On the preservation of invariants in the simulation of solitary waves in some nonlinear dispersive equations

In this paper we generalize some results in the literature concerning the structure of numerical approximations to solitary wave solutions of some nonlinear, dispersive equations is studied. We prove that those time discretizations with the property of preserving, exactly or approximately up to certain order, some invariants of the problems, have a better propagation of the error and provide a more suitable simulation of the solitary waves. The generalization involves the treatment of nonlocal operators and two different kinds of equations.     

An improved two-dimensional unstructured CE/SE scheme for capturing shock waves

In this paper, the accuracy of Chang’s unstructured space–time conservation element and solution element (CE/SE) scheme is analysed for the first time. Based on a redefinition of conservation elements and solution elements, an improved two-dimensional (2D) unstructured CE/SE scheme with an adjustable parameter β is proposed to accurately capture shock waves. The new scheme can be applied to any type of grid without special treatment. Compared with Chang’s original parameter α, larger β dose not cost extra computational resources. Numerical tests reveal that the new scheme is not only clear in physical concept, compact and highly accurate but also more capable of capturing shock waves than the popular fifth-order accurate weighted essentially non-oscillatory scheme.     

Shadow waves for pressureless gas balance laws

The pressureless gas dynamic model with body force as a source is considered. The problem is solved using the procedure used for finding delta shock type solutions to a special conservation laws known as Shadow Waves. If the body force is interpreted as the acceleration constant multiplied by the density, the solution obtained in this paper looks physically reasonable since the velocities of waves are changed accordingly with the acceleration.     

Reflection from a semi-infinite stack of layers using homogenization

A canonical scattering problem is that of a plane wave incident upon a periodic layered medium. Our aim here is to replace the periodic medium by a homogenized counterpart and then to investigate whether this captures the reflection and transmission behaviour accurately at potentially high frequencies.We develop a model based upon high frequency homogenization and compare the reflection coefficients and full fields with the exact solution. For some material properties it is shown that the asymptotic behaviour of the dispersion curves are locally linear near critical frequencies and that low frequency behaviour is replicated at these critical, high, frequencies. The homogenization approach accurately replaces the periodic medium and the precise manner in which this is achieved then opens the way to future numerical implementation of this technique to scattering problems.     

Nucleus-acoustic solitary waves in white dwarfs

The formation of nucleus-acoustic solitary waves (NASWs), and their basic properties in white dwarfs containing non-relativistically or ultra-relativistically degenerate electrons, non-relativistically degenerate light nuclei, and stationary heavy nuclei have been theoretically investigated. The reductive perturbation method, which is valid for small but finite amplitude NASWs, is used. The NASWs are, in fact, associated with the nucleus acoustic (NA) waves in which the inertia is provided by the light nuclei, and restoring force is provided by the degenerate pressure of electrons. On the other hand, stationary heavy nuclei maintain the background charge neutrality condition. It has been found that the presence of the heavy nuclei significantly modify the basic features (polarity, amplitude, width, and speed) of the NASWs. The basic properties are also found to be significantly modified by the effects of ultra-relativistically degenerate electrons and relative number densities of light and heavy nuclei. The implications of our results in white dwarfs are pinpointed.     

Phase and structure state of titanium loaded by spherically converging shock waves

The phase and structural states of titanium spheres loaded by spherical converging shock waves of various intensities were studied layer by layer by means of X-ray diffraction, optical, and transmission electron microscopy. It was established that defects of different types (twins, dislocations, and adiabatic shear bands) are produced during high-rate deformation occurring in materials under such method of pulsed loading. The amount and distribution of the defects depend on the loading intensity. The presence of the ω-phase is revealed only in the layers near the external surface of the titanium sphere after low-intensity loading.     

Forward mode coupling and bandgap formation in magnonic crystals

The bandgap formation for magnetostatic surface spin waves (MSSW?s) is numerically simulated in a lateral magnetic superlattice. It is found that, while the bandgaps opened at Brillouin zone edges is an intrinsic property of the periodic lattice, the exact mechanism of their formation can be revealed by incorporating a real source of MSSW excitation into the system. The obtained results uncover that the bandgaps arise due to the coupling between the MSSW?s travelling away from the excitation source (forward modes). This behavior differs from that provided by Bragg diffraction, which is the well-established mechanism for bandgap formation in periodic lattices.     

Acoustic wave excitation by particle pulsed beams in solids

Abstract

Some experimental investigations of acoustic wave excitation during interaction of pulsed proton beams with solids and pulsed electron beams with YBa₂Cu₃O₇ have been performed.     

Solitary waves for Non-Destructive Testing applications: Delayed nonlinear time reversal signal processing optimization

An original signal processing method called delayed Time Reversal-Nonlinear Elastic Wave Spectroscopy is introduced in the present paper. The method could be used to amplify signal in certain regions of the material under Non Destructive Testing. It allows to optimize and change the shape of the received focused wave in the material, either by making the focusing sharper by decreasing the side lobes or making it wider by modifying the actual focusing peak. It is also possible to use the focused signal as a delta-basis to construct a signal with arbitrary envelope or reduce the side lobes of the focused signal. These concepts are shown to work well in the simulations and the physical experiments. This signal processing method is particularly promising for nonlinear and solitary wave analysis, since it allows to create an interaction of sharp and solitary wave peaks just underneath the receiving transducer. Due to simple and accurate linear prediction of the received interaction signal, any differences of measurements and predictions could indicate the presence of nonlinearities.     

Theoretical study of the generation of screw dislocations for Rayleigh and Lamb waves in isotropic solids

Phase singularities are generic structures which occur in all wave fields, and they are characterised by an inability to assign a value to the phase. Screw dislocations are a particular kind of phase singularity where the phase possesses a helical structure, with the singularity at the centre of the helix. In this paper we show that it is possible to generate screw dislocations on the surface of elastic isotropic solids by means of the interference of three Rayleigh waves or three Lamb waves. The dispersive character of Lamb waves leads to more complicated behaviour, which may in turn result in greater potential for applications.