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Ultrasound can be used to manipulate protein function and activity, as well as for targeted drug delivery, making it a powerful diagnostic and therapeutic modality with wide applications in sonochemistry, nanotechnology, and engineering. However, a general particle-based approach to ultrasound modeling remains challenging due to the significant disparity between characteristic time scales governing ultrasound propagation. In this study, we use open-boundary molecular dynamics to simulate ultrasound waves in liquid water under ambient conditions by employing supramolecular water models, i.e., the Martini 3, dissipative particle dynamics, and many-body dissipative particle dynamics models. We demonstrate that our approach successfully reproduces the solution of the traveling wave equation and captures the velocity dispersion characteristic of high-frequency ultrasound waves. [ABSTRACT FROM AUTHOR]

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

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

This paper explores the long time/large space dynamics of the neural field equation with an exponentially decaying initial data. By establishing a Harnack type inequality, we derive the Hamilton-Jacobi equation corresponding to the neural field equation due to the elegant theory developed by Freidlin [ Ann. Probab. (1985)], Evans and Souganidis [ Indiana Univ. Math. J. (1989)]. In addition, we obtain the exact formula for the motion of the interface by constructing the explicit viscosity solutions for the underlying Hamilton-Jacobi equation. It is then shown that the propagation speed of the interface is determined by the decay rate of the initial value. As an intriguing implication, we find that the propagation speed of interface is related to the speed of traveling waves. Finally, we study the spreading speed of the corresponding Cauchy problem. To the best of our knowledge, it is the first time that the Hamilton-Jacobi approach is used in the study of dynamics of neural field equations. [ABSTRACT FROM AUTHOR]

The Fordy–Gibbons equation is a nonlinear differential equation. Physically, the motion of a damped oscillator with a more complex potential than in basic harmonic motion is described by the Fordy–Gibbons equation. For the equation under consideration, numerous novel families of precise analytical solutions are being successfully found. The soliton solutions are represented as rational and exponential functions. To further illustrate the potential and physical behavior of the equation, the findings are also stated visually. Three approaches are suggested in this paper for solving the Fordy–Gibbons equation. These solutions are new solutions. [ABSTRACT FROM AUTHOR]

This article represents a first step toward understanding the long time dynamics of solutions for the intermediate long wave equation (ILW). While this problem is known to be both completely integrable and globally well-posed in H 3 2 , much less seems to be known concerning its long time dynamics. Here we prove well-posedness at much lower regularity, namely an L2 global well-posedness result. Then we consider the case of small and localized data and show that the solutions disperse up to cubic timescale. [ABSTRACT FROM AUTHOR]

In the bistable reaction-diffusion equation on a half-line there exists an entire solution exhibiting the front propagation from the infinity of the line. We glue an arbitrary number of half-lines at a single point (junction) to build a metric graph and examine the front propagation on given half-lines (upstream branches). The aim of the article is to show the existence of an entire solution which converges to the profile of the traveling wave with an arbitrarily assigned phase-shift in each upstream branch as t → - ∞ . The proof is carried out by constructing an appropriate pair of sub-and super-solutions. [ABSTRACT FROM AUTHOR]

We consider in this paper various theoretical and numerical issues on classical one dimensional models of internal waves with surface tension. They concern the Cauchy problem, including the long time dynamic, localized solitons or multisolitons, the soliton resolution property. We survey known results, present a few new ones together with open questions and conjectures motivated by numerical simulations. A major issue is to emphasize the differences of the qualitative behavior of solutions with those of the same equations without the capillary term. [ABSTRACT FROM AUTHOR]

We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study. [ABSTRACT FROM AUTHOR]

In this article, the Kawahara and modified Kawahara equations with dual-power law nonlinearities are solved based on the sine-cosine method and finite difference schemes. The fourth-order accurate difference scheme for the Kawahara equation and the second-order accurate difference scheme for the modified Kawahara equation have been constructed. The difference schemes have energy conservative properties. The existence, uniqueness, stability, and convergence of the numerical solutions are all well established. Finally, some numerical experiments demonstrate the reliability of the theoretical statements. Also, comparisons between the solutions obtained from the exact solitary wave solutions and the linearly finite difference schemes are made to demonstrate that the present schemes are efficient and reliable, and can simulate the single or multi-solitary waves propagating at a long period. [ABSTRACT FROM AUTHOR]

In recent years, physics-informed neural networks (PINNs) have garnered widespread attentions for the ability of solving nonlinear partial differential equations (PDE) using neural networks. The paper regards PINNs as multitask learning and proposes an adaptive loss weighting algorithm in physics-informed neural networks (APINNs). APINNs could balance the magnitudes of different loss functions during the training process to ensure a balanced contribution of parameters with different magnitudes to loss functions, thereby training solutions that satisfy initial boundary conditions and physical equations. Based on the original PINNs and APINNs, we respectively simulated the solitary wave solution of the Benjamin-Ono equation, the breather wave solution of the Sine-Gordon equation and the breather wave solution of the Mukherjee-Kundu equation. In the experiment of solving the solitary wave solution of the Benjamin-Ono equation, the minimum predict error of PINNs is about 60%, while the minimum predict error of APINNs is about 1%. As for solving breather wave solution, for the Sine-Gordon equation the minimum predict error of PINNs is around 10%, while the minimum predict error of APINNs is around 2%; and for the Mukherjee-Kundu equation, the minimum predict error of PINNs is about 30%, while the minimum predict error of APINNs is about 10%. The experimental results show that compared with PINNs, the predict solutions trained by APINNs have smaller errors. • We regard PINNs as multitask learning. • APINNs are physics-informed neural networks with an adaptive loss weighting algorithm. • APINNs improve the training accuracy significantly. [ABSTRACT FROM AUTHOR]

In this work, we propose an absolute value block α -circulant preconditioner for the minimal residual (MINRES) method to solve an all-at-once system arising from the discretization of wave equations. Motivated by the absolute value block circulant preconditioner proposed in McDonald et al. (2018) [40] , we propose an absolute value version of the block α -circulant preconditioner. Since the original block α -circulant preconditioner is non-Hermitian in general, it cannot be directly used as a preconditioner for MINRES. Our proposed preconditioner is the first Hermitian positive definite variant of the block α -circulant preconditioner for the concerned wave equations, which fills the gap between block α -circulant preconditioning and the field of preconditioned MINRES solver. The matrix-vector multiplication of the preconditioner can be fast implemented via fast Fourier transforms. Theoretically, we show that for a properly chosen α the MINRES solver with the proposed preconditioner achieves a linear convergence rate independent of the matrix size. To the best of our knowledge, this is the first attempt to generalize the original absolute value block circulant preconditioner in the aspects of both theory and performance the concerned problem. Numerical experiments are given to support the effectiveness of our preconditioner, showing that the expected optimal convergence can be achieved. [ABSTRACT FROM AUTHOR]

This paper investigates the well-posedness and the optimally complete regularity of the weak solutions, the existence of strong bi-space global attractors and their upper semicontinuity for the structurally damped wave equation with two nonlinearities on $ \mathbb{R}^{N} (N\geqslant 3): u_{tt}-\Delta u+(-\Delta)^\alpha u_{t}+h(x, u_{t})+g(x, u) = f $, where $ \alpha\in [1/2, 1) $ is a perturbed dissipative index determining the dissipative strength. We show that when the nonlinearity $ g(x, u) $ is of critical growth: $ p^*: = \frac{N+2}{N-2} $ when $ \alpha\in (1/2, 1) $, and optimally subcritical growth $ p0 $, which ensures that the weak solution is just a strong one; (ii) the related dynamical system $ (S^{\alpha}(t), \mathcal{H}) $ possesses a strong $ (\mathcal{H}, \mathcal{H}_{2\alpha}) $-global attractor $ \mathcal{A}_\alpha $ for each $ \alpha\in [1/2, 1) $, where $ \mathcal{H} $ is the energy space while $ \mathcal{H}_{2\alpha} $ is the strong solution space, and it is also the standard global attractor of optimal regularity of $ (S^{\alpha}(t), \mathcal{H}) $; (iii) the family of strong $ (\mathcal{H}, \mathcal{H}_{2\alpha}) $-global attractors $ \mathcal{A}_\alpha $ is upper semicontinuous at each point $ \alpha_0\in (1/2, 1) $ in $ \mathcal{H}_{2\alpha_0} $-topology. [ABSTRACT FROM AUTHOR]

We consider the geometric inverse problem of determining an immersed obstacle in a 2D linear wave equation from overspecified boundary data. We use the topological gradient method to solve this problem. The unknown obstacle is reconstructed using the leading term of the Khon–Vogelius shape function variation. We propose a simple and fast detection algorithm. The efficiency and accuracy of the proposed approach are illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

We investigate the dispersive behaviors of Schrödinger and wave equations with partial inverse-square potentials. The crucial aspect in establishing Strichartz estimates lies in constructing the spectral measure for a Schrödinger operator with partial inverse-square potentials in R 2 + n . This approach differs from the perturbation argument presented in Burq et al. (J Funct Anal 203:519–549, 2003) and Burq et al. (Indiana Univer Math J 53:1665–1680, 2004) because the potentials have no decay in certain directions. As applications, we prove the Strichartz estimates for Schrödinger and wave equations associated with two-particle interaction Schrödinger operators. [ABSTRACT FROM AUTHOR]

We consider a network of coupled strings that contains a cycle. We show that boundary feedback stabilization that uses only data of the state at boundary nodes that are not contained in the cycle is in general impossible. We demonstrate that also with control actuators that act at the interior nodes contained in the cycle with certain standard control laws that use only Neumann control action the situation does not improve and the system is still not stable. We prove that with an actuator that uses also Dirichlet control action and is located within the cycle, exponential stabilization is possible. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exact boundary controllability for a degenerate and singular wave equation in a bounded interval with a moving endpoint. By the multiplier method and using an adapted Hardypoincar é inequality, we prove direct and inverse inequalities for the solutions of the associated adjoint equation. As a consequence, by the Hilbert Uniqueness Method, we deduce the controllability result of the considered system when the control acts on the moving boundary. Furthermore, improved estimates of the speed of the moving endpoint and the controllability time are obtained. [ABSTRACT FROM AUTHOR]

We study the controllability of a semi-discrete system obtained by discretizing in space coupled 1-D wave equations with a boundary control at one extreme. The finite difference and finite element methods introduce high frequency spurious oscillations that lead to non-uniform controllability as the discretization parameter tends to zero, see [3]. To filter these high numerical frequencies we add a numerical vanishing term to each equation, which ensures the convergence of the sequence of discrete controls to a control of the continuous coupled system of wave equations when the mesh size tends to zero. [ABSTRACT FROM AUTHOR]

For a coupled system of wave equations with locally distributed controls, we show that the state variable can be divided into three groups: the first group can exactly reach any given target, the second group can approximately reach any given target, while the last one evolutes following a conservative system, therefore, is independent of applied controls. In particular, the exactly synchronizable state by groups is independent of applied controls if and only if Kalman's minimum rank condition holds. This paper generalizes the previous work [13] from the simple case of exact synchronization to the general case of exact synchronization by groups. [ABSTRACT FROM AUTHOR]

In this paper, we present a summary of fractal calculus and propose the use of Lagrange multipliers for both fractal calculus and fractal variational calculus with constraints. We examine the application of these methods across various branches of physics. By employing fractal variational calculus with constraints, we derive fundamental equations such as the fractal mechanical wave equation, the fractal Schrödinger equation in quantum mechanics, Maxwell’s equations in fractal electromagnetism, and the Lagrange equation for constraints in fractal classical mechanics. Several examples are provided to illustrate these concepts in detail. [ABSTRACT FROM AUTHOR]

In this paper, we solve the vibration problem of a finite bar with a viscously damped boundary in conjunction with a spring on the same side subject to the support motion on the other side. To avoid the computation of complex modes and the difficulty of orthogonal conditions for the partial differential equation (PDE) of a continuous system, two alternatives are employed. One is the analytical derivation by using the diamond rule of the method of characteristics. The advantage is that this method can yield an analytical solution. However, the disadvantage is that the method can be only applied to some simple problems such as a linear system and the one-dimensional (1D) wave equation. The other numerical method, finite element method (FEM), is incorporated into a general-purpose program to solve this problem. The advantage of the FEM is that this methodology can be applied to solve various problems such as different PDEs. Therefore, the solution obtained by using the FEM is compared with that of the analytical solution. Two special cases, only a spring and a damper alone, are also considered by using the FEM. Interestingly, the same silent area is captured in the displacement profile by using the FEM for the three cases. We also find that the displacement profiles have slope discontinuity which only occurs on the characteristic line. After the wave front arrives at the boundary, various responses appear due to different boundaries, spring, damper and both. It is found that the displacement amplitude of the general case is smaller than the other two special cases. This result matches the application of engineering practice. [ABSTRACT FROM AUTHOR]

We investigate a nonlinear shallow water wave equation, which contains the Fornberg–Whitham model. The L 2 conservation law of solutions for the nonlinear equation is derived. Employing the Riccati-type differential inequalities, we give sufficient conditions such that the wave breaking of solutions happens at finite time. The wave-breaking rate and lifespan of solutions are also discussed. [ABSTRACT FROM AUTHOR]

This study investigates the paraxial approximation of the M-fractional paraxial wave equation with Kerr law nonlinearity. The paraxial wave equation is most important to describe the propagation of waves under the paraxial approximation. This approximation assumes that the wavefronts are nearly parallel to the axis of propagation, allowing for simplifications that make the equation particularly useful in studying beam-like structures such as laser beams and optical solitons. The paraxial wave equation balances linear dispersion and nonlinear effects, capturing the essential dynamics of wave evolution in various media. It plays a crucial role in understanding phenomena like diffraction, focusing, and self-phase modulation in optical fibers. It substantially contributes to our comprehension of the special characteristics of optical soliton solutions and the dynamics of soliton in a variety of optical systems. We create a range of wave structures using the powerful extended Jacobian elliptic function expansion (EJEFE) method, including periodic waves, lump-periodic waves, periodic breather waves, kink-bell waves, kinky-periodic waves, anti-kinky-periodic waves, double-periodic waves, etc. These solutions have applications in wave dynamics in different optical systems and optical fibre. Furthermore, we investigate chaotic phenomena by analyzing the model qualitatively. We analyze phase portraits in detail for a range of parameter values to provide insights into the behavior of the system. We also investigate the sensitivity analysis for diverse parametric values of the perturbated coefficient. We may use various strategies, including time series and 3D and 2D phase patterns, to identify chaotic and quasi-periodic phenomena by providing an external periodic strength. The above discussion of the suggested method demonstrates adaptability and usefulness in resolving a broad spectrum of mathematics and physical difficulties, indicating its potential for generating such optical solutions. [ABSTRACT FROM AUTHOR]

Polyethylene pipelines have been widely used in the construction of urban pipelines. Due to the non-magnetic and non-conductive properties of polyethylene pipelines, traditional metal pipe positioning techniques are unable to effectively detect polyethylene pipelines. To improve the positioning accuracy of polyethylene pipelines, this paper proposes a pipeline positioning method considering the attenuation characteristics of acoustic wave. The proposed method combines active acoustic positioning technology with the resonance characteristics of pipelines, which can accurately locate the horizontal position and depth of pipelines simultaneously. Firstly, based on the theory of acoustic wave attenuation, the solution of the acoustic wave equation in non-ideal media is derived. Then, the amplitude attenuation characteristics of acoustic waves in a buried polyethylene pipeline are analyzed using finite element simulation. By observing the maximum amplitude of acoustic signals, the horizontal direction of buried polyethylene pipelines can be effectively determined. To determine the depth of buried pipelines, the resonance phenomenon is utilized and the attenuation of acoustic waves is considered to obtain the mathematical model. The largest error of simulation results is 2% for horizontal direction, and 7.8% for depth position, indicating the proposed method can effectively locate the buried pipeline. Finally, a complete pipeline excitation and data acquisition system is established through an on-site experiment, and the experiments are conducted under various burial depth conditions. The experimental results show that the horizontal positioning error and depth positioning error are both less than 8%, indicating the proposed method can accurately locate the buried pipeline. [ABSTRACT FROM AUTHOR]