Nonlocal Wave Research Articles

Asymptotic profile and optimal decay of solutions of some wave equations with logarithmic damping

We introduce a new model of the nonlocal wave equation with a logarithmic damping mechanism, which is rather weak as compared with frequently studied fractional damping cases. We consider the Cauchy problem for the new model in \(\mathbf{R}^{n}\) and study the asymptotic profile and optimal decay rates of solutions as \(t \rightarrow \infty \) in \(L^{2}\)-sense. The damping terms considered in this paper is not studied so far, and in the low-frequency parameters, the damping is rather weakly effective than that of well-studied power type one such as \((-\Delta )^{\theta }u_{t}\) with \(\theta \in (0,1)\). When getting the optimal rate of decay, we meet the so-called hypergeometric functions with special parameters, so the analysis seems to be more difficult and attractive.

On the Choice of Kernel Function in Nonlocal Wave Propagation

It is a challenge to choose the appropriate kernel function in nonlocal problems. We tackle this challenge from the aspect of nonlocal wave propagation and study the dispersion relation at the analytical level. The kernel function enters the formulation as an input. Any effort to narrow down this function family is valuable. Dispersion relations of the nonlocal governing operators are identified. Using a Taylor expansion, a selection criterion is devised to determine the kernel function that provides the best approximation to the classical (linear) dispersion relation. The criterion is based on selecting the smallest coefficient in magnitude of the dominant term in the Taylor expansion after the constant term. The governing operators are constructed using functional calculus, which provides the explicit expression of the eigenvalues of the operators. The ability to express eigenvalues explicitly allows us to obtain dispersion relations at the analytical level, thereby isolating the effect of discretization on the dispersion relation. With the presence of expressions of eigenvalues of the governing operator, the analysis is clear and accessible. The choices made to obtain the best approximation to the classical dispersion relation become completely transparent. We find that the truncated Gaussian family is the most effective compared to power and rational function families.

Energy dissipation for hereditary and energy conservation for non-local fractional wave equations.

Using the method of a priori energy estimates, energy dissipation is proved for the class of hereditary fractional wave equations, obtained through the system of equations consisting of equation of motion, strain and fractional order constitutive models, that include the distributed-order constitutive law in which the integration is performed from zero to one generalizing all linear constitutive models of fractional and integer orders, as well as for the thermodynamically consistent fractional Burgers models, where the orders of fractional differentiation are up to the second order. In the case of non-local fractional wave equations, obtained using non-local constitutive models of Hooke- and Eringen-type in addition to the equation of motion and strain, a priori energy estimates yield the energy conservation, with the reinterpreted notion of the potential energy. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Fourier Spectral Methods for Nonlocal Models

Efficient and accurate spectral solvers for nonlocal models in any spatial dimension are presented. The approach we pursue is based on the Fourier multipliers of nonlocal Laplace operators introduced in Alali and Albin (Appl Anal :1–21, 2019). It is demonstrated that the Fourier multipliers, and the eigenvalues in particular, can be computed accurately and efficiently. This is achieved through utilizing the hypergeometric representation of the Fourier multipliers in which their computation in n dimensions reduces to the computation of a 1D smooth function given in terms of 2F3. We use this representation to develop spectral techniques to solve periodic nonlocal time-dependent problems. For linear problems, such as the nonlocal diffusion and nonlocal wave equations, we use the diagonalizability of the nonlocal operators to produce a semi-analytic approach. For nonlinear problems, we present a pseudo-spectral method and apply it to solve a Brusselator model with nonlocal diffusion. Accuracy and efficiency of the spectral solvers are discussed.

Nonlocal Dispersion Analysis of a Fluid – Conveying Thermo Elastic Armchair Single Walled Carbon Nanotube Under Moving Harmonic Excitation

In this work, the nonlocal elastic waves in a fluid conveying armchair thermo elastic single walled carbon nanotube under moving harmonic load is studied using Eringen nonlocal elasticity theory via Euler Bernoulli beam equation. The governing equations that contains partial differential equations for single walled carbon nanotube is derived by considering thermal and Lorenz magnetic force. The small scale interactions induced by the nano tubes are simulated by the non-local effects. The time domain responses are obtained by using both modal super position method and Newmarks’s direct integration method. The effect of nonlocal parameter, thermal load, magnetic field of the moving harmonic load on the dynamic displacement of SWCNT is discussed. The results obtained show that the dynamic displacement of fluid conveying SWCNT ratio is significantly affected by the load velocity and the excitation frequency. This type of results presented here, will provide useful information for researchers in structural nano science to understand the small scale response of elastic waves coupled with thermo elasticity and some field forces.

Wave dispersion in nonlocal anisotropic macro/nanoplates made of functionally graded materials

By means of a higher-order shear deformation theory (HSDT) coupled with the nonlocal elasticity theory, this work is attempted to elucidate the size-dependent behavior of functionally graded (FG) Anisotropic macro/nanoplates. In the HSDT, the displacement field is modeled as a combination of an exponential formula and a trigonometric function. This type of theory has only five unknowns but can have precise results in various conditions for thick and thin plates. By employing Hamilton’s principle, the governing equations are derived for FG anisotropic macro/nanoplates. In the case of FG anisotropic materials, all components of elastic stiffness tensor are involved, and the material gradation is considered to be arbitrary in the thickness direction. In order to attain the analytical dispersion relations, an eigenvalue problem is solved. The model’s validation is evaluated by comparing authors’ results with other papers available in the open literature. The influences of nonlocal parameter, exponential factor, geometrical shape, and wave number on the circular frequencies and phase velocities are inspected. From the knowledge of authors, it is the first time that the small-scale influences for FG anisotropic macro/nanoplates based on the present higher-order shear deformation theory are investigated.

Nonstandard bilinearization and interaction phenomenon for PT-symmetric coupled nonlocal nonlinear Schrödinger equations

We develop a nonstandard bilinearization procedure to generate more general bright soliton, breather soliton and quasi-rogue wave solutions for the PT-symmetric coupled nonlocal nonlinear Schrödinger (CNNLS) equations. We achieve some novel solutions by bilinearizing both the CNNLS equations and their associated auxiliary equations in a novel way. The soliton equations are written into bilinear operator forms by means of auxiliary equations, then the 1-soliton solution and 2-soliton solution of PT-symmetric CNNLS equations are constructed. Some novel interaction properties of the PT-symmetric two-soliton solutions are derived, which can present the potential applications to the soliton wave phenomena in nonlocal wave models.

ДИНАМИКА НЕУСТОЙЧИВЫХ РЕШЕНИЙ ВОЛНОВОГО УРАВНЕНИЯ С ИСТОЧНИКАМИ

Two new accurate solutions of the wave equation with sources are obtained. The dynamics of unstable states described by these solutions is studied. Analytical forms are given for the partial derivatives of the required function with respect to the spatial coordinate and time on the plane of independent variables “the required function – time”. This structure of the solution allows us to consider nonstationary analogs of self-similar kinks describing the transition between two equilibrium states of the “medium – source” system. For the classical wave equation, a nonlinear rheonomic source is used, the behavior of which affects the properties of the relaxing kink. The conditions under which the speed of movement of the formed self-similar transfer wave is subsonic or supersonic are determined. An important role of the velocity of the inflection point of a nonselfsimilar kink has been analyzed; the threshold value of the velocity is calculated, which separates the subsonic and supersonic regimes. An unstable version of the presented solution gives a strong discontinuity of the required function with an unlimited increase in time. The stopping of the kink inflection point is an indicator of a strong rupture. An estimate of the value of the moment in time preceding the beginning of the return motion of the inflection point is indicated. A solution to a spatially nonlocal fourth-order wave equation with two additively entering sources is given. One source depends on the desired function in a linear homogeneous way; the second one depends on the modulus of the gradient of the desired function the same way. The solution is an analog of an overthrow wave in an interval with non-stationary boundaries. At each finite moment of time this solution is continuous, and for an infinite time there is a loss of smoothness of the solution, we have the so-called “slow explosion”. In the unstable solution, the isolines of the sought-for function on the concave section (the lower part of the kink) move towards the convex section, which is adjacent to the upper boundary of the kink. In the stable version, the kink degenerates into a homogeneous state. It has been analyzed that for a nonselfsimilar process, the inversion of the sign of the gradient source gives an inversion of the stability conditions for the kink and antikink. An unstable kink/ antikink corresponds to a gradient sink/source.

Analysis of the controllability from the exterior of strong damping nonlocal wave equations

We make a complete analysis of the controllability properties from the exterior of the (possible) strong damping wave equation associated with the fractional Laplace operator subject to the non-homogeneous Dirichlet type exterior condition. In the first part, we show that if 0 <s< 1, Ω ⊂ ℝN(N≥ 1) is a bounded Lipschitz domain and the parameterδ> 0, then there is no control functiongsuch that the following system\begin{align} u_{1,n}+ u_{0,n}\widetilde{\lambda}_{n}^++ \delta u_{0,n}\lambda_{n}=\int_0^{T}\int_{\Omc}(g(x,t)+\delta g_t(x,t))e^{-\widetilde{\lambda}_{n}^+ t}\mathcal{N}_{s}\varphi_{n}(x)\d x\d t,\label{39}\\ u_{1,n}+ u_{0,n}\widetilde{\lambda}_{n}^- +\delta u_{0,n}\lambda_{n}=\int_0^{T}\int_{\Omc}(g(x,t)+\delta g_t(x,t))e^{-\widetilde{\lambda}_{n}^- t}\mathcal{N}_{s}\varphi_{n}(x)\d x\d t,\label{40} \end{align}is exact or null controllable at timeT> 0. In the second part, we prove that for everyδ≥ 0 and 0 <s< 1, the system is indeed approximately controllable for anyT> 0 andg∈D(O× (0,T)), whereO⊂ ℝN\ Ω is any non-empty open set.

Analysis of the nonlocal wave propagation problem with volume constraints

Analysis of the nonlocal wave propagation problem with volume constraints

Hysteresis phenomena in gravity–capillary waves on deep water generated by a moving two-dimensional/three-dimensional air-blowing/air-suction forcing

Hysteresis phenomena in forced gravity–capillary waves on deep water where the minimum phase speed are experimentally investigated. Four kinds of forcings are considered: two-dimensional/three-dimensional air-blowing/air-suction forcings. For a still-water initial condition, as the forcing speed increases from zero towards a certain target speed ( ), there exists a certain critical speed ( ) at which the transition from linear to nonlinear states occurs. When , steady linear localized waves are observed (state I). When , steady nonlinear localized waves, including steep gravity–capillary solitary waves, are observed (state II). When , periodic shedding phenomena of nonlinear localized depressions are observed (state III). When , steady linear non-local waves are observed (state IV). Next, with these state-II, III and IV waves as new initial conditions, as the forcing speed is decreased towards a certain target speed ( ), a certain critical speed ( ) is identified at which the transition from nonlinear to linear states occurs. When , relatively steeper steady nonlinear localized waves, including steeper gravity–capillary solitary waves, are observed. When , linear state-I waves are observed. These are hysteresis phenomena, which show jump transitions from linear to nonlinear states and from nonlinear to linear states at two different critical speeds. For air-blowing cases, experimental results are compared with simulation results based on a theoretical model equation. They agree with each other very well except that the experimentally identified critical speed ( ) is different from the theoretically predicted one.

Well-posedness results for fractional semi-linear wave equations

This work is concerned with well-posedness results for nonlocal semi-linear wave equations involving the fractional Laplacian and fractional derivative operator taken in the sense of Caputo. Representations for solutions, existence of classical solutions, and some \begin{document}$ L^{p} $\end{document} -estimates are derived, by considering a quasi-stationary elliptic problem that comes from the realisation of the fractional Laplacian as the Dirichlet-to-Neumann map for a non-uniformly elliptic problem posed on a semi-infinite cylinder. We derive some properties such as existence of global weak solutions of the extended semi-linear integro-differential equations.

Nanoptera and Stokes curves in the 2-periodic Fermi–Pasta–Ulam–Tsingou equation

This work presents asymptotic solutions to a singularly-perturbed, period-2 FPUT lattice and uses exponential asymptotics to examine ‘nanoptera’, which are nonlocal solitary waves with constant-amplitude, exponentially small wave trains which appear behind the wave front. Using an exponential asymptotic approach, this work isolates the exponentially small oscillations, and demonstrates that they appear as special curves in the analytically-continued solution, known as ‘Stokes curves’ are crossed. By studying the asymptotic form of these isolated oscillations, it is shown that there are special mass ratios which cause the oscillations to vanish, producing localized solitary-wave solutions. The asymptotic predictions are validated through comparison with numerical simulations.

Nonlinear wave equations from a non-local complex backward–forward derivative operator

ABSTRACT In this study, we argue that it is possible to construct non-local nonlinear wave equations with random propagation as extensions to some well-known wave equations found in field theory with cubic interactions like Maxwell–Klein–Gordon equations, Dirac–Klein–Gordon equations, wave maps, solitons and Yang–Mills equations. Our construction is based on the extended complex backward–forward derivative operator which represents the basic differentiable operator tool to deal with non-local Lagrangians.

Direct Measurement of a Nonlocal Entangled Quantum State.

Entanglement and the wave function description are two of the core concepts that make quantum mechanics such a unique theory. A method to directly measure the wave function, using weak values, was demonstrated by Lundeen etal. [Nature 474, 188 (2011)]. However, it is not applicable to a scenario of two disjoint systems, where nonlocal entanglement can be a crucial element, since that requires obtaining weak values of nonlocal observables. Here, for the first time, we propose a method to directly measure a nonlocal wave function of a bipartite system, using modular values. The method is experimentally implemented for a photon pair in a hyperentangled state, i.e., entangled both in polarization and momentum degrees of freedom.

A semi-discrete numerical method for convolution-type unidirectional wave equations

Numerical approximation of a general class of nonlinear unidirectional wave equations with a convolution-type nonlocality in space is considered. A semi-discrete numerical method based on both a uniform space discretization and the discrete convolution operator is introduced to solve the Cauchy problem. The method is proved to be uniformly convergent as the mesh size goes to zero. The order of convergence for the discretization error is linear or quadratic depending on the smoothness of the convolution kernel. The discrete problem defined on the whole spatial domain is then truncated to a finite domain. Restricting the problem to a finite domain introduces a localization error and it is proved that this localization error stays below a given threshold if the finite domain is large enough. For two particular kernel functions, the numerical examples concerning solitary wave solutions illustrate the expected accuracy of the method. Our class of nonlocal wave equations includes the Benjamin–Bona–Mahony equation as a special case and the present work is inspired by the previous work of Bona, Pritchard and Scott on numerical solution of the Benjamin–Bona–Mahony equation.

Nonlocal coupled thermoelastic wave propagation band structures of nano-scale phononic crystal beams based on GN theory with energy dissipation: An analytical solution

This paper deals with the band structures of thermoelastic waves in nano-scale phononic crystal or metamaterial beams considering nano-size effects. The nonlocal coupled thermoelastic governing equations are derived using the Green–Naghdi theory of the generalized coupled thermoelasticity with energy dissipation and Eringen’s nonlocal theory. The derived governing equations are analytically solved and the field quantities including the temperature and the deflection are obtained in the closed forms. Using the proposed analytical solution, the transfer matrix between two unit-cells are obtained using the thermal and mechanical continuity conditions on the interfaces between the unit-cells and between the two sections of each unit-cell. The band structures of the phononic crystals are obtained using the Bloch–Floquet theorem. The detailed discussions are presented for the band structures of nonlocal thermoelastic waves in nano-scale aluminum/epoxy phononic crystal or metamaterial beams. Also, the effects of the small-scale parameter and the thickness of the epoxy layers on the band structures are studied and discussed by using the derived analytical solution.

Dynamics of Upwelling Annual Cycle in the Equatorial Pacific Ocean

A recent work proposed a simple theory based on the framework of Zebiak–Cane (ZC) ocean model, and successfully characterized the equatorial Atlantic upwelling annual cycle as a combination of the local wind-driven Ekman upwelling and nonlocal wind-driven wave upwelling. In the present work, utilizing the same simple framework, we examined the fidelity of the upwelling Pacific annual cycle using observations and simulations from the Coupled Model Intercomparison Project Phase 5 (CMIP5). We demonstrated that the theoretical upwelling annual cycles generally match the original upwelling annual cycles in the equatorial Pacific in both observations and CMIP5 simulations. Therefore, this simple formulation can be used to represent the upwelling annual cycle in the equatorial Pacific. Observationally, the equatorial Pacific upwelling annual cycle is dominated by the local wind-driven Ekman upwelling, while the remote wave upwelling is confined near the eastern boundary with little contribution. In CMIP5 simulations, though the theoretical-reconstructed upwelling well-reproduces the original upwelling, the contribution is totally different compared to the observation. The wave upwelling serves as the main contributor instead of the Ekman upwelling. We further demonstrated that such discrepancy is attributable to the bias of the central to eastern equatorial thermocline depth patterns. This amplified, westward-shift wave upwelling weakened the impacts of the Ekman upwelling, and contributes to the entire Pacific equatorial upwelling annual cycle substantially. This implies that a realistic simulation of the equatorial Pacific upwelling annual cycle in models is very sensitive to the careful simulation of the equatorial thermocline depth annual evolutions.

Nonlinear interaction of domain walls and breathers with a spin-wave field

Abstract We present the analytical solutions of the sine–Gordon model for magnetic solitons in a field of nonlocalized spin waves. We show, that nonlinear interference of solitons and waves leads to oscillations of solitons, relaxing with time in a power-law fashion. We study an interaction of the solitons with the running spin waves of the system: we find the expressions for the velocities and frequencies of the kinks and the breathers in the field of radiation.

A variational scheme for hyperbolic obstacle problems

We consider an obstacle problem for (possibly non-local) wave equations, and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme. We provide the corresponding numerical implementation and raise some open questions.