational shock waves (SGSHWs) in a magnetized super dense degenerate quantum plasma system (DQPS) composed of inertia-less non-relativistic degenerate electrons and inertial non-degenerate extremely heavy nuclei/element. The nonlinear propagation of these SGSHWs in the plasma system under consideration is studied by the standard reductive perturbation technique, which is valid for a small finite amplitude limit. The nonlinear dynamics of the SGSHWs are found to be governed by the Burgers equation. The Burgers equation is derived analytically and solved numerically. It has been found that the considered plasma model supports positive potential shock waves only. The fundamental properties (amplitude, steepness, etc.) of these SGSHWs are significantly modified by the variation of kinematic viscosity, obliqueness and number density of the plasma species. The results of our present investigation can be applied to astrophysical compact objects like neutron stars. Journal of Engineering Science 15(1), 2024, 21-29

A p-Multigrid Hybrid-Spectral Model for Nonlinear Water Waves

AbstractTo improve both scale and fidelity of numerical water wave simulations to study the evolution of wave fields within offshore engineering, it is of key practical interest to achieve high numerical efficiency. We propose a p-multigrid accelerated time-domain scheme for efficient and $${\mathcal {O}}(n \log n)$$ O ( n log n ) -scalable solution of a hybrid-spectral model for the simulation of highly nonlinear and highly dispersive water waves, the accurate calculation of wave kinematics and taking into account varying water depth. To achieve low numerical dispersive and dissipate errors, a high-order hybrid-spectral collocation scheme is implemented to solve the fully nonlinear potential flow (FNPF) equations, and utilizing a p-multigrid iterative solver scheme for solving the Laplace problem. Hereby, the numerical scheme combines the high accuracy of spectral methods, high efficiency of the fast Fourier transform (FFT) algorithm, and multigrid methods. Numerical analysis is performed to evaluate the performance and confirm the spectral accuracy of the scheme. Numerical experiments are considered for steady nonlinear wave propagation. A Fourier-continuation technique is used to extend the scheme from a periodic domain setup to perform benchmarks in a finite domain setup for numerical wave tanks utilizing conventional techniques for wave generation and absorption, and with results in excellent agreement with experimental measurements.