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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 basis-set correction method based on density-functional theory consists in correcting the energy calculated by a wave-function method with a given basis set by a density functional. This basis-set correction density functional incorporates the short-range electron correlation effects missing in the basis set. This results in accelerated basis convergences of ground-state energies to the complete-basis-set limit. In this work, we extend the basis-set correction method to a linear-response formalism for calculating excited-state energies. We give the general linear-response equations as well as the more specific equations for configuration-interaction wave functions. As a proof of concept, we apply this approach to the calculations of excited-state energies in a one-dimensional two-electron model system with harmonic potential and a Dirac-delta electron–electron interaction. The results obtained with full-configuration-interaction wave functions expanded in a basis of Hermite functions and a local-density-approximation basis-set correction functional show that the present approach does not help in accelerating the basis convergence of excitation energies. However, we show that it significantly accelerates basis convergences of excited-state total energies. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the boundary value problem of the wave equation with nonlocal damping ‖ut‖2qut$$ {\left\Vert {u}_t\right\Vert}_2^q{u}_t $$ 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)

We consider an ensemble Kalman inversion scheme for inverse elastic scattering problems in which the unknown quantity is the shape of the scatterer. Assume that the scatterer is a piecewise constant function with known value inside inhomogeneities. The level set method is described as an implicit representation of the scatterer boundary, with Gaussian random fields serving as prior to provide information on the level set functions. The ensemble Kalman filter method is then employed based on the level set functions to reconstruct the shape of the scatterer. We demonstrate the effectiveness of the proposed method using several numerical examples. [ABSTRACT FROM AUTHOR]

We consider wave equations with a critically singular potential ξ ⋅ σ − 2 diverging as an inverse square at a hypersurface σ = 0. Our aim is to construct counterexamples to unique continuation from σ = 0 for this equation, provided there exists a family of null geodesics trapped near σ = 0. This extends the classical geometric optics construction [3] of Alinhac-Baouendi (i) to linear differential operators with singular coefficients, and (ii) over non-small portions of σ = 0 , by showing that such counterexamples can be further continued as long as this null geodesic family remains trapped and regular. As an application to relativity and holography, we construct counterexamples to unique continuation from the conformal boundaries of asymptotically Anti-de Sitter spacetimes for some Klein-Gordon equations; this complements the unique continuation results of the second author with Chatzikaleas, Holzegel, and McGill [12,18,19,24] and suggests a potential mechanism for counterexamples to the AdS/CFT correspondence. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the properties of a five-dimensional Ricci-flat black hole within the framework of the Space–Time–Matter theory. By employing the Klein–Gordon equation, we derive the wave function equations for both the mass dimensional coordinate and the radial dimensional coordinate. We present the analytical solutions for the wave function on the mass dimensional coordinate under two distinct scenarios. Remarkably, we observe the absence of quantum effects on the mass dimensional coordinate, which aligns with the absence of a barrier in the wave function equation. For the radial component, we employ the B-spline method to numerically solve its wave function. Our findings reveal the existence of Hawking radiation between the event horizon and the cosmological horizon, with the transmission and reflection probabilities being determined by the presence of a barrier. Furthermore, we discover a significantly higher probability of particle detection near the cosmological horizon, while the probability near the event horizon approaches zero. [ABSTRACT FROM AUTHOR]

This study deals with the Lonngren wave equation, which is used to model the transmission of electrical signals in a type of semiconductor material called a tunnel diode. The aim of this study is to produce dark soliton, singular soliton, trigonometric, complex rational solutions to the Lonngren wave equation by using the sub-equation method. Graphs representing the Lonngren wave are drawn by giving different values to the variables in the solutions obtained. Graphs are presented using a special package program. In the light of the data obtained as a result of the study, it is thought that this new method used in the research of the solution of the Lonngren wave equation will contribute to the literature. [ABSTRACT FROM AUTHOR]

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes recently developed for the scalar wave and heat equations for heterogeneous materials is extended to a more complex case of a system of the elastodynamics PDEs. Compact 9-point stencils (similar to those for linear finite elements) are used for OLTEM. Compared to our previous results, a new approach is used for the calculation of the right-hand side of the stencil equations due to body forces. It significantly simplifies the analytical derivations of OLTEM for time-dependent problems. There are no unknowns on interfaces between different materials; the structure of the global semi-discrete equations for OLTEM is the same for homogeneous and heterogeneous materials. For the first time we have also developed OLTEM with the diagonal mass matrix. In contrast to many known approaches with some ad-hoc calculations of the diagonal mass matrix, OLTEM offers a rigorous approach which is a particular case of OLTEM with the non-diagonal mass matrix. Another novelty of the article is a new post-processing procedure for the accurate calculations of stresses. It includes the same compact 9-point stencils as those in basic computations and uses the accelerations and the displacements at the grid points along with the PDEs for the stress calculations. OLTEM yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Numerical experiments for elastic heterogeneous materials show: a) at the same number of degrees of freedom (dof), OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; at the engineering accuracy of 0.1 % for the displacements, OLTEM reduces the number of dof by more than 20 times; at the engineering accuracy of 0.1 % for the stresses, OLTEM with the new post-processing procedure reduces the number of dof by more than 104 times compared to linear finite elements; b) at the same number of dof, OLTEM with unfitted Cartesian meshes is even more computationally efficient than high-order finite elements with much wider stencils and conformed meshes. This will lead to a huge reduction in the computation time for elastodynamics problems solved by OLTEM and will allow the direct computations of some complex wave propagation and structural dynamics problems for heterogeneous materials without the scale separation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a finite element approximation of an optimal control problem governed by a time fractional wave equation. We first prove the well-posedness and regularity of the optimal control problem. Furthermore, we discuss the fully discrete scheme and analyze its stability and a priori error estimate. Numerical experiments are given to illustrate the theoretical findings. • The weak solution to the time fractional wave equation is studied under low regular data. • The well-posedness and regularity of the optimal control problem are proven. • Numerical schemes exist the first-order and second-order accuracy in time and space respectively. [ABSTRACT FROM AUTHOR]

In this work, we establish novel solitary wave solutions of three nonlinear evolution dynamical models, namely, Boussinesq model, (3 + 1) -dimensional generalized shallow water wave (SWW) and (4 + 1) -dimensional Fokas dynamical models by applying two-variable (G ′ ∕ G , 1 ∕ G) - expansion technique. The SWW equations are usually appropriate when the fluid is in a state where the vertical length scale is significantly smaller than the length of horizontal scale. In computer simulations for ocean engineering, Boussinesq-type models are frequently used to represent water waves in harbors and shallow seas. In order to understand the physical processes of waves inside and on the surface of water, the Fokas dynamical model plays a crucial part in wave theory. Analytical solutions in different forms such as solitons, solitary waves, trigonometric, hyperbolic, rational function solutions, breathers-type waves and more wave solutions are devised through using the proposed method. The exact solutions are also presented in graphical form having applications in engineering and other areas of applied sciences. The obtained results show that the given technique is universal and efficient. In addition, this technique can also be applied on lots of other nonlinear dynamical wave models occurring in many scientific real-world application domains, including engineering sciences, mathematics physics, and many more. [ABSTRACT FROM AUTHOR]

We use seismic ambient noise recorded by dense ocean bottom nodes (OBNs) in the Gorgon gas field, Western Australia, to compute time‐lapse seafloor models of shear‐wave velocity. The extracted hourly cross‐correlation (CC) functions in the frequency band 0.1–1 Hz contain mainly Scholte waves with very high signal‐to‐noise ratio. We observe temporal velocity variations (dv/v) at the order of 0.1% with a peak velocity change of 0.8% averaged from all station pairs, from the conventional time‐lapse analysis with the assumption of a spatially homogeneous dv/v. With a high‐resolution reference (baseline) model from full waveform inversion of Scholte waves, we present an elastic wave equation based double‐difference inversion (EW‐DD) method, using arrival time differences between the reference and time‐lapsed Scholte waves, for mapping temporally varying dv/v in the heterogeneous subsurface. The time‐lapse velocity models reveal increasing/decreasing patterns of shear‐wave velocity in agreement with those from the conventional analysis. The velocity variation exhibits a ∼24‐hr cycling pattern, which appears to be inversely correlated with the diurnal variations in sea level height, possibly associated with dilatant effects for porous, low‐velocity shallow seafloor and rising pore pressure with higher sea level. This study demonstrates the feasibility of using dense passive seismic surveys and wave‐equation time‐lapse inversion for quantitative monitoring of subsurface property changes in the horizontal and depth domain. Plain Language Summary: Unlike seismic waves generated by earthquakes or human‐made sources, seismic ambient noise (passive seismic) is the ubiquitous background vibration of the solid Earth recorded by seismic sensors. We use seismic interferometry to extract Scholte waves at hourly intervals from ambient noise data collected by dense ocean bottom nodes. Scholte waves travel along the interface between the ocean and the seafloor. This workflow facilitates continuous passive monitoring of the seafloor, without using human‐made seismic sources. Conventional passive monitoring techniques usually assume a spatially homogeneous relative velocity changes. With this assumption, the waveform differences on the extracted Scholte waves reveal temporal variations in the velocity of shear waves up to 0.8%. The velocity variation in this study exhibits a ∼24‐hr cycling pattern, which seems inversely correlated with diurnal variations in sea level height, possibly associated with dilatant effects for porous, low‐velocity shallow seafloor and rising pore pressure with high sea level. Furthermore, we push the limits of passive monitoring with advanced wave‐equation based inversion technique, allowing us to mapping the velocity change into detailed spatial distribution. Therefore we not only infer how velocity changes in time, but also provide insights on where the velocity changes occur in 3‐D beneath the seabed. Key Points: We present a method for space‐time monitoring of subsurface velocity changes in the horizontal and depth domain using seismic ambient noiseWe compute time‐lapse images of seafloor seismic velocity and observe diurnal shear‐wave velocity changes up to 0.8%The wave‐equation based method is suitable for 4‐D subsurface passive monitoring using dense seismic arrays [ABSTRACT FROM AUTHOR]

Accurate seismic wave modeling of viscoelastic anisotropic medium is a fundamental tool for seismic data processing, interpretation and full waveform inversion. Also, free water surface, topographic relief and irregular seabed are often encountered in practical seismic surveys. Thus, basing on the General Maxwell Body, we proposed a generalized matrix form of the velocity-stress seismic wave equation, which becomes valid for composite viscoelastic anisotropic media and satisfies the boundary conditions in presence of topographic free surfaces and irregular fluid–solid interfaces. We theoretically show that the viscoelastic effect of a medium may be considered as the intrinsic body sources accumulated in wavefield history and computed by a recursive convolution formula accurately and efficiently. We also demonstrated that such a generalized viscoelastic wave equation may be solved with the curvilinear MacCormack finite difference method and validated the accuracy and feasibility of the proposed method. The modeling results in homogeneous and heterogeneous media match well with the analytical solutions and the references yielded by the spectral element solutions. [ABSTRACT FROM AUTHOR]

In the present work, we study the global existence and uniform decay rates of solutions to the initial-boundary value problem related to the dynamic behavior of evolution equations accounting for rotational inertial forces along with a linear time-nonlocal Kelvin-Voigt damping arising in viscoelastic materials. By constructing appropriate Lyapunov functional, we show that the solution converges to the equilibrium state exponentially in the energy space. [ABSTRACT FROM AUTHOR]

The article examines the accuracy of homogenization processes in describing the behavior of materials with microstructure. It discusses how composite materials such as laminates or fiber-reinforced plastics are produced by cleverly combining individual material components at the microscopic level. Homogenization allows for the mathematical prediction of the effective behavior of such composite materials. The article also addresses the homogenization of the wave equation for materials with periodic microstructure. It is shown that classical homogenization is unable to capture the long-term change in waveforms. It is emphasized that homogenization results should be viewed with caution as they may not adequately represent reality. [Extracted from the article]

The enduring stability exhibited by solitons is strikingly demonstrated as a soliton pulse traverses an ideal lossless optical fibre, thereby highlighting a compelling attribute for their integration into optical communication systems. In this study, we employ the improved Bernoulli sub-equation function method to systematically derive stable and functionally robust soliton solutions for the Riemann wave equation. The stability of the soliton solutions is demonstrated through their composition involving hyperbolic and exponential functions, among others. The physical significance of these solutions is meticulously analyzed by presenting 2D and 3D graphs that illustrate the behaviour of the solutions for specific parameter values. Additionally, a comprehensive investigation into the influence of the nonlinear parameter on the wave velocity and solution curve is conducted. The study further explores local stability through bifurcation and phase plane analysis. Our findings affirm the reliability of the improved Bernoulli sub-equation function method and suggest its potential application in future endeavours to uncover diverse and novel soliton solutions for other nonlinear evolution equations encountered in the realms of mathematical physics and engineering. [ABSTRACT FROM AUTHOR]

For the Cauchy problem of nonlinear elastic wave equations of three-dimensional isotropic, homogeneous and hyperelastic materials satisfying the null condition, global existence of classical solutions with small initial data was proved in Agemi (Invent Math 142:225–250, 2000) and Sideris (Ann Math 151:849–874, 2000), independently. In this paper, we will consider the asymptotic behavior of global solutions. We first show that the global solution will scatter, i.e., it will converge to some solution of linear elastic wave equations as time tends to infinity, in the energy sense. We also prove the following rigidity result: if the scattering data vanish, then the global solution will also vanish identically. The variational structure of the system will play a key role in our argument. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of fourth-order dispersive wave equations with exponential source term. Firstly, by applying the contraction mapping principle, we establish the local existence and uniqueness of the solution. In the spirit of the variational principle and mountain pass theorem, a natural phase space is precisely divided into three different energy levels. Then we introduce a family of potential wells to derive a threshold of the existence of global solutions and blow up in finite time of solution in both cases with sub-critical and critical initial energy. These results can be used to extend the previous result obtained by Alves and Cavalcanti (Calc. Var. Partial Differ. Equ. 34 (2009) 377–411). Moreover, an explicit sufficient condition for initial data leading to blow up result is established at an arbitrarily positive initial energy level. [ABSTRACT FROM AUTHOR]

The pseudo-differential operator with symbol | k |α has been widely used in seismic modelling and imaging when involving attenuation, anisotropy and one-way wave equation, which is usually calculated using the pseudo-spectral method. For large-scale problems, applying high-dimensional Fourier transforms to solve the wave equation that includes pseudo-differential operators is much more expensive than finite-difference approaches, and it is not suitable for parallel computing with domain decomposition. To mitigate this difficulty, we present a truncated space-domain convolution method to efficiently compute the pseudo-differential operator |$\sqrt{-\nabla ^2}$|⁠ , and then apply it to viscoacoustic reverse-time migration. Although |$\sqrt{-\nabla ^2}$| is theoretically non-local in the space domain, we take the limited frequency band of seismic data into account, and constrain the approximated convolution stencil to a finite length. The convolution coefficients are computed by solving a least-squares inverse problem in the wavenumber domain. In addition, we exploit the symmetry of the resulting convolution stencil and develop a fast spatial convolution algorithm. The applications of the proposed method in Q -compensated reverse-time migration demonstrate that it is a good alternative to the pseudo-spectral method for computing the pseudo-differential operator |$\sqrt{-\nabla ^2}$|⁠ , with almost the same accuracy but much higher efficiency. [ABSTRACT FROM AUTHOR]

The increasing need for petrophysical parameter inversion has spurred the advancement of full-waveform inversion (FWI). This study introduces a new inversion method that can directly estimate petrophysical parameters from seismic waveform data. Specifically, we modified the regular elastic wave equation by including Biot poroelastic parameters and designed an inversion workflow based on the framework of elastic FWI. The newly introduced inversion algorithms have demonstrated their effectiveness and accuracy using the Marmousi model and a synthetic model related to the gas hydrate deposits based on the Shenhu area of the South China Sea. Our work demonstrates the feasibility of directly inverting petrophysical parameters and assessing if a deposit contains gas hydrate. [ABSTRACT FROM AUTHOR]

Material density remains poorly constrained in seismic imaging problems, yet knowledge of density would provide important insight into physical material properties for the interpretation of subsurface structures. We test the sensitivity to subsurface density contrasts of spatial and temporal gradients of seismic ambient noise wavefields, using wave equation inversion (WEI), a form of seismic gradiometry. Synthetic results for 3-D acoustic media suggest that it is possible to estimate relative density structure with WEI by using a full acoustic formulation for wave propagation and gradiometry. We show that imposing a constant density assumption on the medium can be detrimental to subsurface seismic velocity images. By contrast, the full acoustic formulation allows us to estimate density as an additional material parameter, as well as to improve phase velocity estimates. In 3-D elastic media, severe approximations in the governing wave physics are necessary in order to invert for density using only an array of receivers on the Earth's free surface. It is then not straightforward to isolate the comparatively weak density signal from the influence of phase velocity using gradiometric WEI. However, by using receivers both at the surface and in the shallow subsurface we show that it is possible to estimate density using fully elastic volumetric WEI. [ABSTRACT FROM AUTHOR]

To calculate the acoustic problems of relative uniform motion between the acoustic source and the fluid, we propose a boundary element method (BEM) strategy that can calculate various forms of relative uniform motion in subsonic conditions in a unified framework and is simple to implement. The acceleration algorithm for the BEM, like the fast multipole method (FMM), in the relative motionless state between the source and the fluid can be directly used without major modifications to the program. We propose a two-step transformation method to unify the wave equations of different relative motion forms into the classical form. In the first step, we transform the wave equations for various forms of relative motion into the equation where the convective terms are present only in the source part. Then, in the second step, we propose an acoustic-analogy Lorentz (a-a Lorentz) transformation to apply Lorentz covariance further to eliminate the convection term and establish the wave equation with classical form in a-a Lorentz space. We implement the boundary integration in the transformed a-a Lorentz space and derive a transformation method to transform discretized geometry and boundary conditions in the original space to the a-a Lorentz space. The problem that the boundary conditions are difficult to apply when solving the boundary integral equation (BIE) after the time-space coordinate transformation is solved. Numerical validations for the proposed method are performed by comparing with analytical results over a wide range of relative velocities. The results show that the proposed method can efficiently compute such problems with high accuracy and concise formulation. [ABSTRACT FROM AUTHOR]

This article explores the characteristics of Love wave in a piezoelectric layer imperfectly bonded over a heterogeneous fluid-saturated cracked dual porous half-space. The governing equations of the piezoelectric layer are solved analytically to determine mechanical displacement and electrical potential. Dispersion equations have been established using suitable boundary conditions for electrically open and electrically short conditions. In a particular case, the dispersion equation is reduced to the classical Love wave equation, which validates the present mathematical model. The effect of different parameters (like imperfection, dielectric constant, piezoelectric, and inhomogeneity) on phase velocity has been discussed theoretically and graphically, along with a comparative study by considering two piezoelectric materials, namely, PZT-4 and BaTiO3. [ABSTRACT FROM AUTHOR]

In this work we define the generalized φ -pullback attractors for evolution processes in complete metric spaces, which are compact and positively invariant families, that pullback attract bounded sets with a rate determined by a decreasing function φ that vanishes at infinity, called decay function. We find conditions under which a given evolution process has a generalized φ -pullback attractor, both in the discrete and in the continuous cases. We present a result for the special case of generalized polynomial pullback attractors, and apply it to obtain such an object for a nonautonomous wave equation. [ABSTRACT FROM AUTHOR]

In this work, we use two newly powerful and efficient techniques, the unified method and the unified Riccati equation expansion (UREE) method, to derive solitary wave solutions for the (1 + 1)-dimensional longitudinal wave equation (LWE), presenting a systematic approach to reveal the underlying dynamics. By incorporating mathematical analysis and numerical simulations, we investigate the behavior and properties of these solitary wave solutions in various fields of science. The derived solutions are entirely new, and these findings show the essential complexities of the longitudinal wave equation. These obtained solutions shed light on the fundamental mechanisms governing solitary waves in the longitudinal wave equation, contributing to a deeper understanding of nonlinear wave phenomena. We present effective visualizations of the dynamical wave structures using various graphs, such as 3D, 2D, and contour plots in the obtained solutions to improve our comprehensive understanding. Consequently, numerous types of wave profiles, including periodic wave, interaction between periodic wave and kink wave, W-shape wave, kink wave, interaction between breather wave and kink wave, traveling wave, mixed periodic, singular soliton, and some new types of soliton profiles were found. Consequently, this work presents a comprehensive study of solitary wave solutions using the unified method and UREE method for the (1 + 1)-dimensional LWE equation. The outcomes of the current study manifest that the considered methods are significant and systematic in solving nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This work investigates the stochastic nonlinear Schrödinger equation in a more extended form, influenced by multiplicative noise which represents the temporal change of fluctuations. To obtain novel optical soliton solutions, the nonlinear partial differential equation is transformed into a nonlinear ordinary differential equation via symmetry reduction (non-classical symmetry). This stochastic nonlinear problem is proposed to be solved by using the modified generalized Riccati equation mapping method. The resulting stochastic solitons show how waves dispersion throughout transmissions via optical fibers. With the use of this method, we can investigate an expansive range of solutions from important physical standpoints, such as dark, bright, singular, and periodic solitons, as well as their noise term effects related to Brownian motion based on the Itô sense. Both 2D and 3D graphs are used to display the impact of the noise term on the solitons. The comes about and calculations show the importance, exactness, and effectiveness of the strategy. The model under consideration is also examined through the concept of modulation instability analysis. Different stable and unstable nonlinear stochastic differential equations that are found in mathematics, physics, and other connected ranges can be solved by utilizing this technique. Furthermore, the bifurcation of phase portraits are studied. [ABSTRACT FROM AUTHOR]

In this research work, we utilized the auxiliary equation technique and extracted the optical soliton solutions to the nonlinear longitudinal wave equation (NLLWE) in magneto electro elastic (MEE) rod that was spread out in a circle. The NLLWE in MEE systems deals with the mathematical physics of transverse Poisson's effect dispersal and also very important in many engineering fields like sensors and actuators. As a result, we extracted the exact soliton solutions in bright solitons, dark solitons, kink wave solitons, anti-kink wave solitons, combined bright-dark solitons, solitary waves and periodic singular solitons. The physical structure of some extracted solutions visualizing in contour, two, and three dimensions through numerical simulation. The explored soliton solutions are interested, more general and having different physical structure, which may will be helpful to study of physical phenomena in the fields of optical fibers, plasma physics, soliton wave theory, nonlinear optics, ocean engineering, nonlinear dynamics and different branches of applied sciences. The successful extraction of exact solitons shows that this utilized approach is effective, straightforward, concise and powerful can also applicable to other nonlinear partial differential equations that involve in mathematical physics, engineering and applied sciences. [ABSTRACT FROM AUTHOR]

We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, which is also known as the hyperbolic Φ 3 4 -model. This result is the hyperbolic counterpart to seminal works on the parabolic Φ 3 4 -model by Hairer (Invent. Math. 198(2):269–504, 2014) and Hairer-Matetski (Ann. Probab. 46(3):1651–1709, 2018). The heart of the matter lies in establishing local in time existence and uniqueness of solutions on the statistical ensemble, which is achieved by using a para-controlled ansatz for the solution, the analytical framework of the random tensor theory, and the combinatorial molecule estimates. The singularity of the Gibbs measure with respect to the Gaussian free field brings out a new caloric representation of the Gibbs measure and a synergy between the parabolic and hyperbolic theories embodied in the analysis of heat-wave stochastic objects. Furthermore from a purely hyperbolic standpoint our argument relies on key new ingredients that include a hidden cancellation between sextic stochastic objects and a new bilinear random tensor estimate. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with a quasilinear wave equation with nonlinear damping and source terms of variable exponents-type acting in a part of the boundary. Under suitable conditions on the exponents and the initial data, we study the blow-up properties. Firstly, by using Faedo-Galerkin method and Banach-Fixed-Point Theorem, we establish the existence of a weak solution, under suitable assumptions on the variable exponents and the initial data. Secondly, we show a finite time blow-up with lower and upper bound as well. Next, an infinite time blow-up is proved under some conditions in the exponents and the initial data. [ABSTRACT FROM AUTHOR]

In this article, we consider the nonlinear stochastic partial differential equation of fractional order in both space and time variables with constant initial condition: \begin{equation*} \left (\partial ^{\beta }_t+\dfrac {\nu }{2}\left (-\Delta \right)^{\alpha / 2}\right) u(t, x) = \: I_{t}^{\gamma }\left [\lambda u(t, x) \dot {W}(t, x)\right ] \quad t>0,\: x\in \mathbb {R}^d, \end{equation*} with constants \lambda \ne 0 and \nu >0, where \partial ^{\beta }_t is the Caputo fractional derivative of order \beta \in (0,2], I_{t}^{\gamma } refers to the Riemann-Liouville integral of order \gamma \ge 0, and \left (-\Delta \right)^{\alpha /2} is the standard fractional/power of Laplacian with \alpha >0. We concentrate on the scenario where the noise \dot {W} is the space-time white noise. The existence and uniqueness of solution in the Itô-Skorohod sense is obtained under Dalang's condition. We obtain explicit formulas for both the second moment and the second moment Lyapunov exponent. We derive the p-th moment upper bounds and find the matching lower bounds. Our results solve a large class of conjectures regarding the order of the p-th moment Lyapunov exponents. In particular, by letting \beta = 2, \alpha = 2, \gamma = 0, and d = 1, we confirm the following standing conjecture for the stochastic wave equation: \begin{align*} \frac {1}{t}\log \mathbb {E}[|u(t,x)|^p ] \asymp p^{3/2}, \quad \text {for p\ge 2 as t\to \infty.} \end{align*} The method for the lower bounds is inspired by a recent work of Hu and Wang, where the authors focus on the space-time colored Gaussian noise case. [ABSTRACT FROM AUTHOR]

This study focuses on investigating hard-magnetic soft materials, characterized by magnetoactive polymers containing magnetically polarized particles as fillers. The research utilizes the Gent model of hyperelasticity to analyze the propagation of Lamb waves in a magnetically induced deformed compressible plate. In this investigation, we explore both finite deformations and incremental wave propagation in nonlinear hard-magnetic soft materials. The main objective is to formulate the elastic tensor and relevant wave equations within the framework of Lagrangian space. To assess the dispersion characteristics of the guided wave, the study introduces and discusses an extension of the semi-analytical finite element (SAFE) method. Using this numerical approach, the research further examines the effects of magnetic flux densities and its orientation with respect to wave propagation direction on the dispersion characteristics of the fundamental Lamb modes. The study starts by examining the limiting case of the neo-Hookean material model to explain such inherent dependencies. These dependencies are then further emphasized by including the strain-stiffening effect that the Gent material model describes. The research findings reveal the presence of a threshold applied magnetic flux, beyond which the Gent-type material may undergo a snap-through instability, resulting in changes in the dispersion characteristics of the fundamental symmetric Lamb mode. [ABSTRACT FROM AUTHOR]

We investigate the structure of rank-to-rank elementary embeddings at successor rank, working in ZF set theory without the Axiom of Choice. Recall that the set-theoretic universe is naturally stratified by the cumulative hierarchy, whose levels V α are defined via iterated application of the power set operation, starting from V 0 =∅, setting V α+1 =P(V α), and taking unions at limit stages. Assuming that j:V α+1 →V α+1 is a (non-trivial) elementary embedding, we show that V α is fundamentally different from V α+1: we show that j is definable from parameters over V α+1 iff α+1 is an odd ordinal. The definability is uniform in odd α+1 and j. We also give a characterization of elementary j:Vα+2 → Vα+2 in terms of ultrapower maps via certain ultrafilters. For limit ordinals λ, we prove that if j:Vλ→Vλ is Σ1-elementary, then j is not definable over Vλ from parameters, and if β < λ and j:Vβ → Vλ is fully elementary and ∈-cofinal, then j is likewise not definable. If there is a Reinhardt cardinal, then for all sufficiently large ordinals α, there is indeed an elementary j: Vα → Vα, and therefore the cumulative hierarchy is eventually periodic (with period 2). [ABSTRACT FROM AUTHOR]

In this two-paper series, we prove the invariance of the Gibbs measure for a threedimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a paracontrolled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences. [ABSTRACT FROM AUTHOR]

Efficient solution strategies for wave propagation problems with complex geometries are essential in many engineering fields. Immersed boundary methods simplify mesh generation by embedding the domain of interest into an extended domain that is easy to mesh, introducing the challenge of dealing with cells that intersect the domain boundary. We consider the finite cell method that extends the weak form from the physical into the extended domain, multiplied by a small value, to stabilize badly cut cells. Combined with explicit time integration methods, the presence of finite cells with very little support in the physical domain results in tiny critical time step sizes. While the finite cell stabilization limits how small the critical time step size can become, this limit is still restrictive for many applications. Explicit transient analyses commonly use the spectral element method due to its natural way of obtaining diagonal mass matrices through nodal lumping. The resulting method is very efficient since nodal lumping renders the solution of the equation systems trivial while not reducing the accuracy or the critical time step size. The combination of the spectral element and finite cell methods is called the spectral cell method. Unfortunately, a direct application of nodal lumping in the spectral cell method is impossible due to the special quadrature necessary to treat the discontinuous integrand inside the cut cells. The existing approaches to lump the mass matrices of cut cells significantly reduce the accuracy of the approximation. We analyze an implicit-explicit (IMEX) time integration method to exploit the advantages of the nodal lumping scheme for uncut cells on one side and the unconditional stability of implicit time integration schemes for cut cells on the other. In this hybrid, immersed Newmark IMEX approach, we use explicit second-order central differences to integrate the uncut degrees of freedom that lead to a diagonal block in the mass matrix and an implicit trapezoidal Newmark method to integrate the remaining degrees of freedom (those supported by at least one cut cell). The immersed Newmark IMEX approach preserves the high-order convergence rates and the geometric flexibility of the finite cell method while retaining the efficiency of the nodal lumping scheme for uncut cells without compromising the critical time step size. We analyze a simple system of spring-coupled masses to highlight some of the essential characteristics of Newmark IMEX time integration. We then solve the scalar wave equation with immersed boundaries on two- and three-dimensional examples with significant geometric complexity to show that our approach is more efficient than state-of-the-art time integration schemes when comparing accuracy and runtime. While we focus on the finite and spectral cell methods, we expect other immersed methods to benefit equally from Newmark IMEX time integration. [ABSTRACT FROM AUTHOR]

Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative partial differential equations such as the linear Schrödinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space H 2 , a new H 2 -like optimal model reduction problem is introduced and first-order optimality conditions are derived. As in the classical H 2 case, these conditions exhibit a rational Hermite interpolation structure for which an iterative model reduction algorithm is proposed. Numerical examples demonstrate the effectiveness of the new method. [ABSTRACT FROM AUTHOR]

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

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

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

We consider the Galerkin and Collocation Isogeometric approximations of the acoustic wave equation with absorbing boundary conditions, while the time discretization is based on second-order Newmark schemes. A numerical study investigates the properties of the two IGA methods as concerns stability thresholds, convergence errors, accuracy, computational time, and sparsity of the stiffness matrices varying the polynomial degree p , mesh size h , regularity k , and time step Δ t. In order to compare the two IGA methods, we focus on two meaningful examples in the framework of wave propagation simulations: a test problem with an oscillatory exact solution having increasing wave number, and the propagation of two interfering Ricker wavelets. Numerical results show that the IGA Collocation method retains the convergence and stability properties of IGA Galerkin. In all examples considered, IGA Collocation is in general less accurate when we adopt the same choices of parameters p , k , h , and Δ t. On the other hand, regarding the computational cost and the amount of memory required to achieve a given accuracy, we observe that the IGA Collocation method often outperforms the IGA Galerkin method, especially in the case of maximal regularity k = p − 1 with increasing NURBS degree p. [ABSTRACT FROM AUTHOR]

According to the governing wave equations for unsaturated porous media, a theoretical model is proposed to obtain the transient response solution of single-layer unsaturated porous media in one-dimensional conditions with general boundary conditions and arbitrary initial conditions when subjected to arbitrary vertical load. Parameters total stress σ , absolute soil displacement u , wetting fluid pressure pw , relative displacement w , nonwetting fluid pressure pn , and relative displacement v , which are independent of each other, are suggested firstly to express the boundary conditions through a linear combination method in pairs. Various boundary conditions can be obtained through adjusting the linear combination parameters. By utilizing the variable separation method, the eigenvalues and eigenfunctions of the undamped characteristic equation are obtained. Through the method of undetermined coefficients and orthogonality of eigenfunctions, the transient response problem can be converted to an initial value problem, which includes a series of ordinary differential equations. By making use of the precise time-integration method modified by the authors, the semianalytical solutions can finally be approached. Compared with existing research, the semianalytical solutions of this study are more general and can be exactly degenerated into some other boundary conditions. The validity of this method is verified by comparing the results with the current literature. Then, several examples are carried out to study the transient responses characteristic of unsaturated porous media subjected to the step load. The results demonstrate that there are three kinds of compression wave in the dynamic response of unsaturated porous media. In a short time, the semipermeable condition results are similar to permeable and impermeable conditions. However, results become complex over a long time due to the reflection and superposition of the compression waves. [ABSTRACT FROM AUTHOR]

We have developed a numerical scheme for the second‐order acoustic wave equation based on the Lie product formula and Taylor‐series expansion. The scheme has been derived from the analytical solution of the wave equation and in the approximation of the time derivative for a wavefield. Through these two equations, we obtained the first‐order differential equation in time, where the time evolution operator of the analytic solution of this differential equation is written as a product of exponential matrices. The new numerical solution using a Lie product formula may be combined with Taylor‐series, Chebyshev, Hermite and Legendre polynomial expansion or any other expansion for the cosine function. We use the proposed scheme combined with the second‐ or fourth‐order Taylor approximations to propagate the wavefields in a recursive procedure, in a stable manner, accurately and efficiently with even larger time steps than the conventional finite‐difference method. Moreover, our numerical scheme has provided results with the same quality as the rapid expansion method but requiring fewer computations of the Laplacian operator per time step. The numerical results have shown that the proposed scheme is efficient and accurate in seismic modelling, reverse time migration and least‐squares reverse time migration. [ABSTRACT FROM AUTHOR]

Stained migration algorithms have the potential to enhance imaging results in specific geological structures, such as subsalt layers and steeply inclined layers, by incorporating the complex domain. However, conventional stained migration algorithms often require correct staining of the target geological body, which is challenging to achieve in practical applications, leading to inaccurate imaging results. To address this issue, we propose a novel regional staining algorithm that takes into account energy-constrained factors. First, we derive the discrete form of the complex domain wave equation and the imaging formula for the imaginary migration result. We then investigate the impact of stained region boundaries on seismic wave propagation when different migration velocities are used as input. By analyzing the energy difference between the reflected waves generated at the stained region boundary and those generated by the target geological body, we introduce positive and negative energy-constrained factors to control the interference waves caused by stained boundaries. The positive energy-constrained factor effectively filters out weak interference waves, while the negative energy-constrained factor eliminates strong interference waves. Numerical examples using a simple three-layer horizontal model and a complex salt model demonstrate that our new regional staining algorithm, based on energy-constrained factors, enhances the illumination of the target geological body and improves the imaging resolution of the target region, even when the stained areas are incorrect. [ABSTRACT FROM AUTHOR]

This paper concern to determine a local energy decay estimate for synchronizable solutions of a coupled system of linear wave equations. Here, it is shown that the energy for such solutions decays at a rate of t−n$$ {t}&amp;#x0005E;{-n} $$ (n$$ n $$ is the space dimension) when the coupling matrix satisfy the positive C1$$ {C}_1 $$‐compatibility condition. [ABSTRACT FROM AUTHOR]

Despite the significant socioeconomic implications in the link between atmospheric waviness and extreme weather events, future atmospheric waviness trends remain elusive due to uncertainties arising from diverse definitions and insufficient dynamical formalism in existing metrics. This study employs a local wave activity (LWA) metric, whose prognostic equation links wave activity changes to forcing mechanisms, to assess wintertime Northern Hemisphere waviness in ERA5 and HighResMIP data sets. The models generally exhibit high fidelity in reproducing observed waviness, while biases stem primarily from biases in the LWA source, low‐level meridional heat flux, which tend to improve with higher resolutions. Future projections exhibit reduction in LWA, primarily due to suppressed LWA generation, which is mitigated by higher‐resolution models. We found that both biases and reduction of the LWA source are closely associated with sensible heat fluxes from the ocean to the atmosphere, highlighting the potential impacts of resolving ocean currents. Plain Language Summary: This study investigates atmospheric waviness in the Northern Hemisphere during winter, using a measure called local wave activity, which enables identifying causes behind waviness changes. Applying this metric to various climate models with different resolutions and a reanalysis data set, we found that historical simulations by the models generally well capture the observed climatological waviness. Biases are found over Europe and mid‐latitudes in Eurasia and the Pacific, which generally improve with higher‐resolution models. In the future simulation, reduction in waviness over much of the Northern Hemisphere, especially in the Atlantic and Pacific sectors, due to a decreased waviness source is found. Higher‐resolution models mitigate this reduction over North Atlantic. Our findings emphasize the connection between waviness and heat given from the ocean to the atmosphere, highlighting the importance of higher ocean resolution for accurate future waviness projections. This study underscores the significant role of the ocean in shaping atmospheric waviness and its implications for climate predictions. Key Points: Biases in Northern Hemisphere wintertime climatological waviness in historical runs are mostly ameliorated with high‐resolution modelsFuture Northern Hemisphere winter waviness will undergo a general reduction, which is mitigated by high‐resolution models over North AtlanticSensible heat fluxes play a crucial role in both historical biases and future reductions in waviness, highlighting the key role of the ocean [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The shallow water equations are used to describe the behavior of water waves in various shallow regions such as coastal areas, lakes, rivers, etc. These equations are derived by making simplifying assumptions about the water depth relative to the wavelength of the waves. In this paper, the generalized exponential rational function method (gERFM) is used to construct novel wave solutions of the (3+1)-dimensional shallow water wave ((3+1)-dSWW) dynamical model. These solutions encompass distinct kinds of waves, such as solitary waves, solitons, Kink and anti-kink solitons, lump Kink interactional waves, traveling breathers-type waves and multi-peak solitons. The dynamical behavior of these wave solutions is discussed, examining the influence of free parameters on the resulting wave shapes. Furthermore, to provide a scientific elucidation of the obtained results, the solutions are presented graphically, making it easy to distinguish the dynamical features, which have practical implications in different areas of applied sciences and engineering. The stability of this dynamical model is revealed via modulational instability analysis, signifying that all analytical results are stable. The obtained results show that the given technique is universal and efficient. Through comparing the projected technique with the existing techniques, the obtained results demonstrate that the given technique is universal, pithy and efficient. [ABSTRACT FROM AUTHOR]

In this paper, we present the result of global existence of solution for the wave equation with boundary variable damping term. Then, we prove that this global solution is stable. Our study is based on the semi‐groups theory and some integral inequalities. [ABSTRACT FROM AUTHOR]

We are concerned with the description of global quantitative stability of wave equations with linear strong damping and linear or nonlinear weak damping. By giving some energy decay estimates, we obtain several conclusions about the continuous dependence of the global solution on the initial data and the coefficients of the strong damping term and linear or nonlinear weak damping term. This work also establishes a new idea to use the dissipative effect to obtain the better continuous dependence conclusions, which also reflect the dissipative properties of the solution. [ABSTRACT FROM AUTHOR]