Envelope solitary waves and their reflection and transmission due to impurities in a granular material

A nonlinear Schrodinger equation in one-dimensional bead chain is first obtained and an envelope solitary wave of the system is verified numerically in this system. The reflection and the transmission of an incident envelope solitary wave due to impurities has also been investigated. It is found that the magnitudes of both the reflection and the transmission not only depend on the characters of impurity materials, the wave number, the incident wave amplitude, but also on the impurity number. This can be used to detect the character and the number of the impurity materials in the bead chain by measuring the reflection and the transmission of an incident pulse.

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

In the past few decades, the （1 ＋ 1）-dimensional nonlinear Schr6dinger （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 Schrodinger 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.

Groupiness of Sea Waves and Their Characteristic Parameters

Three methods for studying wave groups and their main parameters for describing wave groupiness are reviewed in this paper. Then they are analyzed and compared combined with field data from both aspects of group height and group length. A method and two parameters that can describe wave groupiness are suggested. The groupiness parameters of sea waves at three field stations are given. The effects of groupiness on both distributions of the wave height and the phase of component waves are investigated. The effects of datum length on the calculated value of grouping parameters are also discussed.

Modeling of nonlinear envelope solitons in strongly coupled dusty plasmas:Instability and collision

Modeling of instability and collision of nonlinear dust-acoustic（NDA） envelope solitons in strongly coupled dusty plasmas（SCDPs） is theoretically investigated. The SCDPs consists of strongly correlated negatively variable-charged dust grains and weakly correlated Boltzmann electrons and ions. Using the derivative expansion perturbation technique, a nonlinear Schr dinger-type（NLST） equation for describing the propagation of NDA envelope solitons is derived. Moreover,the extended Poincar′e–Lighthill–Kuo（EPLK） method is employed to deduce the analytical phase shifts and the trajectories after the collision of NDA envelope solitons. In detail, the results show that both modulation instability and phase shift after collision of NDA envelope solitons will modify with the increase in the effects of the viscosity, the relaxation time, and the dust charge fluctuation. Crucially, the modeling of dust-acoustic envelope solitons collision, as reported here, is helpful for understanding the propagation of NDA envelope solitons in strongly coupled dusty plasmas.

Particle-in-Cell Simulation of the Reflection of a Korteweg-de Vries Solitary Wave and an Envelope Solitary Wave at a Solid Boundary

Reflections of a Korteweg-de Vries （KdV） solitary wave and an envelope solitary wave are studied by using the particle-in-cell simulation method. Defining the phase shift of the reflected solitary wave, we notice that there is a phase shift of the reflected KdV solitary wave, while there is no phase shift for an envelope solitary wave. It is also noted that the reflection of a KdV solitary wave at a solid boundary is equivalent to the head-on collision between two identical amplitude solitary waves.

Application of Maximum Entropy Distribution to the Statistical Properties of Wave Groups

The new distributions of the statistics of wave groups based on the maximum entropy principle are presented. The maximum entropy distributions appear to be superior to conventional distributions when applied to a limited amount of information. Its applications to the wave group properties show the effectiveness of the maximum entropy distribution. FFF filtering method is employed to obtain the wave envelope fast and efficiently. Comparisons of both the maximum entropy distribution and the distribution of Longuet-Higgins （1984） with the laboratory wind-wave data show that the former gives a better fit.

Numerical Simulation and Physical Simulation of Sea Wave Groups

Based on field wave data, an empirical formula of wave envelope spectrum is given in this paper. Then the methods of both numerical and physical simulation of sea wave groups with the given spectrum and groupiness parameters are suggested.

New Exact Solutions to NLS Equation and Coupled NLS Equations

A transformation is introduced on the basis of the projective Riccati equations, and it is applied as an intermediate in expansion method to solve nonlinear Schrǒdinger (NLS) equation and coupled NLS equations. Manykinds of envelope travelling wave solutions including envelope solitary wave solution are obtained, in which some arefound for the first time.

Effect of Nonlinearity on Groupiness of Random Wave Fields

Theoretical studies so far on random wave groups have all been in Linear ways. Methods to simulate random wave groups, an important subject in ocean engineering, also employ relationship resulting from a Gaussian process. Many filed measurements have shown that the real sea surface displacement deviates somewhat from Gaussian distribution. Tayfun et al, have further depicted in theory that the envelope spectral peak frequency is constantly zero for a Gaussian process which means that the groupiness factors will be constants, too. In this paper, the effect of nonlinearity on groupiness of a random wave field is examined via the theoretical results derived by Tayfun et al. from an expression of amplitude-modulated Stokes waves. When the surface displacement is treated as a non-Gaussian process, it is found that the group height factors GF(1) and GF(2) proposed by Zhao et al. and Yu et al., respectively, depend on a nonlinearity factor as well as a spectrum-bandwidth factor, deferring from the case of a Gaussion process. Comparison between the theoretical results and the field data shows a favorable agreement in consideration of errors from instrumentation and measuring means. The significance of the results is also discussed.

Spatial and Time Structure of a Gravity Wave in Horizontal Atmosphere of Heterogeneous Stratification

By the use of the WKBJ method combined with the characteristic line method, the asymptotic solution of a gravity wave envelope in the atmosphere of horizontal heterogeneous stratification and time-varying stratification is obtained. The solution shows that not only the variation of amplitude of the gravity wave but also the variation of wavelength and the width of the envelope are affected by the horizontal heterogeneity. As the wave envelope moves from a region, of strong stratification to a weak one, the horizontal wavelength will become shorter, the width of the envelope will narrow and its amplitude will increase. The variation of stratification with time cannot lead to the variation of wavelength and envelope width, but the amplitude of the wave envelope will increase while the amplitude of the wave decreases in time.

Theoretical and experimental investigation on nonlinear interaction among wave groups

Wave group is important in ocean wave theory and applications. In the past, nonlinear interaction among wave groups has been studied on the basis of the nonlinear Sehrrdinger equation. Using this theoretical approach, we found that the nonlinear interaction among wave groups causes asymmetry in the shape of the wave envelope （steeper in the front of the curve of the envelope）. An important consequence of this asymmetry is that the highest wave in a wave group appears one individual wave length ahead of the center of the wave group. Further results show that the degree of envelope asymmetry increases with increasing spectral width and the wave steepness. This theoretical analysis has been supplemented by a systematic experimental study of wind waves. Laboratory and some open sea wave data were analyzed. The results show that the shape of the wind wave envelope of wind waves has the same asymmetry predicted by the theoretical approach. The observed degree of deformation of the envelope also increases with increasing spectral width and the wave steepness as predicted by theory. These conclusions have important ramifications for practical applications of ocean wave theory.

THE ENVELOPE SOLITON OF STRESS WAVE ON THE WALL OF A FLUID-FILLED CIRCULAR TUBE

The reductive perturbation method of multiple-scales is used to investigate the weak nonlinear modulation of the stress wave on the wall of a fluid-filled elastic circular tube. In the case of a single mode, the nonlinear Schrodinger equation which the wave amplitude satisfies and its envelope soliton solution of stress wave are obtained.

NONLINEAR SCHRODINGER EQUATION IN THE ROTATIONAL BAROTROPIC ATMOSPHERE AND ATMOSPHERIC BLOCKING

In this paper,the WKB method is used to obtain the nonlinear Schrdinger equation satisfied by nonlinear Rossby wave in the rotational barotropic atmosphere.It is found that the nonlinear Schrdinger equation has an envelope solitary wave solution under the condition 1≤m≤2(m the zonal wavenumber),and the phase speed of envelope solitary Rossby wave in the atmosphere is related to the square of its amplitude linearly,that is,the larger the amplitude of envelope solitary Rossby wave,the smaller its propagation speed.Farthermore, the blocking high and cut-off low pressures which are consistent with the observations of blocking in the atmo- sphere are obtained by calculating envelope solitary Rossby wave,and the blocking structures persist more than five days,the results demonstrate that the envelope solitary Rossby wave is a possible mechanism about the formation,maintenance and breakout of blocking in the atmosphere.