Wave Equation Research Articles

Nonlinear lower hybrid wave equations in collisional tokamak plasmas

A new set of coupled integro-differential nonlinear lower hybrid (LH) wave equations is derived within the framework of a kinetic theory coupled to the Maxwell equations to study the parametric instabilities (PIs) produced by LH waves in collisional tokamak plasma. Previous models of nonlinear LH wave equations have been significantly improved. The wave equations derived overcome the limits and incorrectness of the standard theory of the PI in inhomogeneous plasma. They allow us to treat the full spectrum in the parallel and poloidal wavenumber of the coupled LH power wave, diffraction effects and possible cascade phenomena, which are elements of the nonlinear LH physics ignored in the standard PI theory. Numerical solutions of the new nonlinear LH wave equations are proposed. The relevant LH frequency spectra produced by PI are calculated, exhibiting characteristic features of PI observed in LH experiments. It is shown that the LH sideband amplification can be overestimated by orders of magnitude by the standard theory of PI. A benchmark of the new model is provided for spatially homogeneous plasmas. The role of the collisions for PI has been assessed. We demonstrate that previous analyses significantly overestimated their stabilization effect.

Stability of coupled wave equations with variable coefficients, localised Kelvin–Voigt damping and time delay

Stability of coupled wave equations with variable coefficients, localised Kelvin–Voigt damping and time delay

Non-linear torsional Alfv\'en waves evolving in stratified viscous plasmas: Coronal hole plumes

We model solar atmospheric structures characterised by parallel structuring. We focus on Alfv\'en waves in the weakly non-linear regime to highlight the efficiency of non-linear wave steepening when dissipative effects are prominent. We also consider the local and equilibrium conditions involved in shock formation and the shock's contributions to coronal seismology. Coronal plumes were modelled analytically by implementing the magnetohydrodynamic (MHD) theory in cylindrical geometry. Here, the stratification and viscosity are present internal to the plume, whilst effects of the external medium, together with equilibrium conditions, are implied where the magnetic fields are parallel to the plume axis. We implemented a second-order thin flux tube approximation to obtain a wave equation that points to effects tied to non-linear, dissipative, and stratification terms, as well as terms representing atmospheric conditions. The impact of shear viscosity on non-linear Alfv\'en waves extracted by the Cohen-Kulsrud-Burgers-type equation proves more efficient when propagated to higher altitudes. The dissipative effects linked to the dimensionless viscosity indicate that the dissipative effects are not linear. Meanwhile, the delay in shock formation enables energy conversions at higher altitudes, thereby maintaining coronal heating at higher levels. The efficiency of parallel structuring and viscous damping is enhanced by such transverse structuring, as it is directly proportional to the external plasma-beta . It is observed that Alfv\'en pulses may undergo a backward shock, either in the lower levels of coronal plasma or as they propagate toward higher regions, implying a conversion of energy occurring at various altitudes. A peak was observed, indicating that the interplay reverses at heights around $1.5$ solar radii. Such effects are shown to play a key role in the context of coronal seismology.

Application of the method of lines to the wave equation for simulating vibrating strings

This paper presents simulations of continuous dynamic models of vibrating strings for several different instruments: a guitar, a piano and a violin. The basis for all three models is the wave equation and is differentiated by the initial and boundary conditions imposed. The models are implemented as part of a Graphical User Interface. The simulations are presented as animations that are conveyed in the paper as a sequence of vibrating string profiles for different times. This paper promotes the idea that using music generation as a framework for teaching dynamics makes the teaching of dynamics a more engaging process. A case is made that there is value in solving the wave equation numerically using the method of lines rather than looking for analytical solutions. The reason for this is the mathematics of the method of lines is simpler to understand than the techniques used to derive analytical solutions to partial differential equations. The numerical approach is also more easily extended to a more sophisticated model for a vibrating string that might include energy dissipation and the stiffness properties of the string.

Analytical and numerical solutions to the generalized Schrödinger equation with fourth-order dispersion and nonlinearity

This study aims to solve the generalized Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity (4-Order-GNLSE) using advanced analytical and numerical methods. The 4-Order-GNLSE is critical in understanding complex wave propagation phenomena in various physical contexts, including optical fibers and fluid dynamics, where higher-order dispersion and nonlinear effects are significant. We employ the Khater II (Khat II) and modified Rational (MRat) methods to construct analytical solutions. To validate these solutions, we utilize the trigonometric-quantic-B-spline (TQBS) method as a numerical scheme, demonstrating a close match between analytical and numerical results. Our findings indicate that the proposed methods effectively capture the intricate dynamics governed by the 4-Order-GNLSE. The results highlight the relevance of these solutions in practical applications, offering new insights into the wave propagation behaviors influenced by higher-order dispersion and nonlinearities. This research provides a significant contribution to the field of nonlinear wave equations, presenting novel solutions and validating methodologies that enhance our understanding and potential applications of these complex systems.

A discontinuous Galerkin based multiscale method for heterogeneous elastic wave equations

In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the interior penalty discontinuous Galerkin (IPDG) to couple the multiscale basis functions that contain important heterogeneous media information. The construction of efficient multiscale basis functions starts with extracting dominant modes of carefully defined spectral problems to represent important media feature, which is followed by solving a constraint energy minimization problem. Then a Petrov-Galerkin projection and systematization onto the coarse grid are applied. As a result, an explicit and energy-conserving scheme is obtained for fast online simulation. The method exhibits both coarse-mesh and spectral convergence as long as one appropriately chooses the oversampling size. We rigorously analyze the stability and convergence of the proposed method. Numerical results are provided to show the performance of the multiscale method and confirm the theoretical results.

The exact solutions for the fractional Riemann wave equation in quantum mechanics and optics

In this paper, the fractional Riemann wave equation with M-truncated derivative (FRWE-MTD) is considered. The Jacobi elliptic function method and the modified extended tanh function method are applied to acquire new elliptic, rational, hyperbolic, and trigonometric functions solutions. Moreover, we expand some earlier studies. The obtained solutions are important in explaining some exciting physical phenomena, since the Riemann wave equation is used in various fields, including quantum mechanics, optics, signal processing, and general relativity. Also, this equation is used to describe the propagation of waves in various dispersive systems, where wave motion is affected by the medium through which it travels. Several 3D and 2D graphs are shown to demonstrate how the M-truncated derivative affects the exact solutions of the FRWE-MTD.

Existence of novel analytical soliton solutions in a magneto-electro-elastic annular bar for the longitudinal wave equation

Existence of novel analytical soliton solutions in a magneto-electro-elastic annular bar for the longitudinal wave equation

Analysis on construction and evolution dynamics for multi-lump solutions of the dispersive long wave equations

Analysis on construction and evolution dynamics for multi-lump solutions of the dispersive long wave equations

A Stable Forward Modeling Approach in Heterogeneous Attenuating Media Using Reapplied Hilbert Transform

In the field of geological exploration and wave propagation theory, particularly in heterogeneous attenuating media, the stability of numerical simulations is a significant challenge for implementing effective attenuation compensation strategies. Consequently, the development and optimization of algorithms and techniques that can mitigate these numerical instabilities are critical for ensuring the accuracy and practicality of attenuation compensation methods. This is essential to reveal subsurface structure information accurately and enhance the reliability of geological interpretation. We present a method for stable forward modeling in strongly attenuating media by reapplying the Hilbert transform to eliminate increasing negative frequency components. We derived and validated new constant-Q wave equation (CWE) formulations and a stable solving method. Our study reveals that the original CWE equations, when utilizing the analytic signal, regenerate and amplify negative frequencies, leading to instability. Implementing our method maintains high accuracy between analytical and numerical solutions. The application of our approach to the Chimney Model, compared with results from the acoustic wave equation, confirms the reliability and effectiveness of the proposed equations and method.

Cylindrical Kadomtsev–Petviashvili equation for ion acoustic waves with double spectral indices distributed electrons

We study theoretically and numerically the nonlinear propagation of ion acoustic waves (IAWs) in plasma. The electrostatic potential in electron-ion (e-i) plasma is mimicked by the cylindrical Kadomtsev–Petviashvili equation using the traditional reductive perturbation approach. By applying the direct integration technique, the soliton solution is obtained. The results of simulations carried out numerically show that the amplitude as well as width of the soliton solution are significantly influenced by the plasma parameters. The exploration of IAWs propagation in laboratory as well as space plasmas may benefit from the current work.

Improved Earthquake Source Parameters with 3D Wavespeed Models in California and Nevada

Abstract Seismic tomography harnesses earthquake data to explore the inaccessible structure of the Earth. Adjoint waveform tomography (AWT), a method of seismic tomography, updates the tomographic model by optimizing the fit between observed earthquake data and synthetic waveforms. The synthetic data are calculated by solving the wave equation through a given 3D model. An important requirement to calculating synthetics is the source information (location, centroid time, depth, and moment tensor). Errors in source information affect the quality of the synthetics produced, which in turn can limit how structure can be inferred in the AWT workflow. To test the effect of updating source information, we used MTTime (Chiang, 2020), a time-domain full-waveform moment tensor inversion code, to calculate the moment tensors and depths of 118 earthquakes that occurred in California and Nevada over a 20-yr period. We calculated 3D Green’s functions using a 3D seismic wavespeed model of California and Nevada (Doody et al., 2023b). We show that the inverted solutions provide better waveform fits than the Global Centroid Moment Tensor catalog and increase usable, well-correlated data by up to 7%. Therefore, we argue that recalculating source parameters should be considered in AWT workflows, particularly for smaller magnitude events (Mw<5.0).

Correction of a High-Order Numerical Method for Approximating Time-Fractional Wave Equation

Correction of a High-Order Numerical Method for Approximating Time-Fractional Wave Equation

2D acoustic equation prestack reverse-time migration based on an optimized combined compact difference scheme

Abstract Reverse-time migration (RTM) is widely regarded as one of the most accurate migration methods available today. A crucial step in RTM involves extending seismic wavefields forward and backward. Compared to the conventional central finite-difference (CFD) scheme, the combined compact difference (CCD) scheme offers several advantages, including a shorter difference operator and the suppression of numerical dispersion under coarse grids. These attributes conserve memory and enhance effectiveness while maintaining the same level of differential precision. In this article, we begin with the five-point eighth-order CCD scheme and utilize the least squares method and Lagrange multiplier method to optimize the difference coefficients. This optimization is guided by the concept of dispersion-relation-preserving (DRP). The result is the acquisition of an optimized combined compact difference (OCCD) scheme, further enhancing the ability to suppress numerical dispersion. We thoroughly compare and analyze dispersion relationships and stability conditions. In addition, we examine several crucial steps in the RTM of the second-order acoustic wave equation. These steps include absorption boundary conditions, boundary storage strategy, and Poynting vector imaging conditions. Finally, we apply both the CCD and OCCD schemes in the RTM of the layered model, graben model, and SEG/EAGE salt model. We compare these results with those obtained from CFD's RTM. Numerical findings demonstrate that, in contrast to the CFD scheme, the CCD scheme effectively suppresses numerical dispersion and enhances imaging accuracy. Moreover, the optimized OCCD scheme further improves the ability to suppress numerical dispersion and can obtain better imaging results, which is an effective RTM method suitable for coarse grid conditions.

Coupled wave equations with monotone nonlinearity: existence of solution and controllability results

In this manuscript, we investigate a coupled semilinear system characterised by two wave equations in which the nonlinear function satisfies the monotonicity condition. Initially, the existence of a solution is established through the application of Schauder's fixed point theorem. Thereafter, the results pertaining to the exact controllability are demonstrated for two different cases by establishing a relation between the reachable set of the semilinear system and its linear counterpart. Additionally, an illustrative example is provided to showcase the practical relevance of our theoretical findings.

On the Continuity Equation in Space–Time Algebra: Multivector Waves, Energy–Momentum Vectors, Diffusion, and a Derivation of Maxwell Equations

Historically and to date, the continuity equation (C.E.) has served as a consistency criterion for the development of physical theories. In this paper, we study the C.E. employing the mathematical framework of space–time algebra (STA), showing how common equations in mathematical physics can be identified and derived from the C.E.’s structure. We show that, in STA, the nabla equation given by the geometric product between the vector derivative operator and a generalized multivector can be identified as a system of scalar and vectorial C.E.—and, thus, another form of the C.E. itself. Associated with this continuity system, decoupling conditions are determined, and a system of wave equations and the generalized analogous quantities to the energy–momentum vectors and the Lorentz force density (and their corresponding C.E.) are constructed. From the symmetry transformations that make the C.E. system’s structure invariant, a system with the structure of Maxwell’s field equations is derived. This indicates that a Maxwellian system can be derived not only from the nabla equation and the generalized continuity system as special cases, but also from the symmetries of the C.E. structure. Upon reduction to well-known simpler quantities, the results found are consistent with the usual STA treatment of electrodynamics and hydrodynamics. The diffusion equation is explored from the continuity system, where it is found that, for decoupled systems with constant or explicitly dependent diffusion coefficients, the absence of external vector sources implies a loss in the diffusion equation structure, transforming it into Helmholtz-like and wave equations.

A semi-analytical solution approach for fuzzy fractional acoustic waves equations using the Atangana Baleanu Caputo fractional operator

A semi-analytical solution approach for fuzzy fractional acoustic waves equations using the Atangana Baleanu Caputo fractional operator

Global well-posedness for a system of quasilinear wave equations on a product space

Global well-posedness for a system of quasilinear wave equations on a product space

PDM relativistic quantum oscillator in Einstein–Maxwell–Lambda space-time

In this paper, we study the relativistic dynamics of quantum oscillator fields within the context of a position-dependent mass (PDM) system in the background of a curved space-time. The chosen curved space-time is generated by a magnetic field incorporating a non-zero cosmological constant called Einstein–Maxwell–Lambda solution. To analyze PDM quantum oscillator fields, we introduce a modification into the Klein–Gordon equation by substituting the four-momentum vector [Formula: see text], where various four-vectors are defined by [Formula: see text], [Formula: see text] with [Formula: see text], and [Formula: see text] is the mass oscillator frequency. The radial wave equation for the modified Klein–Gordon equation is derived and subsequently solve for two distinct scalar multipliers: (i) [Formula: see text] and (ii) [Formula: see text], where [Formula: see text] and [Formula: see text]. The resultant approximate energy levels and wave function for quantum oscillator fields are demonstrated to be influenced by the cosmological constant and the geometrical topology parameter which breaks the degeneracy of the energy spectrum. Furthermore, we observed noteworthy modifications in the approximate energy levels and wave function when compared to the results derived in the flat space.

A Complex-Scaled Boundary Integral Equation for Time-Harmonic Water Waves

A Complex-Scaled Boundary Integral Equation for Time-Harmonic Water Waves