Interaction of a moving dipole with a soliton in the KdV equation

The Korteweg-de Vries equation with the perturbing term εδ'(x −Vt) (a point-like dipole), which models disturbances produced by a small body moving in a liquid layer, is considered. In the case V<0, when the moving dipole emits a quasi-linear monochromatic wave, perturbation of the emission spectrum due to collision of the dipole with a free soliton is investigated. It is demonstrated that prior to the collision (at ft → − ∞) the resultant spectrum's width is exponentially small in ∣t∣, while after the collision (at t → + ∞) the width is ∾t⁻¹. Then it is demonstrated that in the case V>0 (a non-emitting dipole) a soliton may be pinned by the moving dipole. In the adiabatic approximation, the pinned state is stable provided ε < 0. In this case a pair of solitons may also be pinned by the dipole, but that pinned state is unstable. Other types of solitary pinned profiles and their stability are discussed. Oscillations of a soliton near the adiabatically stable pinned state are accompanied by emission of quasi-linear waves. The emission intensity is calculated in a general form, and it is demonstrated that the oscillation are subject to radiative instability due to the fact that the energy of the system is not positive definite. The same model is considered with the Bürgers' dissipative term. The dissipation may compensate the radiative instability and render the pinned state of a soliton completely stable. Besides, it is demonstrated that the Bürgers' term gives rise to multisoliton pinned profiles. A maximum possible number of solitons in the profile is found.     

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.     

Oblique Interaction of Ion-Acoustic Solitary Waves in e-p-i Plasmas

In this study, we investigate the oblique collision of two ion-acoustic waves (IAWs) in a three-species plasma composed of electrons, positrons, and ions. We use the extended Poincare-Lighthill-Kuo (PLK) method to derive the two-sided Korteweg-de-Vries (KdV) equations and Hirota’s method for soliton solutions. The effects of the ratio (δ) of electron temperature to positron temperature and the ratio (p) of the number density of positrons to that of electrons on the phase shift are studied. It is observed that the phase shift is significantly influenced by the parameters mentioned above. It is also observed that for some time interval during oblique collision, one practically motionless composite structure is formed, i.e., when two ion-acoustic waves with the same amplitude interact obliquely, a new non-linear wave is formed during their collision, which means that ahead of the colliding ion-acoustic solitary waves, both the amplitude and width are greater that those of the colliding solitary waves. As a result, the nonlinear wave formed after collision is a new one and is delayed. The oblique collision of solitary waves in a two-dimensional geometry is more realistic in high-energy astrophysical pair plasmas such as the magnetosphere of neutron stars and black holes.     

Solitary waves and rogue waves in a plasma with nonthermal electrons featuring Tsallis distribution

In this Letter, we discuss the electron acoustic (EA) waves in plasmas, which consist of nonthermal hot electrons featuring the Tsallis distribution, and obtain the corresponding governing equation, that is, a nonlinear Schrödinger (NLS) equation. By means of Modulation Instability (MI) analysis of the EA waves, it is found that both electron acoustic solitary wave and rogue wave can exist in such plasmas. Basing on the Darboux transformation method, we derive the analytical expressions of nonlinear solutions of NLS equations, such as single/double solitary wave solutions and single/double rogue wave solutions. The existential regions and amplitude of solitary wave solutions and the rogue wave solutions are influenced by the nonextensive parameter q and nonthermal parameter α. Moreover, the interaction of solitary wave and how to postpone the excitation of rogue wave are also studied.     

Dust ion-acoustic solitary waves in a hot adiabatic magnetized dusty plasma

The basic features of obliquely propagating dust ion-acoustic (DIA) solitary waves in a hot adiabatic magnetized dusty plasma (containing adiabatic inertia-less electrons, adiabatic inertial ions, and negatively charged static dust) have been investigated. The reductive perturbation method has been employed to derive the Korteweg-de Vries (KdV) equation which admits a small amplitude solitary wave solution. The combined effects of plasma particle (electron and ion) adiabaticity, ion-dust collision, and external magnetic field (obliqueness), which are found to significantly modify the basic features of the small but finite-amplitude DIA solitary waves are explicitly examined. The implications of our results in space and laboratory dusty plasmas are briefly discussed.     

Binary collision approximation for solitary waves in a Y-shaped granular chain

We implement a binary collision approximation to study solitary wave propagation in a two-dimensional double Y-shaped granular chain. The solitary wave was transmitted and reflected when it met the interface of the bifurcated branches of the Y-shaped granular chains. We obtain the analytic results of the ratios of the transmitted and reflected speeds to the incident speed of the solitary wave, the maximum force between the two neighbor beads in a solitary wave, and the total time taken by the pulse to pass through each branch. All of the analytic results are in good agreement with the experimental observations from Daraio et al. [Phys. Rev. E 82 036603 (2010)]. Moreover, we also discuss the delay effects on the arrival of split pulses, and predict the recombination of the split waves traveling in branches in the final stem of asymmetric systems. The prediction of pulse recombination is verified by our numerical results.     

Solitary wave interactions in lattice systems with pure higher-order nonlinearity

In this paper, we investigate in detail the interactions of solitary waves in lattice systems with interaction potential V(q)=qⁿ/n, where n is 4,6,8…, through their all possible collision types, and establish a quantitative connection between the scattering property of solitary waves and the chaotic dynamics of the systems. Kink and antikink are excited in such lattice systems and the variation of their scattering effect with n is concerned. After a kink-antikink collision, the dominant interaction in the systems, if n is small, is that solitary waves pass through each other and the scattering effect increases with n; if n is large, solitary waves reflect back sometimes due to the influence of phase and this leads to a decrease of the scattering effect with n. The largest Lyapunov exponents of systems at fixed energy density first increase and then decrease with n, which is consistent with a variation of the scattering effect. The application of the special scattering behaviors between kink and antikink in information propagation is also discussed.     

Solitary wave in a magnetoelastic easy plane Heisenberg ferromagnet

The spin fluctuations of an easy plane Heisenberg ferromagnet are coupled to the transverse acoustical vibrations. In the long wavelength limit, the possibility of a solitary wave with a bell shape is analysed. It is observed that the velocity v of the solitary wave is bounded by the higher and lower velocities of the acousto-magnetic polaritons for this system. The behaviour of the solitary wave is analysed as a function of this velocity where a splitting of the solitary wave occurs at a critical velocity.     

Ion acoustic solitons in magnetized collisional non-thermal dusty plasmas

The oblique propagation of ion-acoustic solitary waves (IASWs) is considered, in a magnetized non-thermal collisional dusty plasma, composed of non-Maxwelian κ-distributed electrons, inertial ions, and stationary dust. The reductive perturbation approach is adopted to derive the damped Korteweg de-Vries (dKdV) equation, and the dissipative oblique ion-acoustic wave properties are investigated in terms of different key plasma parameters via the numerical solution of the dKdV equation. The collisional effect, describing the ion-neutral collision in the plasma, is taken into account, and seen to influence the dynamics of IASWs significantly. The basic features of IASWs are observed to modify, and the polarity of the wave is seen to change due to the variation of dust to that of ion number density and also due to the variation of the supethermality index κ in the considered plasma system.     

mBBM方程的双扭结孤立波及其动力学稳定性

采用双曲函数展开法得到Modified Benjamin-Bona-Mahony（mBBM）方程的一类扭结-反扭结状的双扭结孤立波解,在不同的极限情况下,此孤立波分别退化为mBBM方程的扭结状和钟状孤立波解.对双扭结型单孤子的结构特征进行分析,构造有限差分格式对其动力学稳定性进行数值研究.有限差分格式为两层隐式格式,在线性化意义下无条件稳定.数值结果表明mBBM方程的双扭结型单孤子在不同类型的扰动下均具有很强的稳定性.对双孤立波的碰撞进行数值模拟,发现既存在弹性碰撞也存在非弹性碰撞.     

Interaction of Three Waves for the Extension (2+1)-Dimensional Sine-Gordon Equation

Exact solutions with three-wave form including three solitary wave, breather-type two-solitary wave, doubly breather-type of solitary wave, double-periodic kind of solitary wave are obtained using bilinear form and extended three-wave approach with the aid of Maple. It is important that completed elastic collision, non-completed elastic collision, and fusion of three waves are investigated, respectively.     

Propagation features of head-on collision dust acoustic solitary waves in four-component quantum plasmas

ABSTRACT

In framework of the extended Poincaré–Lighthill–Kuo, the properties of dust acoustic (DA) solitary wave’s interaction are investigated in four-component quantum dusty plasma. Two Korteweg–de Vries equations describing the colliding DA solitary waves are derived by eliminating the secularities. By knowing the explicit form of the solitary wave solutions, the leading phase changes, trajectories and phase shifts are obtained, accordingly. The effects of various physical parameters such as the quantum mechanical parameters, the charge ratio between positive and negative dust particles, the mass ratio between negative and positive dust particles and the ratio of electron to ion temperatures are studied extensively. Our findings showed that these parameters play a significant role on the characteristics and basic features of DA solitary waves such as phase shifts in trajectories due to collision. The obtained results may be beneficial to understand well the collision of DA solitary waves that may occur in laboratory plasmas, space plasma as well as in plasma applications.     

On exact solutions of the Bogoyavlenskii equation

Exact solutions for the Bogoyavlenskii equation are studied by the travelling wave method and the singular manifold method. It is found that the linear superposition of the shock wave solution and the complex solitary wave solution for the physical field is still a solution of the equation of interest, except for a phase-shift. The dromion-like structures with elastic and nonelastic interactions are found.     

Qualitative analysis and traveling wave solutions for the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity

In this Letter, we investigate the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law nonlinearity. All explicit expressions of the bounded traveling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded traveling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Nonlinear Wave Propagation in a Disordered Medium

In this paper we consider the problem of solitary wave propagation in a weakly disordered potential. Through a series of careful numerical experiments we have observed behavior which is in agreement with the theoretical predictions of Kivshar et al., Bronski, and Gamier. In particular we observe numerically the existence of two regimes of propagation. In the first regime the mass of the solitary wave decays exponentially, while the velocity of the solitary wave approaches a constant. This exponential decay is what one would expect from known results in the theory of localization for the linear Schrödinger equation. In the second regime, where nonlinear effects dominate, we observe the anomalous behavior which was originally predicted by Kivshar et al. In this regime the mass of the solitary wave approaches a constant, while the velocity of the solitary wave displays an anomalously slow decay. For sufficiently small velocities (when the theory is no longer valid) we observe phenomena of total reflection and trapping.     

Head-on Collision of Solitary Waves Described by the Toda Lattice Model in Granular Chain

We study the head-on collision of two solitary waves in a precompressed granular chain using the discrete element method.Our study takes the Toda chain solution as the initial condition for the simulations.The simulation covers the dynamical evolution of the collision process from the start of the incident wave to the end of the collision.The interaction has a central collision region of about five-grain width in which two solitary waves merge completely and share only one peak.Four stages,i.e.,the pre-in-phase traveling stage,lag-phase collision state,lead-phase collision state,and post-in-phase traveling stage,are identified to describe the complex collision processes.Our results may be helpful for explaining the existence of long-lived solitary waves seen in the simulations by Takato and Sen [Europhys.Lett.100(2012) 24003].     

Solitary ion-acoustic wave propagation in the presence of electron trapping and background nonextensivity

Solitary ion-acoustic wave propagation in the presence of electron trapping is investigated within the theoretical framework of the Tsallis statistical mechanics. A physically meaningful Schamel-like distribution is outlined. In the small amplitude limit, the nonlinear dispersion relation is derived to analyze the global dependency of the main solitary wave quantities. It is found that for a given amplitude and trapping state, the solitary potential structure speeds up and broadens as the electron nonextensivity strengthens. Our results may be of basic interest for experiments that involve particle trapping. The flexibility provided by the nonextensive q-parameter enables one to obtain a good agreement between theory and experiment.     

Modulational Instability of Dust Ion Acoustic Waves in a Collisional Dusty Plasma

The modulational instability of dust ion accoustic waves in a dust plasma with ion-dust collision effects is studied.Using the perturbation method,a modified nonlinear Schroedinger equation contains a damping term that comes from the effect of the ion-dust collision is derived.It is found that the inclusion of the ion-dust collision would modify the modulational instability of the wave packet and could not admit any stationary envelope solitary waves.     

The problem of the unambiguity of phase-shift analysis I

For the discrimination of phase-shift solutions, the new so-calledτ-criterion, based on the calculation of some functionτ of the experimental data, is used. This method can be applied, in general, to an arbitrary ambiguous phase-shift analysis. Theτ-criterion is more powerful than theχ ²-criterion, e.g. for rejecting a less probable set it is necessary to have a smaller quantity of experimental points (in statistics this means that theτ-criterion gives a smaller Type II error). For the calculation of the quantityτ, the standard programme for phase-shift analysis with a small additional part can be used. A computer time of about only 5–10 min. on a computer of M-20 type is needed. A Type I error is then obtained from the tables of the normal distribution function.     

Ion‐Acoustic Waves Instability in a Three Components Magneto‐Plasma with Nonthermal Electrons

A theoretical investigation has been made on obliquely propagating ion‐acoustic (IA) solitary structures in a three components magneto‐plasma containing cold inertial ions, Boltzmann distributed positrons, and hot non‐thermal electrons. The Zakharov‐Kuznetsov equation has been derived by the reductive perturbation method, and its solitary wave solution has been analyzed. Multi‐dimensional instability has also studied by the small‐k (long wave‐length plane wave) perturbation expansion technique, which is found to exist in such a plasma. The effects of the external magnetic field, nonthermal electrons, obliqueness and temperature ratio have significantly modified the basic properties of small but finite‐amplitude IA solitary waves, such as amplitude, width, instability criterion and the growth rate. The present investigation contributes to the physics of the nonlinear IA waves in space and laboratory electron‐positron‐ion magneto‐plasmas in which wave damping produces an electron tail. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)