Nonlinear Wave Propagation Research Articles

Improved detection of nonlinear waves in thin solid materials using a pulse-echo method based on nonlinear modeling of wave fields

The nonlinearity parameter is a powerful tool for nondestructive evaluation, as it provides a measure of the state of damage of structural components. When carrying out nonlinearity parameter measurements of solid structures, the pulse-echo method is more suitable than the through-transmission method; however, a planar stress-free boundary is known to destructively alter the nonlinear wave generation process, including the second harmonic component, meaning that the pulse-echo nonlinear method is limited for practical applications, especially for thin solids. In this work, an improved detection method is proposed for the enhancement of nonlinear wave measurement in thin solids using a pulse-echo method with a dual-element transducer consisting of an outer ring transmitter and an inner disk receiver. A multi-Gaussian beam model is developed for the ring transmitter to fast simulate the pulse-echo nonlinear wave fields, and based on the results, improvements for the received nonlinear waves are achieved by optimizing the driving frequency. The advantages of the proposed method are that solid samples with different thicknesses can be evaluated using a dual-element transducer, and a high measurement accuracy can also be achieved for thin solid samples. Experiments are performed to investigate the harmonic waves measured in fatigue-damaged thin steel samples, and the results verify the effectiveness of the proposed method.

New exact solutions of the (3+1)-dimensional double sine-Gordon equation by two analytical methods

The (3+1)-dimensional double sine-Gordon equation plays a crucial role in various physical phenomena, including nonlinear wave propagation, field theory, and condensed matter physics. However, obtaining exact solutions to this equation faces significant challenges. In this article, we successfully employ a modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left( \\dfrac{G'}{G^2}\\right) $$\\end{document}-expansion and improved tanϕξ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ an \\left( \\dfrac{\\phi \\left( \\xi \\right) }{2}\\right) $$\\end{document}-expansion methods to construct new analytical solutions to the double sine-Gordon equation. These solutions can be divided into four categories like trigonometric function solutions, hyperbolic function solutions, exponential solutions, and rational solutions. Our key findings include a rich spectrum of soliton solutions, encompassing bright, dark, singular, periodic, and mixed types, showcasing the (3+1)-dimensional double sine-Gordon equation ability to model diverse wave behaviors. We uncover previously unreported complex wave structures, revealing the potential for complex nonlinear interactions within the (3+1)-dimensional double sine-Gordon equation framework. We demonstrate the modified G′G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\left( \\dfrac{G'}{G^2}\\right) $$\\end{document}-expansion and improved tanϕξ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ an \\left( \\dfrac{\\phi \\left( \\xi \\right) }{2}\\right) $$\\end{document}-expansion methods effectiveness in handling higher-dimensional nonlinear partial differential equations, expanding their applicability in mathematical physics. These method offers enhanced flexibility and broader solution categories compared to conventional approaches.

Landau damping effects on dust ion-acoustic solitary waves in a nonthermal pair-ion plasma

The Landau damping effects on the nonlinear propagation of dust ion-acoustic solitary waves (DIASWs) are studied in an electron-free unmagnetized collisionless dusty pair-ion plasma. The two species of nonthermal ions (positive and negative) are described by the kinetic Vlasov equations, whereas the massive charged dust grains form only the background plasma. Using the multi-scale reductive perturbation technique generalized for the applications to the Vlasov equation, we derive a modified Korteweg–de Vries (KdV) equation that governs the evolution of DIASWs with the effects of linear resonance, in particular, linear Landau damping. The properties of the phase velocity and the Landau damping rate of DIASWs are studied with the effects of positive and negative-ion nonthermality parameters ( α p , α n ), the ratios of the positive to negative ion masses (m) and temperatures (T) as well as the dust to negative-ion number density ratio (δ). It is found that, depend on the degree of positive to negative ion mass ratios (m), both of damped and growing oscillations occur in a collisionless dusty pair-ion plasma due to the resonance phenomena between the wave and the nonthermal particles of the system. The properties of the decay rates of the amplitude of the KdV soliton with a small effect of Landau damping are also studied with the above system of parameters. The results may be useful for understanding the localization of solitary pulses and associated resonance damping and/or growing oscillations of the wave in laboratory and space plasmas, in which the number density of free electrons is much smaller than that of ions and highly charged heavy, micron seized dust grains.

A study of the non-linear properties and wave generation of the multi-layer non-hydrostatic wave model SWASH

Despite the increasing use of non-hydrostatic models in the study of wave processes in coastal regions, there is still limited understanding of the non-linear properties of the governing equations and how they improve with increased vertical resolution. In this study, the governing equations of the non-hydrostatic wave model SWASH are analysed and the linear and non-linear solutions up to third-order of all dependent variables are derived, considering one to four vertical layers. The analysis concludes that the model can achieve excellent non-linear properties with respect to the Stokes theory over a large range of water depths using only a few vertical layers. Furthermore, deriving solutions for all variables enables the formulation of improved wave generation and absorption boundary conditions for non-hydrostatic models. A well-known issue of non-linear wave models is related to the generation and propagation of spurious free waves, resulting to non-homogeneous wave fields. In this study, it is proven that by imposing the derived exact mathematical solutions of the governing equations at the model’s boundaries, the target first- and second-order wave profiles can be generated with high accuracy, while the spurious waves can be entirely eliminated.

Numerical Simulation for Solitary Waves of the Generalized Zakharov Equation Based on the Lattice Boltzmann Method

The generalized Zakharov equation is a widely used and crucial model in plasma physics, which helps to understand wave particle interactions and nonlinear wave propagation in plasma. The solitary wave solution of this equation provides insights into phenomena such as electron and ion acoustic waves, as well as magnetic field disturbances in plasma. The numerical simulation of solitary wave solutions to the generalized Zakharov equation is an interesting problem worth studying. This is crucial for plasma-based technology, as well as for understanding nonlinear wave propagation in plasma physics and other fields. In this study, a numerical investigation of the generalized Zakharov equation using the lattice Boltzmann method has been conducted. The lattice Boltzmann method is a new modeling and simulating method at the mesoscale. A lattice Boltzmann model was constructed by performing Taylor expansion, Chapman–Enskog expansion, and time multiscale expansion on the lattice Boltzmann equation. By defining the moments of the equilibrium distribution function appropriately, the macroscopic equation can be restored. Furthermore, the numerical experiments for the equation are carried out with the parameter lattice size m=100, time step Δt=0.001, and space step size Δx=0.4. The solitary wave solution of the equation is numerically simulated. Numerical results under different parameter values are compared, and the convergence and effectiveness of the model are numerically verified. It is obtained that the model is convergent in time and space, and the convergence orders are all 2.24881. The effectiveness of our model was also verified by comparing the numerical results of different numerical methods. The lattice Boltzmann method demonstrates advantages in both accuracy and CPU time. The results indicate that the lattice Boltzmann method is a good tool for computing the generalized Zakharov equation.

Computer simulation of a nonlinear wave model of transport phenomena in media with non-local effects

The main contribution of the presented article is that sufficient assumptions have been established for the modified Witham equation describing nonlinear wave propagation for transport phenomena in physicochemical systems. It is shown that when deriving Witham-type equations, the presence of spatial nonlocality of the medium can play a fundamental role. The article also discusses the issues of physical interpretation of the model under study. The principal presence of non-local effects in the studied systems can be justified and obtained as a result of the presence of domains with a complexly organized spatial structure, as well as the presence of heat and mass sources. Computer modeling was carried out and the results were obtained using the Java programming language.

Wave propagation analysis in the modified nonlinear time fractional Harry Dym equation: Insights from Khater II method and B-spline schemes

This study aims to investigate the modified nonlinear time fractional Harry Dym equation using analytical and numerical techniques. The modified nonlinear time fractional Harry Dym equation is a generalization of the classical Harry Dym equation, which describes the propagation of nonlinear waves in a variety of physical systems. The conformable fractional derivative is used to define the time fractional derivative in the equation, which provides a natural and straightforward approach. The Khater II method, a powerful analytical technique, is employed to obtain approximate solutions for the equation. Additionally, three numerical schemes, namely, Cubic-B-spline, Quantic-B-spline and Septic-B-spline schemes, are developed and implemented to solve the equation numerically. The numerical results are compared with other numerical solutions to assess the accuracy and efficiency of the proposed schemes. The physical meaning of the modified nonlinear time fractional Harry Dym equation is discussed in detail, and its relation to other nonlinear evolution equations is highlighted. The results of this study provide new insights into the behavior of nonlinear waves in physical systems and contribute to a better understanding of the physical characterizations of the modified nonlinear time fractional Harry Dym equation.

Excitation and stability of nonlinear waves by a pair of steady waves in incompressible boundary layers

The research explores the generation and stability of nonlinear waves in incompressible boundary layers, spotlighting the effects of dual steady waves. Using the Benjamin-Ono equation, modified for the incorporation of steady waves, we provide insights into the solitary wave generation process. A perturbed wave oscillator system is used for analyzing the stability of these nonlinear waves, uncovering that the loss of stability in low-order oscillators primarily governs wave transition and energy. Further, the study highlights the enlightening significance of complex terrain on the generation of nonlinear waves. Since the influence of terrain on the fluid passing over it usually appears as the external excitation of stationary waves, our study has important implications for terrain-fluid interactions.

Nonlinear periodic ion-acoustic waves in nonthermal plasmas

A Sagdeev pseudopotential analysis is developed for the propagation of nonlinear periodic ion-acoustic waves in a plasma comprising cold fluid ions and various models of nonthermal electron descriptions. In plasma nonlinear wave studies, whether addressing solitary or periodic modes, the more common nonthermal distributions are the Cairns, kappa, and Tsallis models. A mathematically and physically consistent description incorporates three evident properties: there is conservation per cycle of ion and electron densities in addition to ion flux, the solutions reduce for very small amplitudes to linear waves, and the nonlinear periodic structures are generated by a perturbation of the undisturbed equilibrium. After establishing the corresponding general analytical methodology, a numerical analysis is given, with illustrative graphs, for the nonthermal Cairns, superthermal kappa, and nonextensive Tsallis distributions.

How do various forces affect pressure waves in bubbly flows?

This study investigated the weakly nonlinear propagation of pressure waves in compressible, flowing water with spherical microbubbles, considering various forces. Previous theoretical studies on nonlinear pressure waves in bubbly flows did not consider the forces acting on the bubbles, although the validity of ignoring these forces has not been demonstrated. We focused on every possible force such as drag, gravity, buoyancy, and Bjerknes (acoustic radiation) forces acting on bubbles and studied their effects on pressure waves in a one-dimensional setting. Using a singular perturbation method, the Korteweg–de Vries–Burgers equation describing wave propagation was derived. The following results were obtained: (i) Bjerknes force on the bubbles enhanced the nonlinearity, dissipation, and dispersion of the waves; (ii) Drag, gravity, and buoyancy forces acting on the bubbles increased wave dissipation; (iii) Thermal conduction had the most substantial dissipation effect, followed by acoustic radiation, drag, buoyancy, and gravity. We confirmed that the dissipation due to forces on gas bubbles was quantitatively minor.

Integrability, breather, rogue wave, lump, lump-multi-stripe, and lump-multi-soliton solutions of a (3 + 1)-dimensional nonlinear evolution equation

In this article, we consider a new (3 + 1)-dimensional evolution equation, which can be used to interpret the propagation of nonlinear waves in the oceans and seas. We effectively investigate the integrable properties of the considered nonlinear evolution equation through several aspects. First of all, we present some elementary properties of multi-dimensional Bell polynomial theory and its relation with Hirota bilinear form. Utilizing those relations, we derive a Hirota bilinear form and a bilinear Bäcklund transformation. By employing the Cole–Hopf transformation in the bilinear Bäcklund transformation, we present a Lax pair. Additionally, using the Bell polynomial theory, we compute an infinite number of conservation laws. Moreover, we obtain one-, two-, and three-soliton solutions explicitly from Hirota bilinear form and illustrate them graphically. Breather solutions are also derived by employing appropriate complex conjugate parameters in the two-soliton solution. Choosing the generalized algorithm for rogue waves derived from the N-soliton solution, we directly obtain a first-order center-controllable rogue wave. Lump solutions are formulated by employing a well-established quadratic test function as a solution to the Hirota bilinear form. Further taking the test function in a combined form of quadratic and exponential functions, we obtain lump-multi-stripe solutions. Furthermore, a combined form of quadratic and hyperbolic cosine functions produces lump-multi-soliton solutions. The fission and fusion effects in the evolution of lump-multi-stripe solutions and lump-soliton-solutions are demonstrated pictorially.

The importance of the inertial coupling in the two-fluid model of two-phase flow

The new flux representation of the two-fluid model of two-phase flow, where the mixture is described in terms of the volumetric and drift fluxes, is currently the most consistent formulation to treat the inertial coupling between phases. In this representation, the dynamics of the relative motion between phases is revealed as a non-linear wave propagation equation. It is shown that the character and stability of this equation is determined by the balance between the inertial coupling and the interfacial drag. A novel stability criterion is derived that can be used to assess the interfacial closure laws and as a tool to determine the conditions under which a drift-flux correlation is stable. A family of inertial coupling functions for vertical two-phase flow, based on topologies of bubble's vortical wakes, is derived and the corresponding coupling parameters are assessed using available experimental data. The resulting stability maps reveal the occurrence of an unstable region at intermediate void fractions bound by a bistable condition at low and high void fractions, which can be associated with the slug flow-pattern regime.

Wave transmission in 2D nonlinear granular-solid composite systems

Wave propagation in granular media composed of contacting discrete elastic granules attracted considerable attention due to its highly tunable acoustic properties. While prior investigations focused on granular systems with natural boundaries (e.g., fixed or free), our study delves into two-dimensional nonlinear wave propagation within hybrid metamaterials composed of ordered granular media interacting with linearly elastic solids, employing discrete element modeling for the granular media and finite element analysis for the elastic solids. Challenges resulting from numerical instabilities arise from non-smooth contact nonlinearities and friction at granular and granular–solid interfaces. To tackle the numerical challenges, we developed an interrelated interpolation-iteration algorithm with a self-adaptive time scheme. Special consideration was given to the convergence of contact forces at the granular–solid interface. Specifically, to ensure robust computational modeling without numerical instabilities, we monitored the eigenvalues of appropriately defined local maps governing the iterative computations of the contact forces. We demonstrated the capacity of these hybrid metamaterials for shock mitigation and non-reciprocal acoustics with properties that are passively tuned (self-adaptive with) energy. Such results and computational methods contribute to the predictive modelling and design of 2D granular media with flexible interfaces, with diverse applications, e.g., in shock protectors and acoustic diodes.

An adaptive harmonic polynomial cell method for three-dimensional fully nonlinear wave-structure interaction with immersed boundaries

To accurately simulate wave-structure interaction based on fully nonlinear potential flow theory, a three-dimensional (3 D) high-order immersed-boundary adaptive harmonic polynomial cell (IB-AHPC) method is proposed. Both the free surface and body surface are immersed in background octree cells that are adaptively refined near the boundaries of interest, thereby dramatically reducing computational costs without loss of accuracy. We also propose an easy-to-implement IB strategy to deal with possible instabilities in the time-domain solution arising from the intersection of Dirichlet–Neumann boundaries. For a linearized problem of wave-wall interaction, a matrix-based stability analysis is performed, providing mathematical support for the robustness of the proposed IB strategy. In contrast to the two-dimensional HPC method, compressed cells are found to offer superior stability compared to stretched cells in the vertical direction, while equal mesh aspect ratio in the horizontal plane is superior. Cubic octree cells are, however, still preferred in practice. The free surface is primarily described by a set of massless background wave markers; however, to address the challenges of IB methods in tracking the free surface evolution near the structure, additional body-fitted wave markers are introduced close to the waterline. The information exchange between these two sets of wave markers is realized by radial basis function (RBF) interpolation. While standard RBF schemes have grid-size-dependent filtering performance, we propose a normalized RBF scheme, which is then optimized in terms of the number of neighboring nodes, a smoothing coefficient and the basis functions. Excellent accuracy properties of the proposed 3 D IB-AHPC method are demonstrated by studying fully nonlinear wave propagation. The method is further applied to study relevant fully nonlinear wave-structure interaction problems, including sloshing in 3 D rectangular tanks and wave diffraction of a bottom-mounted cylinder in regular waves. Satisfactory agreement is demonstrated with existing experimental and numerical results, suggesting that the proposed 3 D IB-AHPC method is a promising potential-flow method in marine hydrodynamics.

Nonlinear ion acoustic waves in dense magnetoplasmas: Analyzing interaction solutions of the KdV equation using Wronskian formalism for electron trapping with Landau diamagnetism and thermal excitations

Nonlinear ion acoustic waves in the presence of electron trapping with Landau quantization and thermal excitations induced smearing effects of the Fermi step function are investigated using the two-fluid theory in the vicinity of white dwarfs. In spite of the fact that the Kortweg de Vries (KdV) equation is derived within the confines of small amplitude approximation used in the reductive perturbation theory (RPT), it is found to admit solutions that exhibit outstanding capability of wave enhancement and have applications in the nonlinear wave propagation. It is found that enhancing the quantizing magnetic field and the electron thermal effects modify the spatial scale of the formation of the novel nonlinear structures reported for our model. Interestingly, it is found that, unlike the solitary and periodic structures, the choice of space and time coordinates affects the structure of these solutions. Interaction of these solutions with a stable nonlinear structure are also studied.

New optical wave solutions of M-fractional nonlinear Schrödinger’s complex hyperbolic model by an efficient method

In this paper, we succeed to obtain the new optical wave solutions to the truncated M-fractional (2+1)-dimensional nonlinear Schrödinger’s complex hyperbolic model by utilizing the Sardar sub-equation method. The obtained solutions are in the form of trigonometric, hyperbolic and mixed form. These solutions have many applications in nonlinear optics, fiber optics, and other areas of physics and engineering where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by two-dimensional, three-dimensional and contour plots. The gained solutions are helpful for the further development of concerned model. Finally, this technique is simple, fruitful and reliable to handle the nonlinear PDEs.

Nonlinear wave propagation in graphene incorporating second strain gradient theory

The exploration of graphene has attracted extensive interest owing to its significant structural and mechanical properties. In this research, we numerically investigate wave propagation in defect-free single-layer graphene, considering its geometrically nonlinear behavior through second-strain gradient elasticity. To capture the geometric nonlinearity, firstly, the nonlinear strain–displacement relations are introduced. The governing equation and associated boundary conditions are derived using Hamilton’s principle. Then, the weak form, including the element matrices, is established. The eigenvalue problem is solved for 2D wave propagation through periodic structures theory. Finally, the dynamical properties such as band structures, mode shapes, energy flow, and wave beaming effects are analyzed. The numerical results reveal that the geometric nonlinearity through the second strain gradient influences the wave propagation characteristics in graphene. The findings are significant and contribute to the understanding of graphene’s dynamic response, with implications for the engineering applications of graphene-based nanostructures.

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

Nonlinear dust-ion acoustic solitary waves for collisional electronegative dusty plasma in the presence of trapped electron distribution

We have investigated the characteristics of nonlinear propagation of dust-ion acoustic solitary waves in collisional electronegative unmagnetized dusty plasma, which consists of trapped electrons, Boltzmann negative ions, mobile positive ions, mobile negative dust particulates, and a uniform background of neutral particles. In account of ion-neutral collisions, the modified Korteweg–de Vries relation has been derived by employing the standard reductive perturbation method. Analytical and numerical solutions of the damped Korteweg–de Vries equation has been presented in which finite difference method is used for numerical solution. On the other hand, the dust charging equation has been solved by using Newton’s Raphson method. It is found that the temperature ratio of free to trapped electrons, ion-neutral collision, concentration of negative ions, dust number density, and dust density perturbation modify the basic properties of the dust-ion acoustic solitary waves. The temporal evolution of dust-ion acoustic solitary waves is crucial as it affects the amplitude and width of wave structure. In addition, the analytical and numerical solutions are compared, and their deviation is graphically illustrated.

Nonlinear wave propagation in a metamaterial film bounded by Kerr-type dielectrics

The nonlinear waves propagating in a metamaterial (MM) film bounded by Kerr-type dielectric media are investigated. It is shown that the waves propagate in surface mode at nearly 95% probability. We calculate the power flow in the MM waveguide and find the existence of forward or backward waves propagating in the guiding film. A thorough investigation of the power confinement factor reveals that it can be tuned by a set of parameters pertaining to the incident frequency and the width of the MM film. We expect the theoretical analysis can provide ideas for potential applications in the field of information processing.