Wave Equation Research Articles

Deep-water and shallow-water limits of the intermediate long wave equation

In this paper, we study the low regularity convergence problem for the intermediate long wave equation (ILW), with respect to the depth parameter δ > 0, on the real line and the circle. As a natural bridge between the Korteweg–de Vries (KdV) and the Benjamin–Ono (BO) equations, the ILW equation is of physical interest. We prove that the solutions of ILW converge in the Hs -Sobolev space for s>12 , to those of BO in the deep-water limit (as δ→∞ ), and to those of KdV in the shallow-water limit (as δ → 0). This improves previous convergence results by Abdelouhab et al (1989 Physica D 40 360–92), which required s>32 in the deep-water limit and s⩾2 in the shallow-water limit. Moreover, the convergence results also apply to the generalised ILW equation, i.e. with nonlinearity ∂x(uk) for k⩾2 . Furthermore, this work gives the first convergence results of generalised ILW solutions on the circle with regularity s⩾34 . Overall, this study provides mathematical insights for the behaviour of the ILW equation and its solutions in different water depths, and has implications for predicting and modelling wave behaviour in various environments.

Partial Smoothing of the Stochastic Wave Equation and Regularization by Noise Phenomena

AbstractWe establish partial smoothing properties of the transition semigroup $$(P_t)$$ ( P t ) associated to the linear stochastic wave equation driven by a cylindrical Wiener noise on a separable Hilbert space. These new results allow the study of related vector-valued infinite-dimensional PDEs in spaces of functions which are Hölder continuous along special directions. As an application we prove strong uniqueness for semilinear stochastic wave equations involving nonlinearities of Hölder type. We stress that we are able to prove well-posedness although the Markov semigroup $$(P_t)$$ ( P t ) is not strong Feller.

Acoustic wave simulation in strongly heterogeneous models using a discontinuous Galerkin method

In recent years, the discontinuous Galerkin method (DGM) has been rapidly developed for the numerical simulation of seismic waves. For wavefield propagation between two adjacent elements, it is common practice to apply a numerical flux to the boundary of each element to propagate waves between adjacent elements. Several fluxes, including the center, penalty, Local Lax–Friedrich (LLF), upwind, and Rankine–Hugoniot jump condition-based (RH-condition) fluxes are widely used in numerical seismic wave simulation. However, some fluxes do not account for media differences between adjacent elements. Although different fluxes have been successfully used in DGM for many velocity models, it is unclear whether they can produce sufficiently accurate or stable results for strongly heterogeneous models, such as checkerboard models commonly used in tomographic studies. We test different fluxes using the acoustic wave equation. We analyzed the accuracy of the penalty, LLF, upwind, and RH-condition fluxes based on the results of the numerical simulations of the homogeneous and two-layer models. We conducted simulations using checkerboard models, and the results indicated that the LLF, penalty, and upwind fluxes may have instability problems in heterogeneous models with long-time simulations. We observed instability issues in the LLF, penalty, and upwind fluxes when the wave-impedance contrast was high at the media interface. However, the results of RH-condition flux remained consistently stable. The series of numerical examples presented in this work provide insights into the characteristics and application of fluxes for seismic wave modeling.

Exploring the Exact Solution of the Space-Fractional Stochastic Regularized Long Wave Equation: A Bifurcation Approach

This study explores the effects of using space-fractional derivatives and adding multiplicative noise, modeled by a Wiener process, on the solutions of the space-fractional stochastic regularized long wave equation. New fractional stochastic solutions are constructed, and the consistency of the obtained solutions is examined using the transition between phase plane orbits. Their bifurcation and dependence on initial conditions are investigated. Some of these solutions are shown graphically, illustrating both the individual and combined influences of fractional order and noise on selected solutions. These effects appear as alterations in the amplitude and width of the solutions, and as variations in their smoothness.

The Dynamical and Kinetic Equations of Four-Five-Six-Wave Resonance for Ocean Surface Gravity Waves in Water with a Finite Depth

Based on the Hamilton canonical equations for ocean surface waves with four-five-six-wave resonance conditions, the determinate dynamical equation of four-five-six-wave resonances for ocean surface gravity waves in water with a finite depth is established, thus leading to the elimination of the nonresonant second-, third-, fourth-, and fifth-order nonlinear terms though a suitable canonical transformation. The four kernels of the equation and the 18 coefficients of the transformation are expressed in explicit form in terms of the expansion coefficients of the gravity wave Hamiltonian in integral-power series in normal variables. The possibilities of the existence of integrals of motion for the wave momentum and the wave action are discussed, particularly the special integrals for the latter. For ocean surface capillary–gravity waves on a fluid with a finite depth, the sixth-order expansion coefficients of the Hamiltonian in integral-power series in normal variables are concretely provided, thus naturally including the classical fifth-order kinetic energy expansion coefficients given by Krasitskii.

Topological defects on solutions of non-relativistic equation for extended double ring-shaped potential

Abstract In this study, we focus into the non-relativistic wave equation described by the Schrödinger equation, specifically considering angular-dependent potentials within the context of a topological defect background generated by a cosmic string. Our primary goal is to explore
quasi-exactly solvable problems by introducing an extended ring-shaped potential. We utilize the Bethe ansatz method to determine the angular solutions, while the radial solutions are obtained using special functions. Our findings demonstrate that the eigenvalue solutions
of quantum particles are intricately influenced by the presence of the topological defect of the cosmic string, resulting in significant modifications compared to those in a flat space background. The existence of the topological defect induces alterations in the energy spectra, disrupting degeneracy. Afterwards, we extend our analysis to study the same problem in the presence of a ring-shaped potential against the background of another topological defect geometry known as a point-like global monopole. Following a similar procedure, we obtain the eigenvalue solutions and analyze the results. Remarkably, we observe that the presence of a global monopole leads to a decrease in the energy levels compared to the flat space results. In both cases, we conduct a thorough numerical analysis to validate our findings.

Positive-Energy Dirac Particles and Dark Matter

The relativistic positive-energy wave equation proposed by P. Dirac in 1971 is an old but largely forgotten subject. The purpose of this note is to speculate that particles described by this equation (called here Dirac particles) are natural candidates for the dark matter. The reasoning is based on a fact that the internal structure of such particles simply prohibits their interaction with electromagnetic fields (at least with the minimal coupling) which is exactly what is required for dark matter. Dirac particles have quite unusual properties. In particular, they are transformed by an infinite-dimensional representation of the homogeneous Lorentz group, which clearly distinguishes them from all known elementary particles described by finite-dimensional representations and hints to a physics beyond the Standard Model. To clarify the topic, a brief review of the main features of the above-mentioned Dirac equation is given.

Stabilization of the wave equation on larger-dimension tori with rough dampings

Stabilization of the wave equation on larger-dimension tori with rough dampings

Analytical study of soliton dynamics in the realm of fractional extended shallow water wave equations

In this study, we use the Khater Method (KM) as an efficient analytical tool to solve (3+1)-dimensional fractional extended shallow water wave equations (FESWWEs) with conformable derivatives. The KM transforms fractional partial differential equations to ordinary differential equations (ODEs) via strategic variable transformation. Then, series-form solutions to these ODEs are proposed, which turn them into nonlinear algebraic systems. The solution to this set of algebraic equations yields shock travelling wave solutions expressed in hyperbolic, trigonometric, exponential, and rational functions. The study’s findings are corroborated by 2D, 3D, and contour graphs that show the changing patterns of the detected shock travelling waves. These findings have important significance for the discipline, offering vital insights into the intricate dynamics of FESWWEs. The effectiveness of KM is demonstrated by its capacity to produce varied solutions and contribute to a thorough knowledge of such complex phenomena.

From raw seismic data to quantitative interpretation via cloud-based FWI

Full-waveform inversion (FWI) has evolved in recent years from a technique for recovering long- to intermediate-wavelength updates of acoustic overburden velocities as part of a broader model building workflow to a standalone tool for high-fidelity seismic imaging using raw seismic data – with ever greater resolution and ever more sophisticated physics being sought. In doing so, it has moved notably closer to fulfilling the vision of its original inventors in the late 70s and early 80s. In this work, two novel approaches to FWI along with enabling technologies for their effective deployment are considered. The first of these allows for robust long-wavelength updates to be generated from raw seismic data in the absence of diving wave energy, providing reliable estimates of acoustic velocity at depths equivalent to or greater than the longest offsets available in the data – thereby enabling more effective imaging with data of limited quality. The second allows for the estimation of elastic amplitude-versus-angle behaviour while still using an acoustic wave equation, thereby mitigating the significant computational burden typically associated with elastic FWI. Finally, it is demonstrated that these approaches can be deployed in a reliable and cost-effective manner via the considered use of public cloud resources, with learnings relevant for analogous massively parallel scientific applications. Case studies from two marine environments are provided for discussion, one being from deep-water US Gulf of Mexico and the other from intermediate water depths offshore Western Australia.

Stable blowup for focusing semilinear wave equations in all dimensions

We consider the wave equation with focusing power nonlinearity. The associated ODE in time gives rise to a self-similar solution known as the ODE blowup. We prove the nonlinear asymptotic stability of this blowup mechanism outside of radial symmetry in all space dimensions and for all superlinear powers. This result covers for the first time the whole energy-supercritical range without symmetry restrictions.

The posteriori analysis of the spectral element discretization of the wave equation

This work focuses on discretizing a second-order linear wave equation using the implicit Euler scheme for time discretization and the spectral element method for spatial discretization. We prove that optimal adaptivity can be achieved by combining adaptive time steps with adaptive spectral mesh. We introduce two sets of error indicators for time and space, respectively, and derive optimal estimates.

LOCAL FRACTIONAL SUMUDU DECOMPOSITION METHOD TO SOLVE FRACTAL PDEs ARISING IN MATHEMATICAL PHYSICS

In this paper, we investigate solutions of telegraph, Laplace and wave equations within the local fractional derivative operator by using local fractional Sumudu decomposition method. This method is coupled by the Sumudu transform and decomposition method. The method in general is easy to implement and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the presented method.

Analysis of natural vibration of truncated conical shells partially filled with fluid

AbstractIn this paper, we study the vibrational behavior of shells in the form of truncated cones containing an ideal compressible fluid. The sloshing effect on the free surface of the fluid is neglected. The dynamic behavior of the elastic structure is investigated based on the classical shell theory, the constitutive relations of which represent a system of ordinary differential equations written for new unknowns. Small fluid vibrations are described in terms of acoustic approximation using the wave equation for hydrodynamic pressure written in spherical coordinates. Its transformation into the system of ordinary differential equations is carried out by applying the generalized differential quadrature method. The formulated boundary value problem is solved by Godunov's orthogonal sweep method. Natural frequencies of shell vibrations are calculated using the stepwise procedure and the Muller method. The accuracy and reliability of the obtained results are estimated by making a comparison with the known numerical and analytical solutions. The dependencies of the lowest frequency on the fluid level and cone angle of shells under different combinations of boundary conditions (simply supported, rigidly clamped, and cantilevered shells) have been studied comprehensively. For conical straight and inverted shells, a numerical analysis has been performed to estimate the possibility of finding configurations at which the lowest natural frequencies exceed the corresponding values of the equivalent cylindrical shell.

Analytical insights into the behavior of finite amplitude waves in plasma fluid dynamics

This study introduces innovative analytical solutions for the [Formula: see text]-dimensional nonlinear Jaulent–Miodek ([Formula: see text]) equation, a governing model elucidating the propagation characteristics of nonlinear shallow water waves with finite amplitude. Employing analytical methodologies such as the Khater II and unified methods, alongside the Adomian decomposition method as a semi-analytical approach, series solutions are derived with the primary aim of elucidating the fundamental physics dictating the evolution of [Formula: see text] waves. Within the realm of nonlinear fluid dynamics, the [Formula: see text] equation encapsulates the behavior of irrotational, inviscid, and incompressible fluid flow, wherein nonlinear effects and dispersion intricately balance to yield stable propagating waves. This equation encompasses terms representing nonlinear convection, dispersion, and nonlinearity effects. The analytical methodologies employed in this investigation yield solutions for various instances of the [Formula: see text] equation, demonstrating convergence, accuracy, and computational efficiency. The outcomes reveal that the Adomian decomposition method yields solutions congruent with those obtained through analytical techniques, thereby affirming the precision of the derived solutions. Furthermore, this study advances the comprehension of the physical implications inherent in the [Formula: see text] equation, serving as a benchmark for evaluating alternative methodologies. The analytical approaches elucidated in this research furnish accessible tools for addressing a diverse array of nonlinear wave equations in mathematical physics and engineering domains. In summary, the introduction of novel exact and approximate solutions significantly contributes to the advancement of knowledge pertaining to the [Formula: see text]-dimensional [Formula: see text] equation. The ramifications of this research extend to the modeling of shallow water waves, offering invaluable insights for researchers and practitioners engaged in the field.

Inverse problem in energy-dependent potentials using semiclassical methods

Wave equations with energy-dependent potentials appear in many areas of physics, ranging from nuclear physics to black hole perturbation theory. In this work, we use the semiclassical Wentzel-Kramers-Brillouin (WKB) method to first revisit the computation of bound states of potential wells and reflection/transmission coefficients in terms of the Bohr-Sommerfeld rule and the Gamow formula. We then discuss the inverse problem, in which the latter observables are used as a starting point to reconstruct the properties of the potentials. By extending known inversion techniques to energy-dependent potentials, we demonstrate that so-called width-equivalent or WKB-equivalent potentials are not isospectral anymore. Instead, we explicitly demonstrate that constructing quasi-isospectral potentials with the inverse techniques is still possible. Those reconstructed, energy-independent potentials share key properties with the width-equivalent potentials. We report that including energy-dependent terms allows for a rich phenomenology, particularly for the energy-independent equivalent potentials. Published by the American Physical Society 2024

Optimizing the noise control in a two-layer conduit

In the modern world, noise pollution is a major concern due to its prevalence. This work focuses on optimizing noise control in a two-layer conduit. A conduit comprises an inlet and double-outlet zones (DOZ). The upper walls of the DOZ are lined with perforated absorbent material while the lower walls are layered with fibrous. Mathematically, the physical problem is formulated with a field wave equation together with rigid and impedance boundary conditions in the respective zones. Such governing boundary value problem (BVP) leads to the Sturm-Liouville (SL) category in which standard orthogonality relations (OR) are indispensable. The system of the linear equations of the BVP is acquired with semi-analytical Mode-Matching (MM) approach by implementing the continuity conditions of sound pressures and velocities at the matching junction with the aid of OR. These systems are truncated and solved numerically with computation learning to obtain the reflected and transmitted modal amplitudes in respective zones. Due to the lining's perforated upper walls, fibrous lower walls, and reversal in the DOZ, the analysis of the reflected and transmitted powers versus frequencies is significantly observed and is shown in graphical findings. The algebraic derivation is validated by satisfying the conservation law of power flux and matching continuity conditions for impedance, perforated, and fibrous lined boundaries.

Exponential Attractors for the Sup-Cubic Wave Equation with Nonlocal Damping

Exponential Attractors for the Sup-Cubic Wave Equation with Nonlocal Damping

Existence and Blow-up Study of a Quasilinear Wave Equation with Damping and Source Terms of Variable Exponents-type Acting on the Boundary

Existence and Blow-up Study of a Quasilinear Wave Equation with Damping and Source Terms of Variable Exponents-type Acting on the Boundary

On a quasilinear parabolic–hyperbolic system arising in MEMS modeling

AbstractA 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.