Deltons, peakons and other traveling-wave solutions of a Camassa-Holm hierarchy

In this letter, we study an integrable Camassa-Holm hierarchy whose high-frequency limit is the Camassa-Holm equation. Phase plane analysis is employed to investigate bounded traveling wave solutions. An important feature is that there exists a singular line on the phase plane. By considering the properties of the equilibrium points and the relative position of the singular line, we find that there are in total three types of phase planes. Those paths in phase planes which represented bounded solutions are discussed one-by-one. Besides solitary, peaked and periodic waves, the equations are shown to admit a new type of traveling waves, which concentrate all their energy in one point, and we name them deltons as they can be expressed as some constant multiplied by a delta function. There also exists a type of traveling waves we name periodic deltons, which concentrate their energy in periodic points. The explicit expressions for them and all the other traveling waves are given.     

Lorentz-boosted evanescent waves

Polarization, spin, and helicity are important properties of electromagnetic waves. It is commonly believed that helicity is invariant under the Lorentz transformations. This is indeed so for plane waves and their localized superpositions. However, this is not the case for evanescent waves, which are well-defined only in a half-space, and are characterized by complex wave vectors. Here we describe transformations of evanescent electromagnetic waves and their polarization/spin/helicity properties under the Lorentz boosts along the three spatial directions.     

Propagation of travelling waves in sub-excitable systems driven by noise and periodic forcing

It has been reported that traveling waves propagate periodically and stably in sub-excitable systems driven by noise [Phys. Rev. Lett. 88, 138301 (2002)]. As a further investigation, here we observe different types of traveling waves under different noises and periodic forces, using a simplified Oregonator model. Depending on different noises and periodic forces, we have observed different types of wave propagation (or their disappearance). Moreover, reversal phenomena are observed in this system based on the numerical experiments in the one-dimensional space. We explain this as an effect of periodic forces. Thus, we give qualitative explanations for how stable reversal phenomena appear, which seem to arise from the mixing function of the periodic force and the noise. The output period and three velocities (normal, positive and negative) of the travelling waves are defined and their relationship with the periodic forces, along with the types of waves, are also studied in sub-excitable system under a fixed noise intensity. Electronic supplementary material Supplementary Online Material     

Heterogeneity selected target waves and their competition: Outgoing or ingoing?

Traveling direction (outgoing or ingoing) of target waves sustained by a localized inhomogeneity and their competition rules in the complex Ginzburg-Landau equation (CGLE) are studied. We show that even a local positive (negative) frequency shift is capable of creating ingoing (outgoing) target waves. A novel transition between ingoing and outgoing target waves, as we find, can be switched by the size of the inhomogeneity. The competition rules for wave patterns (e.g., target waves) are also studied systematically. All the numerical findings are found to be in perfect agreement with the theoretical criteria that are derived based on the fundamental conceptions (e.g., group velocity, phase velocity).     

Dynamical System Approach to a Coupled Dispersionless System: Localized and Periodic Traveling Waves

We investigate the dynamical behavior of a coupled dispersionless system describing a current-conducting string with infinite length within a magnetic field. Thus, following a dynamical system approach, we unwrap typical miscellaneous traveling waves including localized and periodic ones. Studying the relative stabilities of such structures through their energy densities, we find that under some boundary conditions, localized waves moving in positive directions are more stable than periodic waves which in contrast stand for the most stable traveling waves in another boundary condition situation.     

On the Quasi-Periodic Wave Solutions and Asymptotic Analysis to a (3+1)-Dimensional Generalized Kadomtsev—Petviashvili Equation

In this paper, a (3+1)-dimensional generalized Kadomtsev—Petviashvili (GKP) equation is investigated, which can be used to describe many nonlinear phenomena in fluid dynamics and plasma physics. Based on the generalized Bell's polynomials, we succinctly construct the Hirota's bilinear equation to the GKP equation. By virtue of multidimensional Riemann theta functions, a lucid and straightforward way is presented to explicitly construct multiperiodic Riemann theta function periodic waves (quasi-periodic waves) for the (3+1)-dimensional GKP equation. Interestingly, the one-periodic waves are well-known cnoidal waves, which are considered as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional that they have two independent spatial periods in two independent horizontal directions. Finally, we analyze asymptotic behavior of the multiperiodic periodic waves, and rigorously present the relationships between the periodic waves and soliton solutions by a limiting procedure.     

Variational treatment of Faraday waves in inhomogeneous Bose-Einstein condensates

Motivated by the recent experimental progress on the collective modes of a Bose-Einstein condensate whose atomic scattering length is tuned via Feshbach resonances, we analyze by variational means the dynamics of Faraday waves in trapped Bose-Einstein condensates. These waves can be excited by modulating periodically either the strength of the magnetic trap or the atomic scattering length. To study their dynamics, we develop a variational model that describes consistently both the bulk part of an inhomogeneous, low-density, cigar-shaped condensate and small-amplitude, small-wavelength Faraday waves. The main ansatz used in the variational treatment is tailored around a set of Gaussian envelopes and we show extensions for the high-density regime using a q-Gaussian function. Finally, we show explicitly that for drives of small amplitude, the two methods of obtaining Faraday waves are equivalent, and we discuss the existing experimental results.     

Near-monochromatic water waves on the sphere

We study analytically and numerically a class of traveling and standing waves in a model of weakly non-linear gravity water waves on the sphere. These waves are ‘near-monochromatic’ in space, i.e. their amplitude consists of one spherical harmonic plus small corrections, and we see numerically that they retain this property for long time. A main feature of the model we consider is that it possesses a Hamiltonian structure. This structure is preserved by our numerical implementation, and we use formal and rigorous arguments from classical perturbation theory to understand the numerical observations.     

Random sound waves in a weakly stratified atmosphere

The influence of random mass density and velocity fields on the frequencies and amplitudes of the sound waves that propagate along a constant gravity field is examined in the limit of weak random fields, small amplitude oscillations and a weakly stratified medium. Using a perturbative method, we derive dispersion relations from which we conclude that the effect of a space-dependent random mass density field is to attenuate sound waves. Frequencies of these waves are higher than in the case of a coherent medium. A time-dependent random mass density field increases frequencies and amplifies the sounds waves. On the other hand, a space-dependent random flow reduces the wave frequencies and attenuates the sound waves. The time-dependent random flow raises the frequencies of the sound waves and amplifies their amplitudes. In the limit of the gravity-free medium the above results are in an agreement with the former findings.     

Mechanism of Thermally Biological Effects of the Millimeter Waves and Its Properties

We think that the thermally biological effects of millimeter waves are caused by the thermal motions of water molecules in the living systems, according to experimental fact that the millimeter waves can heat water, and the skin effect on the surface of the biological tissues arising from the millimeter waves. For clarifying this idea we studied the states and features of the liquid water and calculated the rotational energy-spectra of water molecules in the living systems by quantum mechanics. In fact, there is a large number of water which are polarized and have certain dipole moments in the living systems. This shows that the millimeter waves can interact with the water molecules. Through calculation of quantum rotational energy-spectra of the water molecules, we can confirm that the water molecules can absorb the millimeter waves with certain wavelength to generate the rotations of water molecules according to the principle of resonant absorption. One mechanism of the thermally biological effect of the millimeter waves is just a result produced by disorderly thermal-motions of the water molecules which are transformed from their rotation energy caused by the millimeter waves. Owing to the fact that water has a lot of biological functions and plays an important role in the living activity. Thus the heating waters by the millimeter waves can cause a lot of biological effects and phenomena in the living systems. Another mechanism of the thermally biological effect of the millimeter waves is caused by the Joule-Lenz heat arising from the skin effect of the millimeter waves in the skin layers of human beings and animals and membranes of cells which can facilitate the blood circulation in them. We finally study this effect.PACSnumbers: 87.50.Hj; 05.70.Ce; 87.15.He; 65.50.tm.     

Slow motions as inelastic strain autowaves in ductile and brittle media

Here we provide a review of research on slow motions and strain waves in the Earth and propose a substantiated hypothesis that all stress-strain perturbations in the form of slow waves propagating in solids and geomedia, including plastic waves in metals and waves in faults of different scales, are of common physical nature. Loaded solids and geomedia are active hierarchically organized multiscale systems that display nonlinear dynamics and lose their stability when disturbed by any dynamic processes at block boundaries, e.g., displacements in fault zones. Such a medium cooperatively responds to parametric excitation by generating slow strain waves (autowaves) as a way of its self-organization. In support of the proposed concept, a consistent mathematical model is suggested for describing the evolution of stress-strain states and slow strain autowaves in an unstable elastoplastic medium, and examples of simulations are presented for strain autowaves in ductile materials under tension and quasi-brittle materials and geomedia with a fault zone under compression.     

宽入射角度偏振不敏感高效异常反射梯度超表面

偏振不敏感超表面在实际应用中具有重要意义, 本文提出了一种光通信波段的、对偏振不敏感的异常反射式梯度超表面, 这种超表面对于x-偏振和y-偏振入射光都能够实现高效率的异常反射, 表现出偏振不敏感特性, 为解决传统反射式超表面的偏振敏感性问题提供了一种新途径. 它采用金属(Au)-绝缘层(SiO₂)-金属(Au)结构, 超表面的超晶胞由五个各向同性的、尺寸不同的十字形基本结构单元组成. 仿真结果表明, 这种超表面结构对不同线偏振入射平面光波有几乎相同的相位和振幅响应; 合理的选取五个基本结构单元的尺寸, 在一个超晶胞内实现了2πup 相位的覆盖, 反射光波阵面畸变小, 而且反射光都集中到异常反射级次, 在工作波长1480 nm处具有较高的异常反射率(～ 70%). 此外, 这种结构的超表面在-30°–0°的宽入射角度范围内都具有偏振不敏感的异常反射特性. 在光通信、光信号处理、显示成像等领域具有潜在的应用前景.     

Scaling of waves in the bak-tang-wiesenfeld sandpile model

We study probability distributions of waves of topplings in the Bak-Tang-Wiesenfeld model on hypercubic lattices for dimensions D>/=2. Waves represent relaxation processes which do not contain multiple toppling events. We investigate bulk and boundary waves by means of their correspondence to spanning trees, and by extensive numerical simulations. While the scaling behavior of avalanches is complex and usually not governed by simple scaling laws, we show that the probability distributions for waves display clear power-law asymptotic behavior in perfect agreement with the analytical predictions. Critical exponents are obtained for the distributions of radius, area, and duration of bulk and boundary waves. Relations between them and fractal dimensions of waves are derived. We confirm that the upper critical dimension D(u) of the model is 4, and calculate logarithmic corrections to the scaling behavior of waves in D=4. In addition, we present analytical estimates for bulk avalanches in dimensions D>/=4 and simulation data for avalanches in D相似文献   

(2+1)-dimensional dissipation nonlinear Schrödinger equation for envelope Rossby solitary waves and chirp effect

In the past few decades, the (1+1)-dimensional nonlinear Schrödinger (NLS) equation had been derived for envelope Rossby solitary waves in a line by employing the perturbation expansion method. But, with the development of theory, we note that the (1+1)-dimensional model cannot reflect the evolution of envelope Rossby solitary waves in a plane. In this paper, by constructing a new (2+1)-dimensional multiscale transform, we derive the (2+1)-dimensional dissipation nonlinear Schrödinger equation (DNLS) to describe envelope Rossby solitary waves under the influence of dissipation which propagate in a plane. Especially, the previous researches about envelope Rossby solitary waves were established in the zonal area and could not be applied directly to the spherical earth, while we adopt the plane polar coordinate and overcome the problem. By theoretical analyses, the conservation laws of (2+1)-dimensional envelope Rossby solitary waves as well as their variation under the influence of dissipation are studied. Finally, the one-soliton and two-soliton solutions of the (2+1)-dimensional NLS equation are obtained with the Hirota method. Based on these solutions, by virtue of the chirp concept from fiber soliton communication, the chirp effect of envelope Rossby solitary waves is discussed, and the related impact factors of the chirp effect are given.     

Rogue waves in the basin of intermediate depth and the possibility of their formation due to the modulational instability

The properties of rogue waves in the basin of intermediate depth are discussed in comparison with known properties of rogue waves in deep waters. Based on observations of rogue waves in the ocean of intermediate depth we demonstrate that the modulational instability can still play a significant role in their formation for basins of 20 m and larger depth. For basins of smaller depth, the influence of modulational instability is less probable. By using the rational solutions of the nonlinear Schrodinger equation (breathers), it is shown that the rogue wave packet becomes wider and contains more individual waves in intermediate rather than in deep waters, which is also confirmed by observations.     

Analogien zwischen außerordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen

Within the limits of Linear Optics we treat analogies between ordinary and extraordinary waves in uniaxial media which become conspicuous through a nonorthogonal transformation of coordinates. To any ordinary wave solution in unbounded uniaxial media we can construct a corresponding extraordinary wave solution by interchanging electrical and magnetical field components. Boundary conditions for instance for ideal conducting plane surfaces approximately preserve their original form, if the optical axis or the middle wave vector are normal to the surface. The parabolic approximative equations for slowly varying amplitudes are derived, the polarisation of these waves being considered as a slowly varying quantity. Further these approximative equations are expanded to include frequency dispersion. Through the specified transformation we can simplify problems with extraordinary waves.     

Regge poles in the collective model of the nuclear giant multipole resonances

Giant nuclear resonance states, of higher multipolarity than dipole, have recently been observed in hadron and electron scattering. They may possibly be described by the collective Goldhaber-Teller model as extended to the higher multipolarity case, and as generalized to include spin-isospin vibrations and spin waves. Based on such a model, we show that the multipole resonances, together with the conventional dipole resonances, can be generated by the motion of a few basic Regge poles through the complex angular momentum plane. This viewpoint unifies all the resonances with a given spin-isospin character but arbitrary value of the multipolarity, and suggests the existence of the higher-multipole resonances as a consequence of the existence of the dipole resonance. We also analyze these Regge poles in terms of collective surface waves (“creeping waves”) circumnavigating the nucleus, and we determine their phase velocities and attenuations. Finally, the giant resonances them selves are explained as a resonant reinforcement of phase-matched surface waves.     

Acoustics of shells

We discuss the physical phenomena that arise in the scattering of acoustic waves from fluid-immersed elastic (metal) shells which may be either evacuated or filled with the same or with a different fluid. The phenomena occurring here include the formation of circumferential (peripheral, or “surface”) waves that circumnavigate the shells, propagating either as elastic waves in the shell material or as fluid-borne waves of the Scholte-Stoneley type in the external or the internal fluid. By phase matching along a closed circuit, these waves may lead to prominent resonances in the acoustic scattering amplitude, and we demonstrate how the set of observed resonance frequencies is related to the dispersive phase velocities of the surface waves, so that one can be determined from the other. In addition, we discuss how the dispersion curves (phase velocity plotted vs. frequency) of the various types of surface waves show repulsion phenomena due to their coupling through the boundary conditions. The cases of spherical and cylindrical shells are investigated here as typical examples, and as an introductory topic we additionally mention surface waves on plates where related phenomena also occur. Both the theoretical and the experimental aspects of the present subject will be considered, including the experimental visualization of the surface waves.     

Fragmentation of rods by cascading cracks: why spaghetti does not break in half

When thin brittle rods such as dry spaghetti pasta are bent beyond their limit curvature, they often break into more than two pieces, typically three or four. With the aim of understanding these multiple breakings, we study the dynamics of a bent rod that is suddenly released at one end. We find that the sudden relaxation of the curvature at this end leads to a burst of flexural waves, whose dynamics are described by a self-similar solution with no adjustable parameters. These flexural waves locally increase the curvature in the rod, and we argue that this counterintuitive mechanism is responsible for the fragmentation of brittle rods under bending. A simple experiment supporting the claim is presented.