Analytical Wave Solutions Research Articles

Nonlinear wave dynamics in ferromagnetic media: A study with the Kuralay-IIA equation

In this research, we delve into the exploration of the Kuralay-IIA equation by employing two power full approaches, namely extended hyperbolic function method (EHFM) and the improved modified Sardar-subequation method (IMSSEM). Applications of the Kuralay-IIA equation can be observed in a variety of physical phenomena, particularly in optical fibers, magnetic materials, and nonlinear optics. This equation performs an essential part in understanding gas behavior. Specifically, it offers an image of the relationship between volume, temperature and pressure in a gas system. We proceed through the matter in detail through the use of space curves which demonstrate integrable motion. Using the proposed techniques, our investigation provides an extensive variety of soliton solutions, such as periodic, dark–bright, bright, dark, solitary, and some other solutions. These solutions are in good alignment with previous study findings in this area. Analytical wave solutions are vital as they provide a basic comprehension of the physics or mathematics at play and establish a framework for further studies. The findings obtained from this study may help design models to come. The techniques used in this study are very effective, straightforward, and capable of handling more nonlinear models with high reliability. We use the Mathematica software to validate the accuracy and reliability of our results. Using carefully constructed 2D and 3D graphs generated, the resulting solutions are visually shown.

Optical dromions for M-fractional Kuralay equation via complete discrimination system approach along with sensitivity analysis and quasi-periodic behavior

This paper investigates new analytical wave solutions for the fractional Kuralay-IIB equations (FK-IIBE), including a new definition of the fractional derivative. This model works in disciplines such as ferromagnetic materials, optical fibers, wave mixing, and nonlinear optics. The integrable motion of space curves can be determined by our governing model. This research holds great importance in the area of optical fibers, exact and analytical solitons solution. The polynomial method’s complete discrimination system (CDSPM) is used to identify a variety of exact and analytical soliton solutions such as periodic, hyperbolic, rational, Jacobian elliptic (JE), and trigonometric functions. Additionally, we also convert the JE function into a solitary wave (SW) solution. To provide a comprehensive understanding of the dynamic aspects of the system, we also address bifurcation points, quasi-periodic behavior, sensitivity analysis, Poincarè maps, time series profile, and critical solution conditions. The research offers straightforward, efficient algorithms that can consistently and effectively address FK-IIBE. Machine learning technologies are also utilized to satisfy the criteria defined by dynamic analysis. Python regression learner tools are used for reverse engineering to analyze the mathematical model’s equilibrium based on parameter intervals and thresholds determined by dynamical analysis.

Exploring the New Analytical Wave Solutions for M-Fractional Stochastic Nizhnik-Novikov-Veselov System by an Efficient Analytical Technique

Exploring the New Analytical Wave Solutions for M-Fractional Stochastic Nizhnik-Novikov-Veselov System by an Efficient Analytical Technique

Lie symmetry analysis, traveling wave solutions and conservation laws of a Zabolotskaya-Khokholov dynamical model in plasma physics

This article analyzes the analytic and solitary wave solutions of the one-dimensional Zabolotskaya-Khokholov (ZK) dynamical model which provides information about the propagation of sound beam or confined wave beam in nonlinear media and studies of beam deformation. By the Lie symmetry analysis method, we acquire the vector fields, commutation relations, optimal system, reduction, and analytic solutions to the specified equation by exerting the Lie group method. Moreover, the solitary wave solutions of the ZK model are procured by exerting the new auxiliary equation method (NAEM). The behavior of the acquired outcomes for several cases is exhibited graphically through two and three-dimensional dynamical wave profiles. Furthermore, the conservation laws of the ZK model are acquired by Ibragimov’s new conservation theorem.

Magneto-thermoelastic surface waves phenomenon with voids, gravity, initial stress, and rotation under four theories

This paper addresses a significant research gap in the study of surface waves propagation in a nonhomogeneous, within a magneto-thermoviscoelastic material of higher order, initial stress, rotation, gravity effects and voids. This study provides analytical solutions for surface waves propagating through a medium consisting of a magneto-thermoelastic material with voids under the rotation, electro-magnetic field, gravity field and initial stress. The analytical solutions are derived for the displacement components, volume fraction, temperature to Stoneley and Rayleigh waves are computed numerically and presented graphically considering the external parameters impact. Furthermore, this investigates how magnetic field, voids, gravity, initial stress and fiber-reinforced parameters influence these wave phenomena. This investigation provides valuable insights into the synergistic dynamics among electric constituents, voids, Stoneley and Rayleigh waves propagation, enabling advancements in sensor technology, augmented energy harvesting methodologies, and pioneering seismic monitoring approaches. For certain materials, numerical simulations are provided and graphically displayed. The results of this study reveal several unique cases that significantly contribute to the understanding of Rayleigh and Stoneley waves propagation within this intricate material system, particularly in the presence of voids.

An efficient numerical solver for highly compliant slender structures in waves: Application to marine vegetation

This paper presents a fully explicit coupled wave–vegetation interaction model capable of efficiently solving the coupled wave dynamics and flexible vegetation motion with large deflections. The flow model is formulated using the continuity equation and linearized momentum equations of an incompressible fluid, with additional terms within the canopy region accounting for the presence of vegetation. This linearized flow solver is unconditionally stable and second-order accurate. The flow model is validated and verified against experimental measurements and analytical solutions for waves over a rigid canopy, demonstrating its capability to accurately capture the wave dissipation and flow velocity profiles, even with a relatively coarse grid. A truss-spring model is proposed to capture vegetation motion with substantial deflections, and is proven to be mathematically consistent with the governing equation for the flexible vegetation motion. It allows for explicit time integration with large time steps when dealing with highly compliant vegetation. The truss-spring model is validated and verified by experimental and numerical results for large-amplitude motions of a single elastic blade subjected to waves and sinusoidal oscillatory flows. The coupled model, combining the linearized flow solver and the truss-spring model, is applied to investigate wave propagating over a heterogeneous, suspended, and flexible canopy, showing high efficiency and good agreement with the experiments concerning wave attenuation and the hydrodynamic loads on the vegetation.

Rayleigh wave in nonlocal piezo-thermo-electric semiconductor medium with fractional MGT model

This paper addresses a critical gap in Rayleigh wave propagation studies within piezoelectric semiconductors. Existing research often overlooks the combined effects of piezoelectricity, semiconductivity, and thermal behavior. This work presents analytical solutions for Rayleigh waves in a novel nonlocal thermo-piezo-photo-electric semiconductor half-space, employing the fractional Moore-Gibson-Thompson model. The wave-mode method and Durand-Kerner technique are employed to extract relevant solutions from the characteristic equation, ensuring surface wave decay. Analysis is conducted for waves propagating in a stress-free, photo-excited, and isothermal boundary condition. Numerical simulations unveil phase velocity, attenuation coefficient, and particle motion, leading to a deeper understanding of wave behavior. The influence of nonlocal and fractional order parameters is meticulously investigated. These findings reveal unique phenomena that significantly advance the comprehension of Rayleigh waves in these intricate materials.

Exploring Wave Interactions and Conserved Quantities of KdV–Caudrey–Dodd–Gibbon Equation Using Lie Theory

This study introduces the KdV–Caudrey–Dodd–Gibbon (KdV-CDGE) equation to describe long water waves, acoustic waves, plasma waves, and nonlinear optics. Employing a generalized new auxiliary equation scheme, we derive exact analytical wave solutions, revealing rational, exponential, trigonometric, and hyperbolic trigonometric structures. The model also produces periodic, dark, bright, singular, and other soliton wave profiles. We compute classical and translational symmetries to develop abelian algebra, and visualize our results using selected parameters.

Analytic wave solutions to the beta-time fractional modified equal width equation based on two efficient approaches

By using the two distinct methods known as the exp(-ϕ(η))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (-\\phi (\\eta ))$$\\end{document} and the Expa\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Exp_a$$\\end{document}-function methods, various forms of soliton solutions of the modified equal width wave equation (MEWE) with beta time derivative (BTD) are produced in this study. This model is used as a model in partial differential equations for the simulation of one-dimensional wave transmission in nonlinear media with dispersion processes. The obtained solutions are in the form of rational, trigonometry and hyperbolic trigonometry functions. Using Mathematica software, the resulting solitons are validated. Graphs are also used at the conclusion to explain the findings. These soliton solutions imply that these two methods are more dependable, simple and efficient than other methods. The findings can be used to explain how studious structures and other comparable non-linear physical structures are substantially understood. The obtained results are very helpful in the fields of optics, kinetics solid-state physics and hydro-magnetic waves in cold plasma.

Analyzing bifurcation, stability, and wave solutions in nonlinear telecommunications models using transmission lines, Hamiltonian and Jacobian techniques

This study presents a comprehensive analysis of a nonlinear telecommunications model, exploring bifurcation, stability, and wave solutions using Hamiltonian and Jacobian techniques. The investigation begins with a thorough examination of bifurcation behavior, identifying critical points and their stability characteristics, leading to the discovery of diverse bifurcation scenarios. The stability of critical points is further assessed through graphical and numerical methods, highlighting the sensitivity to parameter variations. The study delves into the derivation of both numerical and analytical wave solutions, aligning them with energy orbits depicted in phase portraits, revealing a spectrum of wave behaviors. Additionally, the analysis extends to traveling wave solutions, providing insights into wave propagation dynamics. Notably, the study underscores the efficacy of the planar dynamical approach in capturing system behavior in harmony with phase portrait orbits. The findings have significant implications for telecommunications engineers and researchers, offering insights into system behavior, stability, and signal propagation, ultimately advancing our understanding of complex nonlinear dynamics in telecommunications networks.

On traveling wave solutions with stability and phase plane analysis for the modified Benjamin-Bona-Mahony equation.

The modified Benjamin-Bona-Mahony (mBBM) model is utilized in the optical illusion field to describe the propagation of long waves in a nonlinear dispersive medium during a visual illusion (Khater 2021). This article investigates the mBBM equation through the utilization of the rational [Formula: see text]-expansion technique to derive new analytical wave solutions. The analytical solutions we have obtained comprise hyperbolic, trigonometric, and rational functions. Some of these exact solutions closely align with previously published results in specific cases, affirming the validity of our other solutions. To provide insights into diverse wave propagation characteristics, we have conducted an in-depth analysis of these solutions using 2D, 3D, and density plots. We also investigated the effects of various parameters on the characteristics of the obtained wave solutions of the model. Moreover, we employed the techniques of linear stability to perform stability analysis of the considered model. Additionally, we have explored the stability of the associated dynamical system through the application of phase plane theory. This study also demonstrates the efficacy and capabilities of the rational [Formula: see text]-expansion approach in analyzing and extracting soliton solutions from nonlinear partial differential equations.

Size-dependent effect on the interaction of surface waves in micropolar thermoelastic medium with dual pore connectivity

This research tackles a critical knowledge gap in Rayleigh surface wave propagation. It offers a comprehensive analysis that surpasses previous limitations. A size-dependent micropolar medium with unique void distributions and thermal effects is considered in this work. The constitutive relations and equations of motion for a nonlocal micropolar thermoelastic medium with double voids (MTMWDV) have been established by using Eringen’s nonlocal elasticity theory. Employing the three-phase-lag thermoelasticity theory (TPLTE), the study utilizes a wave-mode method to derive analytical solutions for Rayleigh waves in a nonlocal MTMWDV. To gain a comprehensive understanding of wave behavior, we solve the characteristic equation and analyze its roots, applying a filter based on the surface wave decay condition. A medium with stress-free and isothermal boundaries is explored through computational simulations to determine the attenuation coefficient and phase velocity. Furthermore, particle motion analysis is conducted to complement the analytical and computational approaches. Moreover, the influence of the nonlocal parameter and various thermoelastic models on these wave phenomena is investigated. The validity of the current mathematical model is confirmed through the derivation of particular scenarios.

Hydrodynamic synchronization of elastic cilia: How surface effects determine the characteristics of metachronal waves.

Cilia are hairlike microactuators whose cyclic motion is specialized to propel extracellular fluids at low Reynolds numbers. Clusters of these organelles can form synchronized beating patterns, called metachronal waves, which presumably arise from hydrodynamic interactions. We model hydrodynamically interacting cilia by microspheres elastically bound to circular orbits, whose inclinations with respect to a no-slip wall model the ciliary power and recovery stroke, resulting in an anisotropy of the viscous flow. We derive a coupled phase-oscillator description by reducing the microsphere dynamics to the slow timescale of synchronization and determine analytical metachronal wave solutions and their stability in a periodic chain setting. In this framework, a simple intuition for the hydrodynamic coupling between phase oscillators is established by relating the geometry of flow near the surface of a cell or tissue to the directionality of the hydrodynamic coupling functions. This intuition naturally explains the properties of the linear stability of metachronal waves. The flow near the surface stabilizes metachronal waves with long wavelengths propagating in the direction of the power stroke and, moreover, metachronal waves with short wavelengths propagating perpendicularly to the power stroke. Performing simulations of phase-oscillator chains with periodic boundary conditions, we indeed find that both wave types emerge with a variety of linearly stable wave numbers. In open chains of phase oscillators, the dynamics of metachronal waves is fundamentally different. Here the elasticity of the model cilia controls the wave direction and selects a particular wave number: At large elasticity, waves traveling in the direction of the power stroke are stable, whereas at smaller elasticity waves in the opposite direction are stable. For intermediate elasticity both wave directions coexist. In this regime, waves propagating towards both ends of the chain form, but only one wave direction prevails, depending on the elasticity and initial conditions.

Dynamical interactions of the magnetic field in a binary mixture of interacting thermoelastic solids

A more focused scientific approach in the context of a mixture of thermoelastic solids would entail investigating the basic ideas guiding the behavior of the complex materials, including constitutive model development, multi-scale modeling at various length scales, and design optimization for thermoelastic solids with their precise properties. The present work is set up to study the effect of the magnetic field as an external effect on the mixture of two interacting thermoelastic solids. The fundamental mathematical equations were shown and solved by the analytical harmonic wave solution to give the solution of the descriptive functions of the discussed problem. The displacements and stresses for the mixture of the two interreacted thermoelastic solids with the mutual temperature were clarified in the 2-D and 3-D. The results are analyzed and discussed. The obtained solution for the functions is finite everywhere. The solutions for the quantities of practical interest are represented graphically with different choices of the wave number of the harmonic wave as the magnetic field with time. In addition to the numerous physical factors in the governing field equations as depicted in the figures, boundary conditions also play a role in the current damping phenomenon.

Stability analysis and conserved quantities of analytic nonlinear wave solutions in multi-dimensional fractional systems

The (3+1)-dimensional generalized nonlinear fractional Konopelchenko–Dubrovsky–Kaup–Kupershmidt [Formula: see text] model represents the propagation and interaction of nonlinear waves in complex multi-dimensional physical media characterized by anomalous dispersion and dissipation phenomena. By incorporating fractional derivatives, this model introduces non-locality and memory effects into the classical [Formula: see text] equations, commonly utilized in phenomena such as shallow water waves, nonlinear optics, and plasma physics. The fractional approach enhances mathematical representations, allowing for a more realistic depiction of the intricate behaviors observed in numerous modern physical systems. This study focuses on the development of accurate and efficient numerical techniques tailored for the computationally demanding [Formula: see text] model, leveraging the Khater II and generalized rational approximation methods. These methodologies facilitate stable time-integration, effectively addressing the model’s stiffness and multi-dimensional nature. Through numerical analysis, insights into the stability and convergence of the algorithms are gained. Simulations conducted validate the performance of these methods against established solutions while also uncovering novel capabilities for exploring complex wave dynamics in scenarios involving complete fractional formulations. The findings underscore the potential of integrating fractional calculus into higher-dimensional nonlinear partial differential equations, offering a promising avenue for advancing the modeling and computational analysis of complex wave phenomena across a spectrum of contemporary physical disciplines.

Baseband modulational instability and interacting localized mixed waves in coherently coupled optical media

We consider a coherently coupled nonlinear Schrödinger equations (model equations) that describe interacting localized waves in coherently coupled optical media. The baseband modulational instability in such regimes is investigated, leading the result that the modulational instability does not necessary imply the baseband modulational instability. By means of a set of linear transformations, the model equations are converted into two independent scalar nonlinear Schrödinger equations. Based on the generalized perturbation (n,N−n)–fold Darboux transformation, we report and discuss analytical mixed localized wave solutions of these scalar equations. A first-order rogue wave solution of the scalar nonlinear Schrödinger equations is also presented. We then show that linear superpositions of different found analytical mixed localized wave solutions of the derived scalar equations results into several kinds of coherent nonlinear mixed structures such as, coexisting bright/dark soliton-breather, rogue wave-breather-soliton, interacting rogue waves, bright/dark breathers, and bright/dark solitons.

Abundant optical soliton solutions to the fractional perturbed Chen-Lee-Liu equation with conformable derivative

This research focuses on the space-time fractional nonlinear perturbed Chen-Lee-Liu model, which describes the propagation behavior of optical pulses in the fields of optical fiber and plasma. The equation is considered with respect to the conformable derivative, and a composite fractional wave transformation is employed to reformulate it into a nonlinear equation with a single variable. The improved tanh method has been applied to derive novel analytical wave solutions for the given equation. Consequently, various types of solitonic wave patterns emerge, including but not limited to periodic, bell-shaped, anti-bell-shaped, V-shaped, kink, and compacton solitonic structures. The acquired solutions could potentially aid in the analysis of signal transmission in optical fibers and the characterization of plasma properties. The physical interpretations of the solutions are investigated using three-dimensional surface plots and two-dimensional density plots. Additionally, combined two-dimensional plots are being used to discuss the effects of the order of the fractional derivative on the generated wave patterns. Moreover, this study demonstrates the efficacy and reliability of the chosen technique.

Higher-order hybrid rogue wave and breather interaction dynamics for the AB system in two-layer fluids

Rogue waves and breathers emerge as significant solitons, result from ubiquitous nonlinear effects of dynamics in the natural world, science and engineering. In this paper, we study the AB system, a model derived from a two-layer fluid that is widely used to investigate nonlinear fluid dynamics. We use the Darboux transformation to construct analytical solutions for higher-order hybrid rogue waves and breathers, which are expressed by symmetric algebraic matrices. We use these solutions and the spectrum parameters to explore the complex interaction dynamics of the hybrid rogue waves and breathers. We find that the interactions cause significant changes in the phases and shapes. Moreover, we discover an interesting phenomenon: the collisions between the rogue waves and breathers are semi-elastic, meaning that only the phases of the rogue waves change, while the speeds, shapes and phases of the breathers remain the same, before and after the collisions. This study provides new insights into the dynamics of the AB system, and helps us understand nonlinear phenomena in various two-layer fluid systems.

Impact of memory-dependent heat transfer on Rayleigh waves propagation in nonlocal piezo-thermo-elastic medium with voids

PurposeThis paper addresses a significant research gap in the study of Rayleigh surface wave propagation within a piezoelectric medium characterized by piezoelectric properties, thermal effects and voids. Previous research has often overlooked the crucial aspects related to voids. This study aims to provide analytical solutions for Rayleigh waves propagating through a medium consisting of a nonlocal piezo-thermo-elastic material with voids under the Moore–Gibson–Thompson thermo-elasticity theory with memory dependencies.Design/methodology/approachThe analytical solutions are derived using a wave-mode method, and roots are computed from the characteristic equation using the Durand–Kerner method. These roots are then filtered based on the decay condition of surface waves. The analysis pertains to a medium subjected to stress-free and isothermal boundary conditions.FindingsComputational simulations are performed to determine the attenuation coefficient and phase velocity of Rayleigh waves. This investigation goes beyond mere calculations and examines particle motion to gain deeper insights into Rayleigh wave propagation. Furthermore, this investigates how kernel function and nonlocal parameters influence these wave phenomena.Research limitations/implicationsThe results of this study reveal several unique cases that significantly contribute to the understanding of Rayleigh wave propagation within this intricate material system, particularly in the presence of voids.Practical implicationsThis investigation provides valuable insights into the synergistic dynamics among piezoelectric constituents, void structures and Rayleigh wave propagation, enabling advancements in sensor technology, augmented energy harvesting methodologies and pioneering seismic monitoring approaches.Originality/valueThis study formulates a novel governing equation for a nonlocal piezo-thermo-elastic medium with voids, highlighting the significance of Rayleigh waves and investigating the impact of memory.

Consistent travelling wave characteristic of space–time fractional modified Benjamin–Bona–Mahony and the space–time fractional Duffing models

Study on solitary wave phenomenon are closely related on the dynamics of the plasma and optical fiber system, which carry on broad range of wave propagation. The space–time fractional modified Benjamin–Bona–Mahony equation and Duffing model are important modeling equations in acoustic gravity waves, cold plasma waves, quantum plasma in mechanics, elastic media in nonlinear optics, and the damping of material waves. This study has effectively developed analytical wave solutions to the aforementioned models, which may have significant consequences for characterizing the nonlinear dynamical behavior related to the phenomenon. Conformable derivatives are used to narrate the fractional derivatives. The expanded tanh-function method is used to look into such kinds of resolutions. An ansatz for analytical traveling wave solutions of certain nonlinear evolution equations was originally a power sequence in tanh. The discovered explanations are useful, reliable, and applicable to chaotic vibrations, problems of optimal control, bifurcations to global and local, also resonances, as well as fusion and fission phenomena in solitons, scalar electrodynamics, the relation of relativistic energy–momentum, electromagnetic interactions, theory of one-particle quantum relativistic, and cold plasm. The solutions are drafted in 3D, contour, listpoint, and 2D patterns, and include multiple solitons, bell shape, kink type, single soliton, compaction solitary wave, and additional sorts of solutions. With the aid of Maple and MATHEMATICA, these solutions were verified and discovered that they were correct. The mentioned method applied for solving NLFPDEs has been designed to be practical, straightforward, rapid, and easy to use.