Wave Equation Research Articles

Stabilization for degenerate equations with drift and small singular term

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.

Plane Wave Finite Element Method for the Dynamic Response of the Periodic Pile-Type Composite Ground Under a Harmonic Concentrated Load

In this study, a simplified model for the pile-type composite ground, namely, the periodic pile-type composite ground (PPCG) model is proposed. The PPCG in this study is assumed to be periodic in the horizontal directions and finite in the vertical direction. For investigation of the dynamic response of the PPCG to a harmonic concentrated load, a plane wave finite element (PWFE) method for the PPCG under the load is developed in this study. To develop the method, the concentrated load is decomposed into its wavenumber domain components by the 2D Fourier transform first. The response of the PPCG to a wavenumber domain load component is represented by a set of plane waves, and the variables associated with each plane wave are discretized via the FEM along the vertical direction of the PPCG. The finite element equations for each plane wave of the PPCG are developed via the virtual work principle. To obtain the overall dynamic response of the PPCG to the concentrated load by synthesizing the responses due to all load components effectively, the mode superposition method is used to construct the response of the PPCG to the load component, and the polar coordinate system-based inverse Fourier transform method is developed for the inversion of the 2D Fourier transform with respect to the wavenumber. With the developed method, some numerical results about the dynamic response of the PPCG to the concentrated load are presented.

Lifespan estimates for semilinear damped wave equation in a two-dimensional exterior domain

Lifespan estimates for semilinear damped wave equations of the form ∂t2u-Δu+∂tu=|u|p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t^2u-\\Delta u+\\partial _tu=|u|^p$$\\end{document} in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the semilinear heat equation ∂tv-Δv=v2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _tv-\\Delta v=v^2$$\\end{document} with the Dirichlet boundary condition and the initial condition v(0)=εf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v(0)=\\varepsilon f$$\\end{document}, the corresponding lifespan can be estimated from below and above by exp(exp(Cε-1))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\exp (\\exp (C\\varepsilon ^{-1}))$$\\end{document} with different constants C. This paper clarifies that the same estimates hold even for the critical semilinear damped wave equation in the exterior of the unit ball under the restriction of radial symmetry. To achieve this result, a new technique to control L1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^1$$\\end{document}-type norm and a new Gagliardo–Nirenberg type estimate with logarithmic weight are introduced.

Recovery of a coefficient in a diffusion equation from large time data

Abstract This paper considers the determination of a spatially varying coefficient in a parabolic equation from time trace data. There are many uniqueness theorems known for such problems but the reconstruction step is severally ill-posed: essentially the problem comes down to trying to reconstruct an analytic function from values on a strip. However, we look at an even more restricted data where the measurements are not made on the whole time axis but only for large values adding further to the ill-conditioning situation. In addition, we do not assume the initial state is known. Uniqueness is restored by making changes to the boundary condition, in particular, to the impedance parameter, for each of a series of measurements. We show that an implementation of the above paradigm leads to both uniqueness and an effective reconstruction algorithm. Extension is also made to the case of fractional model and to replacing the parabolic equation with a damped wave equation.

An Effective Discontinuous Galerkin Method for Solving Acoustic Wave Equations in Heterogeneous Media

Abstract A discontinuous Galerkin (DG) method is introduced for solving 2D and 3D first-order velocity-pressure acoustic wave propagation problems in heterogeneous media. First, acoustic equations are reformulated into a first-order hyperbolic system, which can be integrated into the framework of the DG method. Then, we introduce an effective numerical flux in DG spatial discretization, which preserves physical laws on both sides of interfaces with discontinuous material parameters. It is compared with both the classic Lax-Friedrichs flux and the exact upwind flux, with the latter derived from solving the Riemann problem and satisfying the Rankine-Hugoniot condition. The comparison reveals that while our flux is formally similar to the classic local Lax-Friedrichs flux, its numerical behavior is analogous to the exact upwind flux. The primary advantage of this method lies in its simplicity and straightforward computation yet can maintain physical continuity conditions near interfaces with discontinuous material parameters. Several numerical experiments of acoustic wave propagation in 2D and 3D heterogeneous media are carried out, illustrating the effectiveness of this method.

Longtime Dynamics for a Class of Strongly Damped Wave Equations with Variable Exponent Nonlinearities

Longtime Dynamics for a Class of Strongly Damped Wave Equations with Variable Exponent Nonlinearities

Investigating the fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation: An iterative framework for nonlinear wave dynamics

Abstract This paper addresses the solution of the fractional Caudrey Dodd Gibbon Sawada Kotera (CDGSK) equation using the Conformable Laplace Decomposition Method (CLDM). The CDGSK equation, a fundamental model in wave dynamics and fluid mechanics, is explored for its applications in quantum mechanics and nonlinear optics. By employing fractional calculus, we demonstrate how fractional derivatives influence the physical characteristics of wave propagation in both optical and quantum systems. The exact solutions obtained provide insight into soliton behavior, essential for understanding wave-particle interactions in quantum fields and light-matter interactions in optics. The fractional nature of the equation allows for more accurate modeling of non-integer order dynamics commonly found in optical fibers and quantum waveguides. The CLDM method proves to be highly effective, providing approximate solutions with minimal computational effort. These findings offer significant contributions to the fields of quantum mechanics and nonlinear optics, where the fractional CDGSK equation can be applied to solve complex wave equations with great accuracy.

Fuzzy fractional shallow water wave equations: analysis, convergence of solutions, and comparative study with depth as triangular fuzzy number

Abstract We explore the integration of fuzzy fractional calculus into the modeling framework, recognizing its significance in capturing the inherent uncertainties and complexities present in Shallow Water Wave (ffSWW) dynamics. By incorporating fuzzy fractional calculus, we aim to enhance the accuracy and robustness of ffSWW equations, particularly in representing vague or imprecise parameters such as seabed topography, initial wave conditions, and material properties. In this article, we consider the time derivative as a fractional order instead of the traditional integer order, which allows us to interpret the behavior of the solution for different orders. Further, the sea depth has been considered as a Triangular Fuzzy Number (TFN). We employ the Homotopy Perturbation Transform Method (HPTM) to obtain the solution of ffSWW equations. The convergence of the obtained series solutions has been investigated theoretically and numerically. Also, the acquired results using the current method are validated through the comparison with pre-existing findings concerning integer order. Furthermore, simulation results for various fractional orders, as well as fuzzy lower and upper solutions of depth-averaged velocity and water surface elevation, are provided for triangular fuzzy numbers.

Second Harmonic Generation of High Power Cosh-Gaussian Beam in Thermal Quantum Plasma: Effect of Relativistic and Ponderomotive Nonlinearity

Second harmonic generation (SHG) of high power Cosh-Gaussian(ChG) beam in thermal quantum plasma (TQP) is explored in current investigation. Relativistic and ponderomotive nonlinearities are taken together in present investigation. Combination of relativistic-ponderomotive nonlinearities causes modification in electron mass, thereby producing density gradients inside plasma. This further produces self-focusing of main beam. The well-known WKB and paraxial approaches are used for solving main beam’s wave equation and to obtain 2nd order ordinary differential equation (ODE). There is generation of electron plasma wave (EPW) due to these density gradients. Excited EPW interacts with main beam to generate 2nd harmonics in plasma. Effects of combined relativistic-ponderomotive nonlinearities and established laser - plasma parameters on beam’s waist and SHG yield are studied in present study.

On the dual-phase-lag thermal response in the pulsed photoacoustic effect: A theoretical and experimental 1D-approach

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)] τp. 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 τT; therefore, the whole system is assumed to have a similar thermal relaxation. A second assumption for τT 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 τT<τp, the DPL temperature has a similar thermal profile of the Fourier heat equation; however, when τT≥τp, 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 τT 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 τT; the transference function was also calculated, which allowed us to find the maximum response in frequency for the considered experimental setup.

Damping of spin waves

We show that, in ideal-spin hydrodynamics, the components of the spin tensor follow damped wave equations. The damping rate is related to nonlocal collisions of the particles in the fluid, which enter at first order in ℏ in a semiclassical expansion. This rate provides an estimate for the timescale of spin equilibration and is computed by considering a system of spin-1/2 fermions interacting via a quartic self-interaction as well as via (screened) one-gluon exchange. It is found that the relaxation times of the components of the spin tensor can become very large compared to the usual dissipative timescales of the system. Our results suggest that the spin degrees of freedom in a heavy-ion collision may not be in equilibrium by the time of freeze-out, and thus should be treated dynamically. Published by the American Physical Society 2024

Point neuron learning: a new physics-informed neural network architecture

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.

Exact solitary wave solutions for a coupled gKdV–Schrödinger system by a new ODE reduction method

AbstractA new method is developed for finding exact solitary wave solutions of a generalized Korteweg–de Vries equation with ‐power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with . No solutions for were known previously in the literature. For , four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations as well as to nonlinear ODE systems of generalized Hénon–Heiles form.

Two-dimensional gravity waves at low regularity I: Energy estimates

This article represents the first installment of a series of papers concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on sharp cubic energy estimates. Precisely, we introduce and develop the techniques to prove a new class of energy estimates, which we call balanced cubic estimates . This yields a key improvement over the earlier cubic estimates of Hunter–Ifrim–Tataru (2016), while preserving their scale-invariant character and their position-velocity potential holomorphic coordinate formulation. Even without using any Strichartz estimates, these results allow us to significantly lower the Sobolev regularity threshold for local well-posedness, drastically improving earlier results obtained by Alazard–Burq–Zuily (2014, 2018), Hunter–Ifrim–Tataru (2016) and Ai (2019).

Approximate kink-kink solutions for the ϕ 6 model in the low-speed limit

In this paper, we consider the problem of elasticity and stability of the collision of two kinks with low speed v for the nonlinear wave equation known as the ϕ 6 model in dimension 1 + 1. We construct a sequence of approximate solutions ( ϕ k ( v , t , x ) ) k ∈ N ⩾ 2 for this model to understand the effects of the collision in the movement of each soliton during a large time interval. The construction uses a new asymptotic method which is not only restricted to the ϕ 6 model.

Comparison and parametric study of characteristics of eleven types of anisotropic woods based on the behaviour of Lamb wave propagation

AbstractKnowledge in advance of the nine orthotropic independent elastic constants (Cij) of the wood medium is essential for evaluating its mechanical properties. The most prominent technique to retrieve Cij is the ultrasonic testing technique. This technique uses guided waves that can propagate through the material under test. Accordingly, it is worth noting that the numerical modelling of the phase and group velocities of guided waves is an unavoidable preliminary step before experimentally producing guided wave modes. Therefore, the main goal of the present work is to numerically calculate the phase velocity, group velocity and the relevant optimal incidence angles of Lamb waves in anisotropic wood that can be used as a numerical parametric study for any future experimental setup. Here, Lamb dispersion curves are calculated for eleven types of woods, where the Legendre polynomial method is employed to solve the wave equations. Moreover, the optimal incidence angle for each Lamb mode is calculated according to the Snell–Descartes law. By calculating out the three parameters of phase velocity, group velocity and optimal incidence angle of Lamb modes in eleven types of anisotropic woods, we hope to fast-track the researchers in considering the present work to facilitate their experimental measurements.

Exact Model of Gravitational Waves and Pure Radiation

An exact non-perturbative model of a gravitational wave with pure radiation is constructed. It is shown that the presence of dust matter in this model contradicts Einstein’s field equations. The exact solution to Einstein’s equations for gravitational wave and pure radiation is obtained. The trajectories of propagation and the characteristics of radiation are found. For the considered exact model of a gravitational wave, a retarded time equation for radiation is obtained. The obtained results are used to construct an exact model of gravitational wave and pure radiation for the Bianchi type IV universe.

Two novel linearized energy-conserving finite element schemes for nonlinear regularized long wave equation

In this paper, two linearized second-order energy-conserving schemes for the nonlinear regularized long wave (RLW) equation are introduced, the unconditional superclose and superconvergence results are presented by using the conforming finite element method (FEM). Initially, through a skillful decomposition of the nonlinear term, two linearized second-order fully discrete schemes are developed. Compared to the previous nonlinear approaches, these schemes significantly reduce the number of iterations and improve computational efficiency; moreover, they conserve energy, and ensure the boundedness of the numerical solution in the H1-norm directly, which represents an advancement over the L∞-norm boundedness reported in prior studies. Secondly, based on the boundedness of the FE solution, the Ritz projection operator and high-precision results of the linear triangular element, the error estimates for superclose and superconvergence are derived without any restrictions on the ratio between time step size Δt and spatial mesh size h. Finally, four numerical examples are provided to confirm the accuracy of the theoretical analysis and the effectiveness of the method.

Modeling a pure qP wave in attenuative transverse isotropic media using a complex-valued fractional viscoacoustic wave equation

Seismic anisotropy, characterized by direction-dependent seismic velocity and attenuation, is a fundamental property of the earth, widely observed in field and laboratory measurements. This anisotropy significantly influences seismic wavefield attributes, such as amplitude, phase, and traveltime. Accurate quantification of anisotropy-related seismic wave propagation is essential for global- and regional-scale studies in seismology, such as imaging and inversion. Because the elastic assumption of the earth requires contending with S waves, which are often weak in our seismic recording, we derive a new general complex-valued spatial fractional Laplacian (SFL) viscoacoustic anisotropic pure quasi-P (qP) wave equation (WE) in attenuative transverse isotropic (A-TI) media under the acoustic assumption and use the fixed-order SFL operator to quantify seismic wave propagation in heterogeneous media. Our WE accurately captures anisotropic effects in seismic velocity and attenuation, even in strongly attenuative and anisotropic environments. It also describes the decoupled energy attenuation and velocity dispersion behaviors of the frequency-independent quality factor Q, beneficial for seismic imaging and inversion. Compared with existing viscoacoustic anisotropic pure qP WEs in A-TI media, our WE simplifies computational complexity and is easy to implement, as it decomposes into a complex-valued scalar operator and two differential operators. The evaluated dispersion curves further confirm the accuracy of our approach. Numerical examples in complex geologic settings demonstrate the new equation’s precision, stability, and efficiency in modeling seismic wave propagation.

On propagation characteristics of ultrasonic guided waves in layered fluid-saturated porous media using spectral method.

Fluid-saturated porous media plays an increasingly important role in emerging fields such as lithium batteries and artificial bones. Accurately solving the governing equations of guided wave is the key to the successful application of ultrasonic guided wave nondestructive testing technology in fluid-saturated porous media. This paper derives the Lamb wave equation in layered fluid-saturated porous materials based on Biot theory and proposes the spectral method suitable for solving complex wave equations. The spectral method reconstructs the fundamental wave equations in the form of a matrix eigenvalue problem using spectral differentiation matrices. It introduces boundary conditions by replacing corresponding rows in the wave equation matrix with stress or displacement in matrix form. For complex differential equations, such as the governing equations of guided waves in porous media, the spectral method has the significant advantages of faster computation speed, less root loss, and easier encoding process. The spectral method is used to calculate the acoustic field characteristics under different boundary conditions and environments of the layer fluid-saturated porous media. Results show that the surface treatment details and environment of fluid-saturated porous materials play an important role in the propagation of guided waves.