分层流体中gKdV型孤立波的迎撞

本文采用约化摄动法和PLK方法并通过双参数摄动展开,讨论了分层流体中以推广的Korteweg-de vries方程(gKdV方程)描述的孤立波的迎撞问题,求得了二阶近似解。分析结果表明,gKdV型孤立波碰撞后保持原来的形状不变,在碰撞时最大波幅为两个来碰孤立波的最大波幅的线性叠加。     

Solitary wave interactions and reshaping in coupled systems

The interaction of the components of composite solitary waves governed by nonlinear coupled equations is studied numerically. It is shown how predictions of the known exact traveling wave solutions may help in understanding and explaining the process of reshaping seen as head-on and take-over collisions of individual solitary waves. The most interesting results concern the switch in the sign or the periodic modulation of the amplitude of the solitary wave and the direction of its propagation due to collisions.     

Nonreciprocal Head-on Collision Between Two Nonlinear Solitary Waves in Granular Metamaterials with an Interface

In this work,the head-on collision and transmission with nonreciprocal properties of opposite propagating solitary waves are studied,in which the interface betw...     

Head-on collision of solitary waves in coupled Korteweg–de Vries systems modeling nonlinear transmission lines

The extended Poincaré–Lighthill–Kuo (PLK) method is applied to characterize head-on collisions of solitary waves in a coupled Korteweg–de Vries (KdV) system that has multiple modes supporting solitons. As a simple physically realizable system, we investigate two coupled electrical nonlinear transmission lines (NLTLs), and the proposed method successfully leads to the collision-induced phase shifts and the wave equation that governs the dynamics of the pulses generated by colliding solitary waves.     

General solution for interaction of solitary waves including head-on collisions

A general solution of the Boussinesq equation is presented which solves the problem of interaction of any number of right-going and left-going solitary waves. The solution relies on the exact solution of Gardner, Greene, Kruskal, and Miura (1967), and has the same degree of accuracy as that solution, but has a wider scope of application. It is much simpler than, but as accurate as, Hirota's exact solution (1973) of the Boussinesq equation, to which the present solution is compared for the simplest case of two solitary waves in head-on collision.     

Head-on collision between two mKdV solitary waves in a two-layer fluid system

In this paper, based on the equations presented in [2], the head-on collision between two solitary waves described by the modified KdV equation (the mKdV equation, for short) is investigated by using the reductive perturbation method combined with the PLK method. These waves propagate at the interface of a two-fluid system, in which the density ratio of the two fluids equals the square of the depth ratio of the fluids. The second order perturbation solution is obtained. It is found that in the case of disregarding the nonuniform phase shift, the solitary waves preserve their original profiles after collision, which agrees with Fornberg and Whitham's numerical result of overtaking collision161 whereas after considering the nonuniform phase shift, the wave profiles may deform after collision.     

Study on the head-on collisions of two-dimensional trains of solitons

The head on collisions of trains of solitons induced by a two-dimensional submerged elliptical cylinder at critical speed in shallow water are studied based on velocity potential theory. The boundary value problems are solved through boundary element method (BEM). The nonlinear free surface boundary conditions are satisfied. The mixed Euler–Lagrangian method is adopted to track the free surface through a time stepping scheme. The effects of thickness and velocity of the elliptical cylinder on the evolution of solitary waves have been investigated. Two sets of solitons are truncated from these trains of solitary waves. The head-on collisions of these solitons have been simulated. The wave profiles and velocity fields during collision have been analysed. The propagation of solitary waves is the transmissions of kinetic energy and the collision processes are the results of the dynamic balance of potential energy and kinematic energy.     

The generalized Boussinesq equations and obliquely interacting solitary waves in a stratified fluid

In this paper, on the basis of Boussinesq’s shallow water theory, we establish the basic equations governing the motion of a stratified fluid, a kind of the generalized Boussinesq equations. And then by way of them, we study the weak interaction of two pairs of obliquely colliding solitary waves, give the second-order approximate solutions for wave profiles and maximum amplitudes, as well as conclude that when the included angle between the directions of propagation of impinging solitary waves is less than 120°, the effect of oblique interaction is stronger than that of the head-on one, but when the angle concerned is greater than 120°, the former is slightly weaker than the latter.     

Higher order dispersive effects in regularized Boussinesq equation

In this paper, we consider the higher order Boussinesq (HBq) equation which models the bi-directional propagation of longitudinal waves in various continuous media. The equation contains the higher order effects of frequency dispersion. The present study is devoted to the numerical investigation of the HBq equation. For this aim a numerical scheme combining the Fourier pseudo-spectral method in space and a Runge–Kutta method in time is constructed. The convergence of semi-discrete scheme is proved in an appropriate Sobolev space. To investigate the higher order dispersive effects and nonlinear effects on the solutions of HBq equation, propagation of single solitary wave, head-on collision of solitary waves and blow-up solutions are considered.     

Frontal collision of internal solitary waves of first mode

The dynamics and energetics of a frontal collision of internal solitary waves (ISW) of first mode in a fluid with two homogeneous layers separated by a thin interfacial layer are studied numerically within the framework of the Navier–Stokes equations for stratified fluid. It was shown that the head-on collision of internal solitary waves of small and moderate amplitude results in a small phase shift and in the generation of dispersive wave train travelling behind the transmitted solitary wave. The phase shift grows as amplitudes of the interacting waves increase. The maximum run-up amplitude during the wave collision reaches a value larger than the sum of the amplitudes of the incident solitary waves. The excess of the maximum run-up amplitude over the sum of the amplitudes of the colliding waves grows with the increasing amplitude of interacting waves of small and moderate amplitudes whereas it decreases for colliding waves of large amplitude. Unlike the waves of small and moderate amplitudes collision of ISWs of large amplitude was accompanied by shear instability and the formation of Kelvin–Helmholtz (KH) vortices in the interface layer, however, subsequently waves again become stable. The loss of energy due to the KH instability does not exceed 5%–6%. An interaction of large amplitude ISW with even small amplitude ISW can trigger instability of larger wave and development of KH billows in larger wave. When smaller wave amplitude increases the wave interaction was accompanied by KH instability of both waves.     

Bidirectional soliton street

The bidirectional long-wave model introduced by Wu (1994)^[1] and Yih & Wu (1995)^[2] is applied to evaluate interactions between multiple solitary waves progressing in both directions in a uniform channel of rectangular cross-section and undergoing collisions of two classes, one being head-on and the other overtaking collisions between these solitons. For a binary head-on collision, the two interacting solitary waves are shown to merge during a phase-locking period from which they reemerge separated, each asymptotically recovering its own initial identity while both being retarded in phase from their original pathlines. For a binary overtaking collision between a soliton of height α₁ overtaking a weaker one of height α₁, the two solition peaks are shown to either pass through each other or remain separated throughout the encounter according as α₁/α₂ or <3, respectively. With no phase locking during the overtaking, the two solitary waves re-emerge afterwards with their initial forms recovered and with the stronger wave being advanced whereas the weaker one retarded in phase from their original pathlines. By extension, the theory is generalized to apply to uniform channels of arbitrary cross-sectional shape. The Inaugural Pei-Yuan Chou Memorial Lecture, presented at The Sixth Asian Congress of Fluid Mechanics. Singapore, 21–26 May 1995     

Non-linear water waves on shearing flows

This article is devoted to the study of the propagations of the nonlinear water waves on the shear flows. Assuming μ=kh is small andε/μ ²∼O(1), and the base flow is uniformly sheared, the modified Boussinesq equation is obtained. We calculate propagations of the single solitary wave with vorticity Γ=0,>0 and <0. The influences of the vorticity are manifested. At the end examples of the interactions of two solitary waves, moving in opposite and the same directions, are given. Besides the phase shift, there also occur second wavelets after head-on collision. The project supported by the National Natural Science Foundation of China     

On the Navier–Stokes equations simulation of the head-on collision between two surface solitary waves

The scope of this Note is to show the results obtained for simulating the two-dimensional head-on collision of two solitary waves by solving the Navier–Stokes equations in air and water. The work is dedicated to the numerical investigation of the hydrodynamics associated to this highly nonlinear flow configuration, the first numerical results being analyzed. The original numerical model is proved to be efficient and accurate in predicting the main features described in experiments found in the literature. This Note also outlines the interest of this configuration to be considered as a test-case for numerical models dedicated to computational fluid mechanics. To cite this article: P. Lubin et al., C. R. Mecanique 333 (2005).     

爆轰波对碰加载下平面金属样品动载行为实验研究

进行冲击波对碰加载简易平面金属Sn和W样品实验,采用X射线照相和激光干涉测速系统进行联合诊断,给出了2种材料冲击波对碰区表面微喷及主体破碎物质的直观图像,研究了Sn和W样品对碰区动力学行为,并比较分析2种材料对碰区特征的异同,给出了定性物理解释,实验结果可为爆轰波对碰加载下材料动力学特性的理论研究提供数据支撑。     

A parametric study of the head-on collision of normal shock waves in dusty gases

The effect of the various flow parameters, namely: the diameter of the solid particles, the material density of the solid particles, and the loading ratio of the solid particles on the flow field which is obtained when two normal shock waves collide head-on in a two phase dust-gas suspension has been investigated numerically, using the modified random choice method (RCM). The results were compared with those appropriate to the dust physical parameters used recently by Elperin, Ben-Dor and Igra in their study of the head-on collision of normal shock waves in dusty gases.     

Bilinear form and soliton interactions for the modified Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics

In this paper, we investigate the modified Kadomtsev–Petviashvili (mKP) equation for the nonlinear waves in fluid dynamics and plasma physics. By virtue of the rational transformation and auxiliary function, new bilinear form for the mKP equation is constructed, which is different from those in previous literatures. Based on the bilinear form, one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. Propagation and interactions of shock and solitary waves are investigated analytically and graphically. Parametric conditions for the existence of the shock, elevation solitary, and depression solitary waves are given. From the two-soliton solutions, we find that the (i) parallel elastic interactions can exist between the (a) shock and solitary waves, and (b) two elevation/depression solitary waves; (ii) oblique elastic interactions can exist between the (a) shock and solitary waves, and (b) two solitary waves; (iii) oblique inelastic interactions can exist between the (a) two shock waves, (b) two elevation/depression solitary waves, and (c) shock and solitary waves.     

Criticality manifolds and their role in the generation of solitary waves for two-layer flow with a free surface

The role of criticality manifolds is explored both for the classification of all uniform flows and for the bifurcation of solitary waves, in the context of two fluid layers of differing density with an upper free surface. While the weakly nonlinear bifurcation of solitary waves in this context is well known, it is shown herein that the critical nonlinear behaviour of the bifurcating solitary waves and generalized solitary waves is determined by the geometry of the criticality manifolds. By parametrizing all uniform flows, new physical results are obtained on the implication of a velocity difference between the two layers on the bifurcating solitary waves.     

Formation of solitary waves on gas-sheared liquid layers

The origin of solitary waves on gas-liquid sheared layers is studied by comparing the behavior of the wave field at sufficiently low liquid Reynolds number, R_L, where solitary waves are observed to form, to measurements at higher R_L where solitary waves do not occur. Observations of the wave field with high-speed video imaging suggest that solitary waves, which appear as a secondary transition of the stratified gas-liquid interface, emanate from existing dominant waves, but that not all dominant waves are transformed. From measurements of interface tracings it is found that for low R_L, waves which have amplitude/substrate depth (a/h) ratios of 0.5–1 occur while for higher R_L, no such waves are observed. A comparison of amplitude/wavelength ratios shows no distinction for different R_L. Consequently, it is conjectured that solitary waves originate from waves with sufficiently large a/h ratios; this change of form being similar to wave breaking. The dimensionless wavenumber is found to be smaller at low R_L, where solitary waves are observed. This suggests that perhaps, larger precursor (to solitary wave) waves are possible because the degree of dispersion, which acts to break waves into separate modes, is lower.     

Head-on collision of normal shock waves in dusty gases

The head-on collision of normal shock waves in dusty gases has been investigated numerically, using the modified random-choice method. The results concerning the various flow field properties as well as the waves configuration were compared with those of a pure gas case.     

Dynamics of stress wave propagation in a chain of photoelastic discs impacted by a planar shock wave; Part I, experimental investigation

The propagation of stress waves through a chain of discs has been studied experimentally. Optically transparent 20-mm diameter discs, made of epoxy, were loaded dynamically by head-on collision with an incident planar shock wave. The loading was done in a vertical shock tube. The head-on collision between the punch-plate, placed on top of the chain of discs, and the incident shock wave resulted in a head-on reflected shock wave inducing behind it a fairly uniform step-wise pressure pulse having duration of about 6 ms. The recorded fringe patterns of the stress field, in the discs-chain, show that the input pressure pulse was broken into several oscillating cycles. The back and forth bouncing of stress waves gave rise to two different modes of the contact stress oscillations, which continued until the overall stress reaches equilibrium with the input conditions. The registered propagation velocity of the stress wave was significantly lower than the appropriate speed of sound in the material from which the discs were made.