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In this paper, the integrable (4 + 1)-dimensional Fokas equation is investigated. Exploiting a set of non-singular local multipliers, we present a set of local conservation laws for the equation. The nonlocally related partial differential equation (PDE) systems are found. Nine nonlocally related systems are discussed reveal thirty five interesting closed form solutions of the equation. These solutions contain different types of wave solutions, double soliton, multi-solitons, kink and periodic wave solutions. Some of the resulting solutions are graphically illustrated. Furthermore, we apply the sine-cosine method to find other traveling wave solutions for this equation. [ABSTRACT FROM AUTHOR]

Considering the importance of correctly understanding the dynamics of microstructure materials for their applications in related technologies, by eliminating the shortcomings and some overlooked physical concepts in the existing asymmetric elastic theories, we have presented an asymmetric elastodynamic model based on a $ U(1) $ U (1) gauge theory with quantum electrodynamics (QED) structure. Accordingly, we have shown that there is a correspondence between an elastic theory, which can explain the behavior of elastic waves within an asymmetric elastic medium and QED. More specific, we have indicated that the corresponding elastic wave equations are somehow analogous to QED ones. In this regard, by adding vibrational degrees of freedom and introducing a gauge property of the waves of displacement for the waves of rotation, we have generalized and modified the related Cosserat theory (CT) for an elastic environment. Thus on macro scales, the elastic waves can possess the QED treatment. This analogy provides a new paradigm of fermions and bosons. Also, from experimental point of view, we have shown that the behavior of elastic waves in a granular medium is equivalent to behavior of light in dispersive media, which can be explained using QED. Hence, contrary to the Cosserat and discrete models, this amended CT has qualitatively been indicated to be consistent with the corresponding empirical observations. [ABSTRACT FROM AUTHOR]

In this paper, analytic solutions and dynamic properties of a variety of Boussinesq-type equations are established via the complete discrimination system for polynomial method. All the existing single traveling wave solutions to these equations as well as some new solutions are shown, and the Hamiltonian and topological properties to these equations are also presented. Considering the significance of the Boussinesq-type equations, our results would have wide applications in ocean engineering and fluid mechanics, like describing and predicting the solitary and periodic waves in various shallow water models. [ABSTRACT FROM AUTHOR]

The paper studies the influence of the initial inhomogeneous thermal stresses on the dispersion of the axisymmetric longitudinal waves propagating in the bi-layered hollow cylinder within the scope of the 3D linearized theory of elastic waves in initially stressed bodies. The initial static thermal stresses are determined within the scope of the uncoupled classical linear theory of thermo-elasticity and it is assumed that the temperature change in the cylinder is caused by the heating or cooling of the inner and outer surfaces of the cylinder. The discrete analytical solution method is employed for the solution to the corresponding equations of the wave propagation. Concrete numerical results are obtained for the cylinder, the materials of the layers of which are steel and aluminum, as well for the cylinder made of steel only. Numerical results on the influence of the initial inhomogeneous thermal stresses on the dispersion curves related to the first, second and third modes are presented and discussed. In particular, it is established that the cooling or the heating of the outer and inner surfaces of the bi-layered hollow cylinder influences the dispersion curves not only in the quantitative sense, but also in the qualitative sense. [ABSTRACT FROM AUTHOR]

Hyperbolic equations are used in several physical phenomena to describe dynamical processes where information propagates with a finite speed. Recently, the wave equations with damping terms turned out to be fundamental hyperbolic equations in certain branches of physics mainly scattering processes and fractal medium. The aim of the present study is double shooting, first to prove that damped quantum wave equations may be obtained using the notion of non-standard Lagrangians and second to show that linear and nonlinear damping terms may be obtained if the concept of 'two-occurrences of integrals' is used, hence reducing the damped quantum wave equation to the conventional quantum wave equation known as the Klein-Gordon equation. This study supports the idea of non-standard Lagrangians and its usefulness in the theory of partial differential equations. [ABSTRACT FROM AUTHOR]

Lump solutions are empirical rational function solutions found in all directions in space. One of the essential solutions to both linear and nonlinear partial differential equations is lump solutions. The current work studies a class of lump interaction phenomena to the generalized (3 + 1) -dimensional nonlinear-wave equation with time-dependent-coefficient. Variable-coefficient nonlinear partial differential equations offer us with more real aspects in the inhomogeneities of media and nonuniformities of boundaries than their counterparts constant-coefficient in many physical cases. The Hirota bilinear form is the fundamental concept that has been used to derive the novel lump-periodic and breather wave solutions. The acquired solutions are constructed using symbolic computations called Maple. The physical characteristics of the acquired solutions are shown with three-dimensional and contour plots in order to shed more light on the acquired novel solutions. [ABSTRACT FROM AUTHOR]

The present paper deals with the propagation of Rayleigh surface waves in a nonlocal thermoelastic orthotropic layer which is lying over a nonlocal thermoelastic orthotropic half space. The problem is considered in the context of Green-Naghdi type III model of hyperbolic thermoelasticity and Eringen's nonlocal elasticity theory in the presence of voids. The problem is solved using eigenfunction expansion method. Frequency equation of Rayleigh waves is derived and different particular cases are also deduced. The effects of voids and nonlocality on different characteristics of Rayleigh waves are presented graphically. [ABSTRACT FROM AUTHOR]

In situ stress is an important feature of underground rock masses. This paper extends the time-domain recursive method to the study of S-wave propagation through in situ stressed rock masses containing parallel joints. A linear elastic model and hyperbolic nonlinear slip model (HNSM) are first used to establish equations for wave propagation across jointed rock masses under a combination of gravitational and tectonic stress. Then, a comparison is made of the waveforms generated using the HNSM and Mohr–Coulomb slip model. Their differences are investigated and the wave propagation equation verified. Finally, parametric studies are conducted to investigate the effect of joint angle, joint number, in situ stress, and lateral pressure coefficient. The results show that the HNSM can describe the dynamic changes in the stress and their effect on the deformation behavior of the joint during the propagation of the S-wave. The effect of in situ stress on wave propagation is related to the joint angle and lateral pressure coefficient, which determine the initial stress state and contact state of the joint. [ABSTRACT FROM AUTHOR]

Rock joints with unequal close-open behavior have significant influences on stress wave propagation across them. A stress wave of sufficiently large amplitude does not only mobilize the nonlinear deformation of joints, but also force the joints to close and open. In the present study, the Bandis-Barton model and the Coulomb-slip model are adopted to describe the normal and tangential behavior, respectively, of the closed joints. However, when the joints are open, both sides of the joints behave as two free surfaces. The improved time-domain recursive method is employed to derive wave propagation equations for three cases, i.e. the joint is closed but does not slide, the joint is closed and slides, and the joint is open. Then, the normal and tangential relative displacements of joints, as well as the energy attenuation coefficient, are calculated. A comparison of transmitted waves based on different methods is conducted to verify the validity of the present approach. Eventually, parametrical studies show that the higher in-situ stress causes the smaller energy attenuation coefficient, and the wave energy attenuation is also dependent on the frequency and amplitude of the incident wave, the incident angle, the joint spacing and the joint number. [ABSTRACT FROM AUTHOR]

In this study, we argue that it is possible to construct non-local nonlinear wave equations with random propagation as extensions to some well-known wave equations found in field theory with cubic interactions like Maxwell–Klein–Gordon equations, Dirac–Klein–Gordon equations, wave maps, solitons and Yang–Mills equations. Our construction is based on the extended complex backward–forward derivative operator which represents the basic differentiable operator tool to deal with non-local Lagrangians. [ABSTRACT FROM AUTHOR]

In contrast to conventional metallic antennas, the resonant wavelength of a plasma dipole antenna is very much different from the predicted ∼λ/2 rule. In this paper, an implicit relation is derived to calculate the effective wavelength λeff and thus the resonant wavelength of a single element 3-layered plasma dipole antenna made of a glass tube filled with the plasma gas. An explicit relation is then derived in terms of the λ using reasonable assumptions. It is shown that this relation could be modeled by a linear formula λeff = m1+m2λ in which the coefficients m1 and m2 depend on the geometry and material properties. Comparison of the results from the full-wave electromagnetic simulation, the numerical solution of the surface wave dispersion equations and the derived relations shows a reasonable match, verifying the theoretical approach. [ABSTRACT FROM AUTHOR]

Rock joints, usually with nonlinear deformation, widely exist in many natural rock masses, which greatly affect stress wave propagation. In practical engineering, the generated stress wave is taken as a cylindrical wave when an explosion with cylindrical charge or a tunneling explosion occurs. The cylindrical wave propagation in jointed rock mass is a great concern of underground structure safety. In this paper, the time-domain recursive method (TDRM) is extended to investigate cylindrical stress wave propagation through a rock joint with nonlinear normal deformation. Based on the conservation of momentum at the wave fronts and the displacement discontinuity method, a quantitative analysis for the interaction between wave and the joint is carried out and the wave propagation equation is accordingly deduced by considering the attenuation. Verification of the TDRM-based results is conducted by comparison with the simulated results from UDEC code. Finally, parametric studies are carried out, which include the influences of joint stiffness, the maximum allowable closure of joint and the frequency of the incident wave on the transmission and reflection coefficients. [ABSTRACT FROM AUTHOR]

The present paper employs variable separable technique to investigate the effect of corrugated boundary surface along with other affecting parameters on the phase and damped velocity of SH-wave in a three medium mathematical model, i.e. a heterogeneous viscoelastic layer with corrugated boundary surface sandwiched between a heterogeneous isotropic half-space and an anisotropic magnetoelastic monoclinic half-space. The dispersion and absorption equations have been obtained in the closed form. Some special cases of the dispersion equation have been obtained, which are in good agreement with the Classical Love Wave Equation. The absence and presence of corrugated frontier surface are determined by the term initial flatness parameter. Considerable effects of some parameters such as initial flatness parameter, heterogeneity, magneto elasticity as well as viscoelasticity associated with the layers have been investigated for both the curves, i.e. dispersion curve and attenuation curve which corresponds to the dispersion equation and the absorption equation, respectively. Vital strides have been made by the integration of both the numerical computation and graphical plots to demonstrate those effects to accomplish the objective of the problem. [ABSTRACT FROM AUTHOR]

In this paper, the symmetry analysis and conservation laws of time fractional coupled Ablowitz–Kaup–Newell–Segur (AKNS) equations have been presented. Initially, the Lie symmetry method has been used for similarity reduction of time fractional coupled AKNS equations. Here, the Erdélyi–Kober differential operators have been used for symmetry reduction. Also, the new conservation vectors for time fractional coupled AKNS equations have been derived by using the new conservation theorem with formal Lagrangian. [ABSTRACT FROM AUTHOR]

This paper addresses the (2+1)-dimensional cubic-quintic nonlinear Schrödinger equation (CQNLS) that serves as the model to study the light propagation through nonlinear optical media and non-Kerr crystals. A dark–bright optical solitary wave solution of this equation is retrieved by adopting the complex envelope function ansatz. This type of solitary wave describes the properties of bright and dark optical solitary waves in the same expression. The integration naturally lead to a constraint condition placed on the solitary wave parameters which must hold for the solitary waves to exist. Additionally, the modulation instability (MI) analysis of the model is studied based on the standard linear stability analysis and the MI gain spectrum is got. Numerical simulation and physical interpretations of the obtained results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the CQNLS. [ABSTRACT FROM AUTHOR]

In a previous paper, by combining three techniques, i.e. Subdomain Decomposition Iterative Method (SDIM), Adaptive Cross Approximation (ACA), and Forward-Backward Spectral Acceleration (FBSA), from the Method of Moments (MoM), a high-efficiency calculation of the propagation and scattering in ducting maritime environments has been proposed. In this paper, this algorithm is updated by adding a perfectly conducting object above the sea surface, assumed to be highly conducting, which makes the environment very complex. Then, to quantify the effect of the object on the total scattered field, the coherent and incoherent powers, with and without object, are simulated by considering a surface of 800,000 unknowns (length of 6 km and a frequency of 5 GHz). [ABSTRACT FROM AUTHOR]

The article studies the dynamics of solitons in electrical microtubule model, which describes the propagation of waves in nonlinear dynamical system. Microtubules are not only a passive support of a cell but also they have highly dynamic structures involved in cell motility, intracellular transport and signaling. The underlying model has been considered with constant and variable coefficients of time function. The solitary wave ansatz has been applied successfully to extract these solitons. The corresponding integrability criteria, also known as constraint conditions, naturally emerge from the analysis of these models. [ABSTRACT FROM AUTHOR]

Wave propagation modeling as a vital tool in seismology can be done via several different numerical methods among them are finite-difference, finite-element, and spectral-element methods (FDM, FEM and SEM). Some advanced applications in seismic exploration benefit the frequency domain modeling. Regarding flexibility in complex geological models and dealing with the free surface boundary condition, we studied the frequency domain acoustic wave equation using FEM and SEM. The results demonstrated that the frequency domain FEM and SEM have a good accuracy and numerical efficiency with the second order interpolation polynomials. Furthermore, we developed the second order Clayton and Engquist absorbing boundary condition (CE-ABC2) and compared it with the perfectly matched layer (PML) for the frequency domain FEM and SEM. In spite of PML method, CE-ABC2 does not add any additional computational cost to the modeling except assembling boundary matrices. As a result, considering CE-ABC2 is more efficient than PML for the frequency domain acoustic wave propagation modeling especially when computational cost is high and high-level absorbing performance is unnecessary. [ABSTRACT FROM AUTHOR]

This study reports a theoretical investigation of the propagation of SH-wave in a piezoelectric layer superimposed on a self-reinforced layer overlying an isotropic gravitational half-space. The expressions of the dispersion relation of SH-wave have been established for electrically open and electrically short conditions in closed form. For the purpose of numerical computation, lithium niobate piezoelectric material has been considered. The dispersion curves have been depicted graphically and the prominent impacts of piezoelectric constant, dielectric constant, reinforced parameter, width ratio, and Biot’s gravity parameter on the phase velocity of SH-wave have been unraveled for both the electrical conditions. As a special case of the problem, it is found that the obtained dispersion relation concurs with classical Love wave equation for both the electrical conditions. Moreover, some important peculiarities have also been traced out through numerical computations for both the electrical cases. [ABSTRACT FROM PUBLISHER]

A mathematical model is developed in which the effect of imperfect bonding between the constituents of layer and half-space on the phase velocity and damped velocity of SH-wave is discussed. The model consists of a micropolar elastic half-space bonded imperfectly with a heterogeneous viscoelastic layer. The dispersion equation and damping equation of SH-wave propagation in the said model is obtained in the closed form analytically. The effects of imperfect bonding, internal friction, heterogeneity, micropolarity, and complex interface stiffness parameters highlighted through numerical computation and graphical demonstrations. Standard Love-wave equation and dispersion equation as well as damping equation for perfectly bonded micropolar half-space with heterogeneous viscoelastic layer is obtained as a special case of the problem. Through comparative study of homogeneity with heterogeneity in the layer; imperfect bonding of layer and half-space with their welded (perfect) contact; and presence of micropolarity in half-space with its absence in half-space are compared meticulously. [ABSTRACT FROM PUBLISHER]

This paper investigates the impact of corrugated boundary surfaces, reinforcement on the propagation of Love-type wave in prestressed corrugated heterogeneous fiber-reinforced layer resting over a void pores half-space. The heterogeneity in the upper corrugated layer is caused due to exponential variation in the elastic constants with respect to the space variable pointing positively downwards. The dispersion equation in the complex form has been derived using method of separation of variables. The real and imaginary parts of the complex dispersion equation were separated and found in well agreement with the classical Love wave equation. Also, the attenuation of the Love waves has been discussed. The study reveals that such a medium transmits two fronts of Love waves. The first front depends upon the change in volume fraction of the pores and the second front depends upon the modulus of rigidity of the elastic matrix of the medium. The substantial influence of corrugation parameters, reinforcement, undulatory parameter, initial stress, heterogeneity parameter and position parameter on the phase velocity, and attenuation of Love-type wave have been observed and depicted by means of graph. It has been observed that the phase velocity decreases with the increase in initial stress parameters, heterogeneity, and reinforcement in upper layer. [ABSTRACT FROM PUBLISHER]

While taking into account thermal motion of electrons, scattering of electromagnetic waves with low frequency from hot plasma spheroids is investigated. In this theoretical research, ions are heavy torespondto electromagnetic fluctuations. The solution of scalar wave equation in spheroidal coordinates for electric potential inside the plasma spheroids are obtained. The variations of resonance frequencies vs. Debye length are studied and consistency between the obtained results in this paper and the results for the well-known plasma objects such as plasma column and spherical plasma have been proved. [ABSTRACT FROM PUBLISHER]

The secular equation of Rayleigh propagating in an orthotropic half-space coated by an orthotropic layer has been obtained by Sotiropolous [Sotiropolous, D. A. (1999), The e®ect of anisotropy on guided elastic waves in a layered half-space, Mechanics of Materials 31, 215–233] and by Sotiropolous & Tougelidis [Sotiropolous, D. A. and Tougelidis, G. (1998), Guided elastic waves in orthotropic surface layer, Ultrasonics 36, 371–374]. However, it is not totally explicit and some misprints have occurred in this secular equation in both papers. This secular equation was derived by expanding directly a six-order determinant originated from the traction-free conditions at the top surface of the layer and the continuity of displacements and stresses through the interface between the layer and the half-space. Since the expansion of this six-order determinant was not shown in both two papers, it has been difficult to readers to recognize these misprints. This paper presents a technique that provides a totally explicit secular equation of the wave. The technique makes clear the way from the traction-free and continuity conditions to the secular equation and enables us to recognize the misprints appearing in the reported secular equation. The technique can be employed to obtain explicit secular equations of Rayleigh waves for many other cases. Moreover, the paper introduces a transfer matrix in explicit form for an orthotropic layer that is much simpler in form than the one obtained previously. [ABSTRACT FROM AUTHOR]

In this article, the bright, dark, and singular solitons are being constructed for nonlinear longitudinal wave equation with dispersion caused by transverse Poisson’s effect in a magneto-electro-elastic circular rod. The solitary wave ansatz is used to carry out these solutions. The constraint conditions, for the existence of the soliton solutions, are also listed. This article provides a lot of encouragement for the researchers in this era. [ABSTRACT FROM AUTHOR]

This paper introduces a systematic approach to investigate a higher order nonlinear dispersive wave equation for modeling different wave modes. We present both the conventional KdV-type soliton and anomaly type solitons for the equation. We also show the conservation laws and Hamiltonian structures for the equation. Our results suggest that the underlying equation has more interacting soliton phenomena than one would have known for the classical KdV and Boussinesq equation. [ABSTRACT FROM PUBLISHER]

In this work, we establish exact solutions for (2 + 1)- and (3 + 1)-dimensional shallow water wave equations. The -expansion method is used to construct travelling wave solutions of nonlinear evolution equations. The travelling wave solutions are expressed in terms of hyperbolic functions, trigonometric functions and rational functions. This method presents a wider applicability for handling nonlinear wave equations. [ABSTRACT FROM AUTHOR]

A hybrid finite element–boundary integral–characteristic basis function method (FE-BI-CBFM) is proposed for an efficient simulation of electromagnetic scattering by random discrete particles. Specifically, the finite element method (FEM) is used to obtain the solution of the vector wave equation inside each particle and the boundary integral equation (BIE) using Green's functions is applied on the surfaces of all the particles as a global boundary condition. The coupling system of equations is solved by employing the characteristic basis function method (CBFM) based on the use of macro-basis functions constructed according to the Foldy–Lax multiple scattering equations. Due to the flexibility of FEM, the proposed hybrid technique can easily deal with the problems of multiple scattering by randomly distributed inhomogeneous particles that are often beyond the scope of traditional numerical methods. Some numerical examples are presented to demonstrate the validity and capability of the proposed method. [ABSTRACT FROM AUTHOR]

Frequency-domain finite-difference (FDFD) modelling is widely used for multi-source experiments modelling and full waveform tomography. In this paper, a frequency-domain finite-difference package written in MATLAB is presented which solves 2D visco-acoustic wave equation. The mixed-grid stencil is used as a state-of-the-art finite-differencing approach and SuitSparseQR solver is utilised for solving the large linear system of equations. Because of the independence of frequency components and the use of TBB-enabled SuitsparseQR solver, the package benefits from parallel computation in multi-core machines. Using MATLAB, codes became more readable and using different visualisation facilities inside MATLAB made this package very useful for research purposes. This package uses a PML absorbing boundary and supports anti-time aliasing and reduction velocity technique. Different attenuation mechanisms can easily be implemented. The performance of codes are examined on simple and complicated models which proved satisfactory in terms of accuracy and required CPU time, both in single and multi-source cases. [ABSTRACT FROM AUTHOR]

The method of small perturbations has been developed for the wave equation of the most general type, and applied for studying the propagation, scattering and transformation of scalar and vector waves in weakly inhomogeneous dispersive media. Both periodic and random inhomogeneities have been considered. [ABSTRACT FROM AUTHOR]

The propagation of waves through a doubly-periodic array of identical rigid scatterers is considered in the case that the field equation is the two-dimensional Helmholtz equation. The method of matched asymptotic expansions is used to obtain the dispersion relation corresponding to wave propagation through an array of scatterers of arbitrary shape that are each small relative to both the wavelength and the array periodicity. The results obtained differ from those obtained from homogenization in that there is no requirement that the wavelength be much smaller than the array periodicity, and hence it is possible to examine phenomena, such as band gaps, that are associated with the array periodicity. [ABSTRACT FROM AUTHOR]

The propagation of a transient electromagnetic pulse over irregular terrain is considered. We model the wave propagation using the parabolic wave equation, which is valid for near-horizontal propagation. We model the effect of scattering from the rough terrain by introducing a surface-flattening coordinate transform. This coordinate transform simplifies the boundary condition of our problem, and introduces an effective refractive index into our wave equation. As a result, the problem of propagation over an irregular surface becomes equivalent to the problem of propagation through random media. The parabolic equation is solved analytically using the path integral method. Both vertically polarized and horizontally polarized signals are treated. Cumulant expansion is introduced to obtain an approximate expression for the two-frequency mutual coherence function. From the mutual coherence function, spatial and temporal dependence of the propagating signal can be determined. It can be shown that scattering from the irregular surface can cause broadening of the transient signal. This can have a significant impact on the performance of radio communication systems. [ABSTRACT FROM AUTHOR]

Research is carried out into scattering of waves by random strong anisotropic inhomogeneities when the inhomogeneities are in the distant zone according to one (transverse) scale and at the same time they are in the near zone according to another (longitudinal) scale. To analyse the formulas of the single scattering the stationary phase method in the longitudinal coordinate integral is used. It is shown that the angle sensitivity of strong anisotropic scattering, unlike weak anisotropic scattering, strongly depends on the longitudinal statistical homogeneity of the medium. [ABSTRACT FROM AUTHOR]

Electromagnetic localization and the existence of gap solitons in nonlinear metamaterials, which exhibit a stop band in their linear spectral response, is theoretically investigated. For a self-focusing Kerr nonlinearity, the equation for the electric field envelope with carrier frequency in the stop band—where the magnetic permeability µ(?) is positive and the dielectric permittivity Ƹ(?) is negative—is described by a nonlinear Klein–Gordon equation with a dispersive nonlinear term. A family of standing and moving localized waves for both electric and magnetic fields is found, and the role played by the nonlinear dispersive term on solitary wave stability is discussed. [ABSTRACT FROM AUTHOR]

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law. [ABSTRACT FROM AUTHOR]