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In a recent work, assuming a Beer–Lambert optical absorption and a Gaussian laser time profile, it was shown that the exact solutions for a 1D photoacoustic (PA) boundary-value-problem predict a null pressure for optically strong absorbent materials. In order to overcome this inconsistency, a heuristic correction was introduced by assuming that heat flux travels a characteristic length during the duration of the laser pulse [M. Ruiz-Veloz et al., J. Appl. Phys. 130, 025104 (2021)] τ p. In this work, we obtained exact analytical solutions in the frequency domain for a 1D boundary-value-problem for the Dual-Phase-Lag (DPL) heat equation coupled with a 1D PA-boundary-value-problem via the acoustic wave equation. Temperature and pressure solutions were studied by assuming that the sample and its surroundings have a similar characteristic thermal lag response time τ T ; therefore, the whole system is assumed to have a similar thermal relaxation. A second assumption for τ T is that it is considered as a free parameter that can be adjusted to reproduce experimental results. Solutions for temperature and pressure were obtained for a one-layer 1D system. It was found that for τ T < τ p , the DPL temperature has a similar thermal profile of the Fourier heat equation; however, when τ T ≥ τ p , this profile is very different from the Fourier case. Additionally, via a numerical Fourier transform, the wave-like behavior of DPL temperature is explored, and it was found that as τ T increases, thermal wave amplitude is increasingly attenuated. Exact solutions for pressure were compared with experimental PA signals, showing a close resemblance between both data sets, particularly in time domain, for an appropriated value of τ T ; the transference function was also calculated, which allowed us to find the maximum response in frequency for the considered experimental setup. [ABSTRACT FROM AUTHOR]

Transient reflectivity measurements are used to probe the strain waves induced by ultrashort laser pulses in bulk [100] germanium. The measurement signals are compared to purely analytical model functions based on the known material parameters for germanium. The modeling includes (i) a derivation of analytical solutions to the wave equation for strain waves coupled to the diffusion equation for heat and charge carriers and (ii) an expression for the impact on reflection coefficients that are caused by perturbations to the dielectric function but extended to cover a non-isotropic, uniaxial dielectric tensorial form. The model is held up against transient reflectivity measurements with an s- and a p-polarized probe and with a probe wavelength in the range of 502–710 nm. Excellent agreement is found when comparing the oscillatory shape of the measurement signals to the models. As for the magnitude of the oscillations, the models reproduce the overall trends of the experiment when using the previously published values for the elasto-optical tensor measured under static strain. [ABSTRACT FROM AUTHOR]

The contribution provided in this paper is to investigate the dynamic response of bidirectional graded material beams with porosities. Higher order shear deformation beam theory for wave propagation in porous functionally graded cantilevered beam are developed, taking into account the bidirectional distribution, which is mainly represented in the density and the Young's modulus. The material properties of the porous FG beam are graded through the thickness and width using a power law distribution. The governing equations of the wave propagation in the porous FG beam are derived by employing the Hamilton's principle. The analytic dispersion relation of the bidirectional porous FG beam is obtained by solving an eigenvalue problem. Two porosities models approximating the even and uneven distributions of porosity are considered. The results obtained for bidirectional FG beam were compared with those reported in the literature, and the results were very agreement. The effects of the volume fraction distributions, the number of wave propagation, various parameters and the porosity type on dynamic of functionally graded beam are discussed in details. The numerical results show that the power law index, number of wave, wave velocity, and porosity distribution model affect the dynamic behavior of the beam significantly. The results show that the uneven porosity predicts higher natural frequencies than the even porosity. [ABSTRACT FROM AUTHOR]

In this study, the modified exponential approach is used to investigate the fractional order longitudinal wave equation in a magneto-elastic circular rod, which represents the nonlinear interplay between dispersion and longitudinal wave velocity depending on the rod's material and geometry. The time-fractional order is used in the standpoint of conformable derivative. Comprehensive and descriptive waveform solutions, including kink-shaped, bell-shaped, and bright bell-shaped solitons, are obtained. The produced soliton solutions are advanced, unique, developed, and crucial. To demonstrate the physical appearance of the obtained solutions, 2D, 3D, and contour graphs of some of the obtained solutions are plotted. The extracted solutions demonstrate the physical significance of the model and express the pressure and electrostatic potential for the longitudinal wave equation and can be used to elucidate more complicated nonlinear models of classical and fractional order. The obtained results are helpful in finding the amplitude of tsunami waves which is of great interest to researchers in the field of ocean engineering. If a tsunami extends across a large area, it becomes a manifestation of solitons. The comparison of the obtained solutions is also given in the form of remarks 1 and 2 which shows the novelty of our work. [ABSTRACT FROM AUTHOR]

Nonlinear partial differential equations (NLPDEs) have emerged as a major focus of study in a variety of nonlinear science disciplines. This is because in general, mathematical, physics and engineering problems may be stated by NLPDEs. It may be noted that a specific kind of exact/analytical solution called traveling wave solutions for NLPDEs has great significance. In this regard, this paper addresses the traveling wave solution to the new coupled Konno–Oono equation (NCKOE), a set of NLPDE. A traveling wave transformation and the Riccati–Bernoulli sub-ode method (RBSOM) are used here to retrieve the exact/analytical traveling wave solution of the NCKOE. By using the above-mentioned approach, NLPDEs can be updated toward a collection of algebraic equations. Further, different types of solitary wave solitons, Trigonometric function solutions, Hyperbolic soliton solutions and dark bright solitons to the NCKOE have been successfully retrieved. Importantly, the newly derived solutions adhere to the main equation when they are inserted into the governing equations. Furthermore, through an appropriate selection of parameters, three-dimensional and two-dimensional figures are presented for physical illustration of the obtained solutions. These visual representations serve to showcase the effectiveness, conciseness and efficiency of the applied techniques. [ABSTRACT FROM AUTHOR]

In this paper we study the stability of cylinder front waves and the propagation of solutions of a nonlocal Fisher-type model describing the propagation of a population with nonlocal competition among bounded and continuous phenotypic traits. By applying spectral analysis and separation of variables we prove the spectral and local exponential stability of the cylinder waves with the noncritical speeds in some exponentially weighted spaces. By combining the detailed analysis with the spectral expansion and the special construction of sub-supersolutions, we further prove the uniform boundedness of the solutions and the global asymptotic stability of the cylinder waves for more general nonnegative bounded initial data, and prove that the spreading speeds and the asymptotic behavior of the solutions are determined by the decay rates of the initial data. Our results also extend some classical results on the stability of planar waves for Fisher-KPP equation to the nonlocal Fisher model in multi-dimensional cylinder case. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the asymptotic behavior of solutions to the non-autonomous stochastic wave lattice systems with random viscosity and multiplicative white noise. Under proper conditions, we first prove the existence and uniqueness of the pullback random attractors and then show their backward compactness in a weighted space. Finally, we establish the asymptotically autonomous robustness of pullback random attractors when the nonlinearity is bounded and the time-dependent forcing term converges to the time-independent counterpart. [ABSTRACT FROM AUTHOR]

We consider the CH–KP-I equation. For this equation, we prove the existence of steady solutions, which are solitary in one horizontal direction and periodic in the other. We show that such waves bifurcate from the line solitary wave solutions, i.e. solitary wave solutions to the Camassa–Holm equation, in a dimension-breaking bifurcation. This is achieved through reformulating the problem as a dynamical system for a perturbation of the line solitary wave solutions, where the periodic direction takes the role of time, then applying the Lyapunov–Iooss theorem. [ABSTRACT FROM AUTHOR]

In this paper, we shall deal with the long‐time dynamics of a nonlinear system of wave equation with a fractional damping and a small delay. Inspired by the works of Chueshov and Lasiecka on the property of quasi‐stability of dynamic systems, we show first the quasi‐stability property holds for the problem considered here and then prove the existence of global and exponential attractors. We shall also prove the upper semicontinuities of global attractors with respect to the fractional exponents. Finally, we shall show the continuities of global attractors with respect to a small parameter multiplying the delay term in a residual dense set and their upper semicontinuities in a complete metric space. [ABSTRACT FROM AUTHOR]

We prove that the uniform radius of spatial analyticity σ(t)$$ \sigma (t) $$ of solution at time t$$ t $$ to the one‐dimensional fourth‐order nonlinear Schrödinger equation i∂tu−∂x4u=|u|2u$$ i{\partial}_tu-{\partial}_x&amp;#x0005E;4u&amp;#x0003D;{\left&amp;#x0007C;u\right&amp;#x0007C;}&amp;#x0005E;2u $$cannot decay faster than 1/t$$ 1/\sqrt{t} $$ for large t$$ t $$, given that the initial data are analytic with fixed radius σ0$$ {\sigma}_0 $$. The main ingredients in the proof are a modified Gevrey space, a method of approximate conservation law, and a Strichartz estimate for free wave associated with the equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the boundary value problem of the wave equation with nonlocal damping ‖ut‖2qut$$ {\left\Vert {u}_t\right\Vert}_2&amp;#x0005E;q{u}_t $$ and nonlinear source term. The main purpose of this paper is to provide a systematic research on the dynamic behavior of the solutions with three different energy levels. More precisely, we prove the global existence, energy decay estimate and blow‐up of solution at both subcritical (E(0)

In this paper, we consider a wave equation with nonlinearities of variable exponents and acoustic boundary conditions. Using the multiplier method, we establish general stability results to the equation when the damping term has a time‐dependent coefficient and the exponent has extended range. [ABSTRACT FROM AUTHOR]

The general solutions for nonlinear waves in a bubble-liquid mixture are obtained from the Kudryashov–Sinelshchikov (KS) equation under different parametric regimes. The Chiellini integrability condition has been used for constructing exact general solutions. In the non-dissipative case, we find that the solutions may be expressed in terms of Jacobi elliptic and Weierstrass functions. On the other hand, in the dissipative case, only implicit solutions are obtained for the KS equation. For an alternative perspective, the notion of the Jacobi Last Multiplier has been utilized to obtain the corresponding Lagrangian and Hamiltonian description of the reduced equation for the bubble-liquid mixture system. [ABSTRACT FROM AUTHOR]

In situ stress is a significant characteristic of underground rock masses. In this work, based on the time-domain recursive method (TDRM), a new theoretical approach is proposed to study the oblique propagation of an S-wave across an in situ stressed joint wherein the normal and shear deformations are both treated nonlinearly. Equations are first established for the propagation of stress waves across a rock mass subjected to a combination of gravitational and tectonic stress. The veracity of the wave propagation equations is then investigated by comparing our theoretical results with those obtained using the Universal Distinct Element Code (UDEC). A comparison of the waveforms predicted using the continuously yielding and existing models is also presented to investigate their differences. Finally, parametric studies are conducted to investigate the effects of joint angle, in situ stress, lateral pressure coefficient, and incident wave amplitude on the propagation of the wave. The results show that the effect of in situ stress on wave propagation is related to the joint angle and lateral pressure coefficient because these factors decide the initial stress and contact states of the joint. [ABSTRACT FROM AUTHOR]

We address a system of equations modeling an incompressible fluid interacting with an elastic body. We prove the local existence when the initial velocity belongs to the space H 1.5 + ϵ and the initial structure velocity is in H 1 + ϵ , where ϵ ∈ (0 , 1 / 20) . [ABSTRACT FROM AUTHOR]

The asymmetric wave equation encompasses the influence of the actual fine structure inside the medium on the wave field, which can better represent the complex seismic wavefield excited by the complex source and reflect the scale effects of the seismic wave response under equal computational power. However, when the finite-difference (FD) operator is applied to implement the numerical modelling using the asymmetric wave equation, numerical dispersion appears due to the use of difference operator to approximate the differential operator, which negatively affects the analysis of the seismic wavefield. To suppress the numerical dispersion, this paper proposes an improved Dung Beetle optimization (IDBO) algorithm to obtain the optimized FD operators. The IDBO algorithm adopts an improved Tent map and the opposition-based learning strategy to initialize the population, which improves the diversity of the population. The nonlinear function adaptive control strategy is introduced to adjust the population allocation ratio and boundary selection parameter R to achieve an adequate balance between global exploration and local exploitation. In addition, adaptive weights and the Levy flight mechanism are combined to improve the ball-rolling dung beetle position updating strategy to avoid falling into local extremes. Numerical dispersion analysis and numerical modelling results demonstrate that the optimization of FD operators based on the IDBO algorithm can effectively suppress numerical dispersion. It is of great significance to extract the wave field perturbation caused by heterogeneity due to the complex microstructure in the medium and analyse the influence of the microstructural properties in the medium on seismic wave propagation. [ABSTRACT FROM AUTHOR]

The aim of this work is to construct and analyze a discretization process that preserves exponential stability at the discrete level for a wave propagation problem with boundary damping when a high-order spectral finite element approximation is used. The main idea is to add a stabilizing term to the wave equation that damps the spurious oscillatory components of the solutions. This term is based on a discrete multiplier analysis that ensures the exponential stability of the semi-discrete problem at any order without compromising the rate of convergence of the discretization strategy. Possible applications of this work concern the solution of boundary control problems or data assimilation problems whose efficiency is based on an exponential stability property of an underlying discrete error system. [ABSTRACT FROM AUTHOR]

We investigate a variable-coefficient wave equation with interior local frictional damping and memory-type dynamic boundary conditions. The innovation of the paper lies in the presence of dynamic boundary conditions, thus we need some special techniques to deal with the high-order terms on the boundary. By the Riemannian geometry method and the multiplier technique, we establish the stable results, and the energy decay result of system is described by employing the solution of a stable first-order ordinary differential equation. The stability depends on the frictional damping function and kernel function, thereby improving and generalizing the results of prior references. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the well-posedness and stabilization of the generalized Rao-Nakra beam equation. The system consists of four wave equations for the longitudinal displacements and the shear angle of the top and bottom layers and one Euler-Bernoulli beam equation for the transversal displacement. First, we prove that the system is well-posed. Second, we show that the analytic stability holds when all the displacements are globally damped through Kelvin-Voigt damping. Third, we establish the case where the damping acts only on the shear angle displacements of the top and bottom layers. In this case, we prove, by using frequency domain arguments combined with the multiplier method, that the energy of the system decays polynomially with a decay rate $ t^{-\frac{1}{2}} $. [ABSTRACT FROM AUTHOR]

Mathematical simulation has significantly broadened with the advancement of parallel computing, particularly in its capacity to comprehend physical phenomena across extensive temporal and spatial dimensions. Highperformance parallel computing finds extensive application across diverse domains of technology and science, including the realm of acoustics. This research investigates the numerical modeling and parallel processing of the two-dimensional acoustic wave equation in both uniform and non-uniform media. Our approach employs implicit difference schemes, with the cyclic reduction algorithm used to obtain an approximate solution. We then adapt the sequential algorithm for parallel execution on a graphics processing unit (GPU). Ultimately, our findings demonstrate the effectiveness of the parallel approach in yielding favorable results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the existence and stability of standing waves with prescribed mass for the Hartree-Fock type nonlocal elliptic equation with rotation$ \begin{equation*} \begin{cases} i\partial_{t}u = -\frac{1}{2}\Delta u+V(x)u-\Omega L_{z}u+\lambda u-\mu(J_{\alpha}\ast|u|^{p})|u|^{p-2}u,\ (t,x)\in \mathbb{R}\times\mathbb{R}^{N},\\ u(0,x) = u_{0}(x), \end{cases} \end{equation*} $where $ N\geq3 $, $ \mu>0 $, $ V(x) = \frac{1}{2}\sum\limits_{j = 1}^{N}\gamma_{j}^{2}x_{j}^{2} $, $ L_{z} $ is the angular momentum operator with a rotational speed $ \Omega>0 $, $ \gamma_{j}>0 $ is the trap frequencies, and the constant $ \lambda $ is unknown Lagrange multiplier. We establish the existence and orbital stability results of prescribed mass standing waves for the equation under two distinct scenarios: one in the subcritical case where $ \frac{N+\alpha}{N}

This paper is concerned with the long term dynamics of the supercritical stochastic wave equation driven by multiplicative noise on unbounded domains. By introducing an appropriate variable, we convert the stochastic equation into a pathwise deterministic wave equation and prove the existence and uniqueness of solutions by the uniform Strichartz estimates. Then we prove the existence and uniqueness of pullback random attractors of the non-autonomous random dynamical system associated with the stochastic equation. The pullback asymptotic compactness of solutions is established by the idea of uniform tail-ends estimates as well as the decomposition method of the solution operator in order to overcome the non-compactness of Sobolev embeddings on unbounded domains. [ABSTRACT FROM AUTHOR]

A coupled system consisting of a quasilinear parabolic equation and a semilinear hyperbolic equation is considered. The problem arises as a small aspect ratio limit in the modeling of a MEMS device taking into account the gap width of the device and the gas pressure. The system is regarded as a special case of a more general setting for which local well-posedness of strong solutions is shown. The general result applies to different cases including a coupling of the parabolic equation to a semilinear wave equation of either second or fourth order, the latter featuring either clamped or pinned boundary conditions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the time‐dependent acoustic–elastic interaction problem associated with a bounded elastic body immersed in a homogeneous air or fluid above an unbounded rough surface. The well‐posedness and stability of the problem are first established by using the Laplace transform and the energy method. A perfectly matched layer (PML) is then introduced to truncate the interaction problem above a finite layer containing the elastic body, leading to a PML problem in a finite strip domain. We further establish the existence, uniqueness, and stability estimate of solutions to the PML problem. Finally, we prove the exponential convergence of the PML problem in terms of the thickness and parameter of the PML, based on establishing an error estimate between the DtN operators of the original problem and the PML problem. [ABSTRACT FROM AUTHOR]

This investigation is concerned with the 2D acoustic scattering problem of a plane wave propagating in a non‐lossy, isotropic, homogeneous fluid host and soliciting a linear, isotropic, macroscopically homogeneous, generally lossy, flat‐plane layer in which the mass density and wavespeed are different from those of the host. The focus is on the inverse problem of the retrieval of the layer mass density. The data are the transmitted pressure field, obtained by simulation (resolution of the forward problem) in exact, explicit form via the domain integral form of the Bergman wave equation. This solution is exact and more explicit in terms of the mass‐density contrast (between the host and layer) than the classical solution obtained by separation of variables. A perturbation technique enables the solution (in its form obtained by the domain integral method) to be cast as a series of powers of the mass density contrast, the first three terms of which are employed as the trial models in the treatment of the inverse problem. The aptitude of these models to retrieve the mass density contrast is demonstrated both theoretically and numerically. [ABSTRACT FROM AUTHOR]

A generalized rod equation including the celebrated Camassa–Holm shallow water model is considered. The blow-up features of the equation are discussed on the line R. Using the method to construct the Lyapunov functions, local-in-space wave breaking criteria to the equation are established. Our wave breaking results are only involved in the initial values V 0 (x 0) and V 0 ′ (x 0) at a single local point x 0 ∈ R. [ABSTRACT FROM AUTHOR]

The nonlinear Schrödinger equation in the weakly nonlinear regime with random Gaussian fields as initial data is considered. The problem is set on the torus in any dimension greater than two. A conjecture in statistical physics is that there exists a kinetic time scale depending on the frequency localization of the data and on the strength of the nonlinearity, on which the expectation of the squares of moduli of Fourier modes evolve according to an effective equation: the so‐called kinetic wave equation. When the kinetic time for our setup is 1, we prove this conjecture up to an arbitrarily small polynomial loss. When the kinetic time is larger than 1, we obtain its validity on a more restricted time scale. The key idea of the proof is the use of Feynman interaction diagrams both in the construction of an approximate solution and in the study of its nonlinear stability. We perform a truncated series expansion in the initial data, and obtain bounds in average in various function spaces for its elements. The linearized dynamics then involves a linear Schrödinger equation with a corresponding random potential whose operator norm in Bourgain spaces we are able to estimate on average. This gives a new approach for the analysis of nonlinear wave equations out of equilibrium, and gives hope that refinements of the method could help settle the conjecture. [ABSTRACT FROM AUTHOR]

Internal solitary waves in polar regions have attracted much interest recently. It is important to understand how sea ice affects them as this may have a profound influence on human activities and the environment. In this study, experiments on internal solitary waves with and without two types of sea ice (ice sheet and ice keel) are presented, as well as corresponding simulations using the Korteweg-de Vries (KdV) equation, the Benjamin-Ono (BO) equation, and the variable-coefficient Korteweg-de Vries (vKdV) equation, which is a derivation of the KdV equation. Comparison between experiments without sea ice and simulations using the KdV and BO equations proves the suitability of the former over the latter for this study. The experiments with sea ice and theoretical simulations using the vKdV equation provide evidence for wave deformation, oscillation occurring in the rear of the wave, and a decrease in amplitude. The latter suggests possibilities of energy dissipation or the emission of small amplitude linear waves. The sharp vertices of the ice result in occasional inconsistency with the vKdV predictions. Nonetheless, the vKdV equation is still suitable for modeling internal solitary waves under sea ice, giving generally accurate results that can assist further studies. This is the first time the vKdV equation has been applied to investigate the impacts of sea ice on internal solitary waves. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the initial-boundary value problem for the radiative full Euler equations, which are a fundamental system in the radiative hydrodynamics with many practical applications in astrophysical and nuclear phenomena, with the non-slip boundary condition on an impermeable wall. Due to the difficulty from the disappearance of the velocity on the impermeable boundary, quite few results for compressible Navier-Stokes equations and no result for the radiative Euler equations are available at this moment. So the asymptotic stability of the rarefaction wave proven in this paper is the first rigorous result on the global stability of solutions of the radiative Euler equations with the non-slip boundary condition. It also contributes to our systematical study on the asymptotic behaviors of the rarefaction wave with the radiative effect and different boundary conditions such as the inflow/outflow problem and the impermeable boundary problem in our series papers including [5,6]. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the Boussinesq equations with magnetohydrodynamics convection, in which the temperature obeys degenerate damped wave equations. Under the influence of an equilibrium, we prove that the 2D magnetic Bénard system remains stable in the whole space ℝ2$$ {\mathrm{\mathbb{R}}}&#x0005E;2 $$. In particular, we obtain the decay rate and large‐time behavior. Without the coupling, the 2D MHD system with only vertical dissipation is not known to be stable. It is revealed that both the thermal damping and the horizontal background magnetic field are indispensable. [ABSTRACT FROM AUTHOR]

We consider a degenerate/singular wave equation in lone dimension, with drift and in presence of a leading operator that is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary damping at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated Cauchy problem. [ABSTRACT FROM AUTHOR]

We study the motion of an elastic rigid rod which is radially free-falling towards a Schwarzschild black hole. This is accomplished by reducing the corresponding free-boundary partial differential equation problem to a sequence of ODEs, which we integrate numerically. Starting with a rod at rest, we show that it is possible to choose its initial compression profile so that its midpoint falls substantially faster, or slower, than a free-falling particle with the same initial conditions. This seems to be a purely kinematic effect, since on average there is no net transfer of elastic energy to mechanical energy. [ABSTRACT FROM AUTHOR]

Machine learning and neural networks have advanced numerous research domains, but challenges such as large training data requirements and inconsistent model performance hinder their application in certain scientific problems. To overcome these challenges, researchers have investigated integrating physics principles into machine learning models, mainly through (i) physics-guided loss functions, generally termed as physics-informed neural networks, and (ii) physics-guided architectural design. While both approaches have demonstrated success across multiple scientific disciplines, they have limitations including being trapped to a local minimum, poor interpretability, and restricted generalizability beyond sampled data range. This paper proposes a new physics-informed neural network (PINN) architecture that combines the strengths of both approaches by embedding the fundamental solution of the wave equation into the network architecture, enabling the learned model to strictly satisfy the wave equation. The proposed point neuron learning method can model an arbitrary sound field based on microphone observations without any dataset. Compared to other PINN methods, our approach directly processes complex numbers, offers better interpretability, and can be generalized to out-of-sample scenarios. We evaluate the versatility of the proposed architecture by a sound field reconstruction problem in a reverberant environment. Results indicate that the point neuron method outperforms two competing methods and can efficiently handle noisy environments with sparse microphone observations. [ABSTRACT FROM AUTHOR]

Pseudomodes of non-self-adjoint Schrödinger operators corresponding to large pseudoeigenvalues are constructed. The approach is non-semiclassical and extendable to other types of models including the damped wave equation and Dirac operators. [ABSTRACT FROM AUTHOR]

The main objective of this work is to apply a novel and accurate algorithm for solving the second-order and fourth-order fractional diffusion-wave equations (FDWEs). First, the desired equation is reduced to the corresponding Volterra integral equation (VIE). Then, the collocation method is applied, for which the Chebyshev cardinal functions (CCFs) have been considered as the bases. In this paper, the CCFs based on a Lobatto grid are introduced and used for the first time to solve these kinds of equations. To this end, the derivative and fractional integral operators are represented in CCFs. The main features of the method are simplicity, compliance with boundary conditions, and good accuracy. An exact analysis to show the convergence of the scheme is presented, and illustrative examples confirm our investigation. [ABSTRACT FROM AUTHOR]

In this paper, we derive a higher‐order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a "flat‐surface" assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long‐wave, and small‐amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations. [ABSTRACT FROM AUTHOR]

This work is devoted to the numerical solution of the Helmholtz equation with variable wave number and including a point source in appropriately truncated infinite domains. Motivated by a two‐dimensional model, we formulate a simplified one‐dimensional model. We study its well posedness via wave number explicit stability estimates and prove convergence of the finite element approximations. As a proof of concept, we present the outcome of some numerical experiments for various wave number configurations. Our experiments indicate that the introduction of the artificial boundary near the source and the associated boundary condition lead to an efficient model that accurately captures the wave propagation features. [ABSTRACT FROM AUTHOR]

Near-surface in seismic exploration generally refers to the low deceleration zone and a section of stratum below it, which can be described in geophysical language as "low velocity", "low Q", and "Free surface", etc. Due to the presence of low-velocity layers in near-surface, shallow seismic simulation is plagued by numerical dispersion. Numerical dispersion affects the accuracy of seismic wave simulation, especially severe in shallow areas containing low-velocity layers. To improve the simulated quality, denser grids or higher-order difference schemes are used, but both of which would seriously reduce computational efficiency, especially for frequency-domain numerical modeling. Differentiation is replaced by difference in wave field simulation, which would inevitably result in numerical errors. These numerical errors are manifested as the difference between the phase velocity of the discretized wave field and the medium velocity. The waves with different wavenumber components have different phase velocities, and the larger the wavenumber, the more that its phase velocity lags the group velocity. Based on this theoretical analysis, a regularization factor is added to the frequency-domain acoustic wave equation to correct the phase velocity of high wavenumber components. According to the Von Neumann stability requirements, a regularization factor that adaptively changes with simulation parameters is derived. Compared to the fixed regularization factor, adaptive regularization factor can better match the velocity field and protect the effective wave field to the maximum extent while suppressing numerical dispersion. The improved acoustic wave equation can effectively suppress numerical dispersion without increasing the number of grids. Different numerical models demonstrate the effectiveness and efficiency of the improved acoustic equation with the adaptive regularization factor for suppressing numerical dispersion. [ABSTRACT FROM AUTHOR]

Engineering rock masses typically feature natural discontinuities that span different scales, with filled joints as macroscopic discontinuities and microdefects as mesoscopic ones. However, the mechanism of wave propagation in the filled joints has been unclear. This study develops a novel dual-scale equivalent rock model to investigate the wave propagation in jointed rock masses. Firstly, two combination models at different scales were developed by using thin layers and contact interfaces to simulate filled joints and microdefects, respectively. Subsequently, a wave propagation equation was derived through a time-domain recursive method, and validated against the experimental data and the predictions of traditional models. Finally, a parametric analysis was conducted on the frequency response of wave propagation in jointed rock masses. The results show that this novel model can describe the wave dispersion and attenuation in dual-scale discontinuous rock masses, and its equation exhibits robustness. It is revealed that two independent mechanisms induce the wave dispersion in a single joint: multiple wave reflections inside the joint and the viscosity of the filling material. The effect of microdefects on wave attenuation mainly depends on propagation distance, while the influence of filled joints on wave propagation can be determined by the quality factor, thickness, effective wave velocity, and density of the filling. The wave velocity of the filling has a significantly higher influence on the transmission coefficient than the filling density. This challenges the conventional concept that the wave impedance of filling is a basic parameter for estimating the transmission coefficient. Highlights: A novel dual-scale equivalent model was proposed for wave propagation in jointed rock masses. The multiple reflection effects at rock joints were revealed to be the main cause of wave dispersion. Using wave impedance of joint filling to predict the transmission coefficient was found to be inaccurate. The frequency response difference in macroscopic and mesoscopic discontinuities of rock masses was revealed. [ABSTRACT FROM AUTHOR]

Based on the elastic wave equation, a pseudoelastic pure P‐mode wave equation has been recently derived by projecting the wavefield along the wavefront normal direction. This pseudoelastic pure P‐mode wave equation offers an accurate simulation of P‐wave fields with accurate elastic phase and amplitude characteristics. Moreover, considering no S‐waves are involved, it is computationally more efficient than the elastic wave equation, making it an excellent choice as a forward simulation engine for P‐wave exploration. Here, we propose a new pseudoelastic pure P‐mode wave equation and apply the stress image method to it to implement the free surface boundary condition. The new pseudoelastic wave equation offers significantly improved computational efficiency compared to the previous pseudoelastic wave equation. Additionally, the wavefields simulated by this new pseudoelastic wave equation exhibit clear physical interpretations. We evaluate the accuracy of the new wave equation in simulating elastic P‐waves by employing a model with high‐velocity contrasts. We find that this new equation, which purely admits P‐waves, though having exact amplitude and phase behaviour as the elastic waves for transmission components, the amplitudes slightly suffer in the scattering scenario. The difference in amplitude between the elastic and our pseudoelastic increases as the contrast in velocity at the interface (interlayer velocity ratio) increases, especially the S‐wave velocities. This has negative implications on scattering from the free surface boundary condition or the sea bottom interface, especially if the shear wave velocity below the surface or the sea bottom is high. However, in cases where, like for land data in the Middle East, the transition to a free surface is smoother, the accuracy of the pseudoelastic equation is high. In all cases, regardless of the interlayer velocity ratio, the accuracy of the pseudoelastic wave equation in simulating the elastic case, for scattered waves, exceeds that of the acoustic wave equation in phase and amplitude. [ABSTRACT FROM AUTHOR]

In seismic frequency‐domain finite‐difference modelling, the affine mixed‐grid method effectively eliminates the spatial sampling restriction associated with square meshes of the rotated mixed‐grid method. Nevertheless, the affine mixed‐grid method makes a weighted average of the entire elastic wave equations, resulting in reduced accuracy compared to the average‐derivative method in the case of rectangular meshes. It is worth noting, however, that the average‐derivative method is presently inapplicable to free‐surface scenarios, whereas the affine mixed‐grid method is applicable. By performing weighted averages of the derivative terms instead of the entire elastic wave equations in Cartesian and affine rotated coordinate systems, we have developed an improved affine mixed‐grid method for elastic‐wave frequency‐domain finite‐difference modelling. The proposed improved affine mixed‐grid method 9‐point scheme overcomes the drawback that the accuracy of affine mixed‐grid method is lower than that of average‐derivative method for unequal directional grid intervals. Moreover, the improved affine mixed‐grid method 6‐point scheme provides much higher numerical accuracy than the affine mixed‐grid method 6‐point scheme at either equal or unequal directional grid intervals. On the other hand, the proposed improved affine mixed‐grid method simplifies the coding complexity for implementing free‐surface condition in elastic‐wave frequency‐domain finite‐difference modelling by modifying the elastic parameters of the free‐surface layer and thus constructing the impedance matrix containing the free‐surface condition directly. [ABSTRACT FROM AUTHOR]

We propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter 0<ε≤1$$ 0<\varepsilon \le 1 $$ and carry out rigorous error estimates. Actually, the equation propagates wave with O(ε2)$$ O\left({\varepsilon}^2\right) $$ wavelength in time, while the amplitude of the leading oscillation is O(ε4)$$ O\left({\varepsilon}^4\right) $$ for well‐prepared initial data, and O(ε2)$$ O\left({\varepsilon}^2\right) $$ for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ε∈(0,1]$$ \varepsilon \in \left(0,1\right] $$) NPI‐SP scheme is proposed with the optimal uniform error bounds at O(τ2)$$ O\left({\tau}^2\right) $$ in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in H1$$ {H}^1 $$‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates. [ABSTRACT FROM AUTHOR]

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This paper is concerned with the well-posedness and long-time dynamics for a class of wave equations with a nonlocal structural energy damping. The main result establishes that for each exponent β ∈ (0 , 1 / 2 + ϵ) of the structural term, for a small ϵ > 0 , the corresponding problem has a compact global attractor A β , which coincides with the unstable manifold M β (N) emanating from the set N of stationary points. This class of problems whose dissipation intensity depends on the energy of the system has connection with flight structure models, see NASA-AirForce reports (Balakrishnan in A theory of nonlinear damping in flexible structures. Stabilization of flexible structures, 1988; Balakrishnan and Taylor in Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical implementation of the 2D wave equation in isotropic-heterogeneous media. The stability analysis of the scheme using the von Neumann stability method has been studied. We conducted a study on modeling the propagation of acoustic waves in a heterogeneous medium and performed numerical simulations in various heterogeneous media at different time steps. Developed parallel code using Compute Unified Device Architecture (CUDA) technology and tested on domains of various sizes. Performance analysis showed that our parallel approach showed significant speedup compared to sequential code on the Central Processing Unit (CPU). The proposed parallel visualization simulator can be an important tool for numerous wave control systems in engineering practice. [ABSTRACT FROM AUTHOR]

We introduce a simplified model for wave turbulence theory—the Wick nonlinear Schrödinger equation, of which the main feature is the absence of all self‐interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial correlation is translation invariant, we obtain a wave kinetic equation similar to the one predicted by the formal theory. In the inhomogeneous setting, we obtain a wave kinetic equation that describes the statistical behavior of the wavepackets of the solutions, accounting for both the transport of wavepackets and collisions among them. Another wave kinetic equation, which seems new in the literature, also appears in a certain scaling regime of this setting and provides a more refined collision picture. [ABSTRACT FROM AUTHOR]

This study is concerned with a nonlinear viscoelastic wave equation with logarithmic source term. By considering the nonlinear distributed delay feedback influenced by acoustic and fractional boundary conditions at the boundary coupling, we establish the global existence and general decay results under appropriate assumptions, even in the case of a general kernel. This result extends and complements some previous results. [ABSTRACT FROM AUTHOR]

The initial-boundary and the inverse coefficient problems for the semilinear hyperbolic equation with strong damping are considered in this study. The conditions for the existence and uniqueness of solutions in Sobolev spaces to these problems have been established. The inverse problem involves determining the unknown time-dependent parameter in the right-hand side function of the equation using an additional integral type overdetermination condition. [ABSTRACT FROM AUTHOR]

We consider how the quasinormal spectrum for the conformal wave operator on the static patch of de Sitter changes in response to the addition of a small potential. Since the quasinormal modes and co-modes are explicitly known, we are able to give explicit formulae for the instantaneous rate of change of each frequency in terms of the perturbing potential. We verify these exact computations numerically using a novel technique extending the spectral hyperboloidal approach of Jaramillo et al. (2021). We propose a definition for a family of pseudospectra that we show capture the instability properties of the quasinormal frequencies. [ABSTRACT FROM AUTHOR]