Nonlocal Wave Equations Research Articles

Optimal error estimates of the Fourier spectral method for a class of nonlocal, nonlinear dispersive wave equations

In this work, a Fourier spectral method for a class of nonlinear, nonlocal dispersive wave equations with periodic boundary conditions is analyzed. Optimal error estimates in H 1 / 2 -norm are obtained for the both semi-discrete and fully discrete schemes.

Toward a nonlocal theory of gravitation

AbstractThe nonlocal theory of accelerated systems is extended to linear gravitational waves as measured by accelerated observers in Minkowski spacetime. The implications of this approach are discussed. In particular, the nonlocal modifications of helicity‐rotation coupling are pointed out and a nonlocal wave equation is presented for a special class of uniformly rotating observers. The results of this study, via Einstein's heuristic principle of equivalence, provide the incentive for a nonlocal classical theory of the gravitational field.

On global well-posedness for a class of nonlocal dispersive wave equations

We prove global well-posedness in Sobolev spaces with weighted low frequencies for a class of non local dispersive wave equations.

Wave Breaking in a Class of Nonlocal Dispersive Wave Equations

The Korteweg de Vries (KdV) equation is well known as an approximation model for small amplitude and long waves in di!erent physical contexts, but wave breaking phenomena related to short wavelengths are not captured in. In this work we consider a class of nonlocal dispersive wave equations which also incorporate physics of short wavelength scales. The model is identified by a renormalization of an infinite dispersive di!erential operator, followed by further specifications in terms of conservation laws associated with the underlying equation. Several well-known models are thus rediscovered. Wave breaking criteria are obtained for several models including the Burgers-Poisson system, the Camassa-Holm type equation and an Euler-Poisson system. The wave breaking criteria for these models are shown to depend only on the negativity of the initial velocity slope relative to other global quantities.

The Cauchy Problem for a Differential-Difference Nonlocal Wave Equation

The Cauchy Problem for a Differential-Difference Nonlocal Wave Equation

Error estimates for a fully discrete spectral scheme for a class of nonlinear, nonlocal dispersive wave equations

We analyze a fully discrete spectral method for the numerical solution of the initial- and periodic boundary-value problem for two nonlinear, nonlocal, dispersive wave equations, the Benjamin–Ono and the Intermediate Long Wave equations. The equations are discretized in space by the standard Fourier–Galerkin spectral method and in time by the explicit leap-frog scheme. For the resulting fully discrete, conditionally stable scheme we prove an L 2 -error bound of spectral accuracy in space and of second-order accuracy in time.

Infinite Dimensional Lie Algebra and Symmetries of Nonlinear Equations in Three Dimension — An Overview

Symmetries of nonlinear systems integrable via inverse scattering transform in three dimension are analysed from a different point of view. Actually the classical approach of Lie Backlund transform is not so effective due to the complexity of the computations involved. So we review here the new approach for finding symmetries on the basis of Lax algebra. In the first part we mention in short the earlier techniques for the study of symmetries and in the second part the present method is reviewed in detail following the earlier work of Li Yi Shen et al., with an eye for application to nonlinear systems in three dimension. The approach is highly effective in finding symmetries of non-local, nonlinear in equations in three dimension, which is otherwise impossible to find out. The methodology is illustrated with the help of three examples. (1) the Harry Dym equation in three dimension, (2) the non-local KP equation and (3) dispersive water wave equation in three dimension. Lastly we demonstrate that these symmetries generate an infinite dimensional Lie algebra through a suitably defined commutator.

A nonlinear nonlocal wave equation arising in combustion theory

The initial value problem for the equation (∂2 / ∂t2 − ∂2 / ∂x2) ∂T / ∂t = (γ ∂2 / ∂t − ∂2 / ∂x2) eT, γ>1, is considered. It is proved that under some restrictions on the initial data there is a curve, denoted by t=φγ(x), which is positive, Lipschitz continuous, and satisfies |φ′γ(x)| 0. The case of γ=1 is also studied. The solution for γ=1 blows up on t = φ¯¯ (x), and it is proved that under certain conditions the solutions for γ>1 converge to the one for γ=1 as γ→1 and lim inf γ→1 φ γ (x)≥φ¯¯(x).

On the structure of solitary waves in molecular chains

The structure of solitary waves in a quantum molecular chain, with the contribution of phonons with different wavenumbers taken into account, is investigated. The problem is solved on the basis of a non-linear non-local wave equation. It is shown that there exist two types of localised collective excitations which depend on the excitation velocity nu and the carrier wave phase velocity nu c. The criteria are considered of continuity and stability of excited bound multiparticle states and the estimation of their validity for the typical parameter values of a molecular chain is presented.

Erratum: Green's Function for the Nonlocal Wave Equation

Erratum: Green's Function for the Nonlocal Wave Equation

A nonlocal infinite-component wave equation with a physical mass spectrum

An infinite-component wave equation is studied, which involves the nonlocal operatorW/p2, whereW is the Pauli-Lubansky invariant. A method of quantization is proposed, which leads to a local commutation relation for the field. In particular a wave equation with a linear mass spectrum and without unphysical solutions is considered. A suitable choice of the Lagrangian gives rise to an algebra of conserved currents free of Schwinger terms in the time-time commutators.

Green's Function for the Nonlocal Wave Equation

The Green's function for the nonlocal wave equation in a semi-infinite medium is calculated using the Wiener-Hopf technique and shown to be given by an integral whose integrand contains only the source-free solution of that same equation.

Transmission of Electromagnetic Waves through a Conducting Slab. II. The Peak at Cyclotron Resonance

This is the second in a series of three papers concerned with the propagation of electromagnetic waves through a metallic slab of finite thickness. In the situation studied here, the relation between the current in the slab and the electric field which drives it is highly nonlocal because of the finite electron mean free path. We calculate the transmission amplitude under these conditions, assuming that there is a steady magnetic field normal to the faces, and assuming also that electrons within the slab scatter diffusely at the surfaces. A transmission peak is predicted at cyclotron resonance which, in the limit of slab width much larger than an electron mean free path, has a line shape which approaches the square root of a Lorentzian and a phase which shifts by $\ensuremath{\pi}$ across the line. All of this is superposed on the Gantmakher-Kaner oscillations which are also present. This resonant behavior, which is absent when the electrons are assumed to be scattered specularly from the surface of the slab, can be understood in terms of electron transport from Fresnel zones on the Fermi sphere which expand at resonance so that the first zone covers the entire hemisphere. The absence of this resonance when the electrons are scattered specularly is tentatively ascribed to a fortuitous cancellation which arises because the equatorial electrons, whose response gives rise to currents which shield the interior of the metal, themselves undergo a resonance which excludes the field from the metal. In carrying out these calculations using the two-sided Wiener-Hopf technique for the nonlocal wave equation, we found that the technique itself can be simplified considerably if the quantity of interest is the transmission amplitude (the field at the emergent face of the slab) rather than the full-field amplitude in the slab. We show that this simplification arises when certain short-range parts of the field are eliminated.

A Non-local Wave Equation and its Connection with the Dirac Particle

A Non-local Wave Equation and its Connection with the Dirac Particle

A Non-local Wave Equation and its Connection with the Dirac Particle

A Non-local Wave Equation and its Connection with the Dirac Particle