Nonlocal Wave Research Articles

Propagation, reflection, and transmission of nonlocal waves at the interface between semiconductor isotropic and diffusive porous semiconductors through a higher-order fractional derivative study

The current research paper investigates the transmission of nonlocal waves occurring at the interface between a semiconductor isotropic medium and a diffusive porous solid. A higher-order fractional-order derivative heat conduction model is utilized to account for thermal effects on wave propagation, reflection, and transmission. The interface generates five coupled and one independent transmitted wave in the lower medium, along with three coupled transmitted waves and one independent SV wave. Graphical analyses are conducted to examine the propagation speed of reflected waves in both porous and non-porous media. The impact of elastic non-locality and the fractional-order parameter on the propagation speed of all waves within the media is also assessed. The amplitude ratios are computed for the incident longitudinal displacement wave at the free boundary. The research explores the influence of physical parameters on reflection and transmission coefficients, and the results are validated through the conservation of energy. Additionally, the analysis yields deductions for certain special cases.

Long-time behavior of solutions to the general class of coupled nonlocal nonlinear wave equations

Long-time behavior of solutions to the general class of coupled nonlocal nonlinear wave equations

Nonlinear tunneling of partially nonlocal dark-dark annular sneaker waves under a parabolic potential

Annular rogue waves have been reported, but little is known about how to excite and control them, particularly their tunneling effect. From the projecting relation between the vector partially nonlocal nonlinear Schrödinger equation possessing different transverse-directional diffractions and the parabolic potential and vector constant-coefficient nonlinear Schrödinger equation, via the Darboux method, partially nonlocal dark-dark annular sneaker wave approximate solutions are found. Two aspects of the study are carried out: (i) four excitations of partially nonlocal dark-dark annular sneaker waves including complete, delayed, valley-maintained and prohibitive excitations and (ii) nonlinear tunneling of partially nonlocal dark-dark annular sneaker waves are studied. The influence of two parameters R and l on dark-dark annular sneaker waves is also analyzed. The partially nonlocal characteristics of dark-dark annular sneaker waves passing through the nonlinear well implies that the layer of annular sneaker wave structures increases in xyz-coordinate when the value of the Hermite parameter adds. These findings will further our understanding of the partially nonlocal nonlinear wave in the disciplines of cold atom and optical communication.

Nonseparable wave evolution equations in quantum kinetics

A nonseparable wave-like integro-differential equation for the time evolution of the Wigner distribution function in phase space is educed from the corresponding separable kinetic equation. By employing the quantum hydrodynamical description, a non-local evolution wave equation is also derived by synthesizing the Hamilton‐Jacobi equation with that of continuity, which predicts the generation of nonlocal and quadrupole quantum phenomena in the propagation of the spatial probability density.

Non-local gravity wave turbulence in presence of condensate

The $k^{-23/6}$ wave action spectrum with an inverse cascade is one of the fundamental Kolmogorov–Zakharov solutions for gravity wave turbulence, which is part of the citation for the Dirac Medal in 2003. Instead of confirming this solution, however, several existing simulations and experiments suggest a spectrum of $k^{-3}$ in set-ups corresponding to the inverse cascade. We provide a theoretical explanation for the latter, considering the condensate that naturally forms in finite domains of experiments/simulations. Our new theory hinges on: (1) derivation of a spectral diffusion equation when non-local interactions with the condensate become dominant, for the first time systematically formulated for quartet-interaction systems; and (2) careful analysis of the asymptotics of interaction coefficient with a remarkable cancellation of all leading-order terms.

Linearly Implicit Conservative Schemes with a High Order for Solving a Class of Nonlocal Wave Equations

Linearly Implicit Conservative Schemes with a High Order for Solving a Class of Nonlocal Wave Equations

Damping Characteristics of Nonlocal Strain Gradient Waves in Thermoviscoelastic Graphene Sheets Subjected to Nonlinear Substrate Effects

Damping Characteristics of Nonlocal Strain Gradient Waves in Thermoviscoelastic Graphene Sheets Subjected to Nonlinear Substrate Effects

Well-posedness and inverse problems for semilinear nonlocal wave equations

This article is devoted to forward and inverse problems associated with time-independent semilinear nonlocal wave equations. We first establish comprehensive well-posedness results for some semilinear nonlocal wave equations. The main challenge is due to the low regularity of the solutions of linear nonlocal wave equations. We then turn to an inverse problem of recovering the nonlinearity of the equation. More precisely, we show that the exterior Dirichlet-to-Neumann map uniquely determines homogeneous nonlinearities of the form f(x,u) under certain growth conditions. On the other hand, we also prove that initial data can be determined by using passive measurements under certain nonlinearity conditions. The main tools used for the inverse problem are the unique continuation principle of the fractional Laplacian and a Runge approximation property. The results hold for any spatial dimension n∈N.

Riemann-Hilbert approach and multi-soliton solutions of nonlocal Newell-type long wave-short wave equation

This article’s purpose is to investigate the inverse scattering transform of the nonlocal long wave-short wave (LW-SW) equation and its multi-soliton solutions via Riemann-Hilbert (RH) approach. By using spectral analysis to the Lax pair of LW-SW equation, the RH problem can be constructed. However, we consider spectral analysis from the time part rather than the usual space part, since it is hard to obtain the analyticity of the space part. Then the RH problem can be solved and the formula of the soliton solutions can be given. We provide several special soliton solutions including Y-shaped solitons, V-shaped solitons, bound-state solitons and mixed four-soliton solutions. Compared with the local case, the solutions of nonlocal LW-SW equation exhibit distinct characteristics that (i) these soliton solutions are strictly symmetric with respect to x = 0 under special parameter conditions, (ii) the mixed four-soliton solution, which combines Y-type and bound-state solitons, is novel.

Characteristics of nonlocal fractional magneto-thermoviscoelastic waves in a micro-rod heated by a moving heat source

Characteristics of nonlocal fractional magneto-thermoviscoelastic waves in a micro-rod heated by a moving heat source

Solitary Wave Solutions to the General Class of Nonlocal Nonlinear Coupled Wave Equations

In this paper, we study a general class of nonlocal nonlinear coupled wave equations that includes the convolution operation with kernel functions. For appropriate selections of the kernel functions, the system becomes well-known nonlinear coupled wave equations, for instance Toda lattice system, coupled improved Boussinesq equations. A numerical scheme is proposed for the solitary wave solutions of the system using the Pethiashvili method. Using the different kernels, the validity of the numerical method has been tested.

The Kuznetsov and Blackstock Equations of Nonlinear Acoustics with Nonlocal-in-Time Dissipation

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm.

Derivation and well-posedness for asymptotic models of cold plasmas

In this paper we derive three new asymptotic models for a hyperbolic–hyperbolic–elliptic system of PDEs describing the motion of a collision-free plasma in a magnetic field. The first of these models takes the form of a non-linear and non-local Boussinesq system (for the ionic density and velocity) while the second is a non-local wave equation (for the ionic density). Moreover, we derive a unidirectional asymptotic model of the latter which is closely related to the well-known Fornberg–Whitham equation. We also provide the well-posedness of these asymptotic models in Sobolev spaces. To conclude, we demonstrate the existence of a class of initial data which exhibit wave breaking for the unidirectional model.

Ring-like partially nonlocal extreme wave of a (3+1)-dimensional NLS system with partially nonlocal nonlinearity and external potential

This paper aims to study extreme waves localized in (3+1)-dimensional space based on a (3+1)-dimensional nonautonomous partially nonlocal NLS system under the linear and parabolic potentials, which is simplified into a (2+1)-dimensional autonomous equation via a converting formula. Based on solutions of the autonomous NLS system, an approximate solution of ring-like extreme wave is found. Characteristics and evolution of ring-like extreme wave are explored in an exponential system. The influence of the magnitude and exponential parameters of diffraction on ring-like extreme wave is studied. Moreover, the impact of the Hermite polynomial parameter q, the radius parameter R and the thickness parameter w on ring-like extreme wave is also discussed.

Temporal evolution of sound pulse in shallow water under conditions of horizontal refraction. Vertical modes and space-time horizontal rays

As is known, to describe horizontal refraction in a shallow water waveguide, the “vertical modes and horizontal rays” approach (Burridge & Weinberg, 1974) is used, within which the two-dimensional eikonal equation for the modal amplitude includes the refractive index determined by the eigenvalues depending on the horizontal coordinates (x,y) and frequency (f). In other words, we have an effective two-dimensional inhomogeneous dispersive medium, to describe the propagation of signals in which, the authors propose to use the so-called space-time rays (Rytov, 1940, Connor & Felsen, 1974, Hillion, 1993) applied for the horizontal plane (STHR). The set of STHRs in a given situation is constructed for each mode. Within the framework of this approach, it is possible to obtain the evolution of the pulses of each mode from the solution of the nonstationary eikonal equation in the horizontal plane or from the corresponding Hamilton-Jacobi equations, obtained using non-local integral wave equation. Examples of constructing STHR in some typical situations (in particular in a wedge) are considered, some specific effects during signal propagation are considered, in particular, the so-called the pulse front tilt, previously discovered in laser optics (Bor & Rasz, 1985). Work was supported by ISF, grant 946/20.

The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation

The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation

(3+1)-dimensional partially nonlocal ring-like bright-dark monster waves

With the help of solutions for a vector nonlinear Schrödinger system via the Darboux approach, we find analytical approximate ring-like bright-dark monster wave solutions of the (3+1)-dimensional nonautonomous partially nonlocal vector equation with different values of two transverse diffractions. We study the excited mechanism of partially nonlocal bright-dark monster wave of this vector nonlinear Schrödinger system. Moreover, we discuss the effect for initial values of two transverse diffractions, radius and thickness parameters on the formation of all excitations for bright-dark monster waves in the exponential system.

Exact controllability for nonlocal wave equations with nonlocal boundary conditions

Exact controllability for nonlocal wave equations with nonlocal boundary conditions

A Non-Local wave equation with General fractional derivatives and time delay

We analyze, spatially one dimensional, non-local elastic body with deformation measure given in the form of General fractional derivative of Riesz type with truncated power-law kernel. It is assumed that stress at the point x and time t depends on the strain at point ξ at time t−|x−ξ|/c, where c is a constant having dimension of time. We formulate spatially one dimensional wave equation for such a body and show the influence of the parameter c on the solution.

Wave breaking phenomenon in the unidirectional non-local wave model

In this paper, we investigate wave breaking phenomenon in the unidirectional non-local wave model. Making use of the L2-conservation law and fine structure of the model, we show a local-in-space wave breaking condition for the unidirectional non-local wave model.