Head-on collision and overtaking collision between an envelope solitary wave and a KdV solitary wave in a dusty plasma

Head-on collision and overtaking collision between a KdV solitary wave and an envelope solitary wave are first studied in present paper by using Particle-in-cell (PIC) method in a dusty plasma. There are phase shifts of the KdV solitary wave in both head-on collision and the overtaking collision, while no phase shift is found for the envelop solitary wave in any cases. The remarkable difference between head-on collision and the overtaking collision is that the phase shift of KdV solitary wave increases as amplitude of KdV solitary wave increases in head-on collision, while it decreases as amplitude of the KdV solitary wave increases in the overtaking collision. It is found that the maximum amplitude during the collision process is less than sum of two amplitudes of both solitary waves, but is larger than either of the amplitude.

However, no report is given on collision between a KdV solitary wave and an envelope one. In the present paper we try to discuss this question by using the PIC simulation method. Both the head-on collision and overtaking collision between two kinds of waves will be studied in this paper.
It is found that there are phase shifts for a KdV solitary wave during the collision between KdV solitary wave and envelope solitary wave, which is similar to that between two KdV solitary waves in which there are phase shifts for both colliding solitary waves 38 . However, there is no phase shift for the envelope solitary wave after the collision.

Results
Analytical solutions of KdV and NLSE by using the perturbation method. The interaction between a KdV solitary wave and an envelope solitary wave in a dusty plasma will be studied in the present paper by using the one dimensional (1D) PIC method in an infinite background plasma. First we give the 1D dimensionless normalized equations of motion for system 38,40 where β = T i /T e is the ratio of the ion and electron temperatures, s = 1/(μ + νβ), μ and ν are the normalized ion and electron number densities, respectively, n d and u d refer to the density, the velocity of the dust grains respectively, φ is the electrostatic potential. In the simulation, we take β = 0.1, μ = 1.1 and ν = 0.1. The spatial coordinate x, the time t, the velocity and the electrostatic potential φ are normalized by the Debye length λ D = (T eff /4πZ d n d0 e 2 ) 1/2 , the inverse of effective dust plasma frequency ω π . By introducing the appropriate stretched coordinates ξ = ε(x − ct) and τ = ε 3 t, we can obtain a KdV equation as follows NLSE can also be obtained as follows by using the different transformations 40 where ξ′ = ε(x − u s t), τ′ = ε 2 t, u s is group velocity of envelope solitary wave. P and Q are functions of system parameters. In the simulation, we choose system parameters as follows: wave number of envelope wave k n = 0.1, then we obtain: P = −0.146, Q = −3.65 40 . One of the envelope solitary wave solution of Eq. (5) is: . φ nls , u nls and n nls are electrostatic potential, velocity and density of dust grains corresponding to envelope solitary wave, respectively. The initial conditions of the KdV solitary wave are also chosen from the analytical solution: , u kdv = φ kdv and n kdv = 1 + φ kdv . φ kdv , u kdv and n kdv are electrostatic potential, velocity and density of dust grains corresponding to KdV solitary wave, respectively. The dust particles are represented as superparticles, while the ions and the electrons are modeled as Boltzmann distributed background. The area weighting technique is used to deposit the charge on the 2th nearest grid points in order to obtain the spatial distribution of the particles on the mesh. At the start of simulations, 100 superparticles with different weighting parameters are allocated symmetrically in each grid. Weighting parameters and initial velocities of superparticles are carefully chosen to make sure that the density and velocity distribution of dust particles are given as those obtained by the reductive perturbation method. The electric field is determined by the In Fig. 2 the colliding trajectories are pictured. There are no phase shift of the envelope solitary wave, i.e., the propagation velocity of the envelope solitary wave is a constant. However, the propagation velocity of the KdV solitary wave is not a constant during the collision. During the collision, the propagation velocity of the KdV solitary wave first becomes larger, then smaller and finally become a constant as that of the previous one. Furthermore, a phase delay of KdV solitary wave is observed. The definition of the phase shift is shown in Fig. 2. In order to further understand the dependence of the phase shifts of both solitary waves on their amplitudes, the dependence of phase shifts of the KdV solitary waves on the parameters of both ε 1 and ε 2 are given in Fig. 3. It is noted that the phase shifts of KdV solitary waves only depend on its amplitudes, i.e., ε 1 , but not that of the  envelope wave, i.e., ε 2 . Furthermore, It seems that there is no phase shift for the envelope solitary wave after the collision in any case.
It is observed that a maximum amplitude will be reached during the collision between two solitary waves. It indicates, however, that the maximum value is less than the sum of two amplitudes of both solitary waves, but it is larger than either of the amplitude. In order to understand this situation in detail, the dependence of the maximum amplitudes in the colliding process between two solitary waves on both of their initial amplitudes is shown in Fig. 4. It is noted that the maximum amplitude in colliding process increases as the amplitudes of both colliding solitary waves increase. It seems that the maximum amplitude is less than the sum of two amplitudes of both the

Discussion
Both the head-on collision and overtaking collision between a KdV solitary wave and an envelope solitary wave are investigated by using PIC method in a dusty plasma. It is found that there are phase delay for KdV solitary wave for either the head on collision or the overtaking collision between a KdV solitary wave and an envelope solitary wave. It is observed that there are no phase delay for the envelop solitary wave in any cases. The remarkable difference between the head-on collision and the overtaking collision is that the phase shift of the KdV solitary wave increases as the amplitude of the KdV solitary wave increases in the head-on collision, while it decreases as the amplitude of the KdV solitary wave increases in the overtaking collision. It is found that the maximum amplitude during the collision process is less than the sum of two amplitudes of both solitary waves, but is larger than either of the amplitude. It is noted that the maximum amplitude in colliding process increases as the amplitudes of both colliding solitary waves increase. Meanwhile, PIC simulation provides a more realistic description of the dynamics of nonlinear dust acoustic waves that can be usefully applied to various low frequency phenomena observed in laboratory as well as space plasmas. The results has potential applications in the instability study of the space plasmas, and the fusion plasmas.

Methods
Numerical experiment is performed by using the one-dimensional PIC simulation method to study interaction between a KdV solitary wave and an envelope solitary wave in a dusty plasma in present work. During the simulation, The dust particles are represented as kinetic particles, while ions and electrons are modeled as Boltzmann distributed background. As well known, the real systems always contain very large amount of particles. In order to make simulations efficient or at least possible, so-called super-particles(SPs) are used. Each SP has a weight factor S specifying the number of real particles contained. Therefore, the equation of motion of the system is the Newton's equation as follows where m j , q j , x j are the mass, charge and position of the jth SP, respectively. E j is the electric field at the position of the jth SP. As the dust particles follow their trajectories, they continually exchange information with the background grid. Each dust particle contributes its charge to the corners of its instantaneous host cell. Therefore, the simulation region is divided to contain several grid cells during the PIC simulation. At each time step, the velocities, the positions of SPs are weighted to all the grids to calculate the charge density ρ g (or electric current density J g ). Once ρ g is obtained, the Maxwell's equations (electromagnetic model) or Poisson-Boltzmann equation (electrostatic model) will be solved numerically to derive the value of E at each grid. In electrostatic model, B g = 0.
Then the field imposed on each SP can be worked out and each SP will be driven by electric field according to Eq. (6), which will be solved numerically via the leap-frog algorithm. At last, the new positions and velocities are obtained, the procedure come to repeat until the simulation completed. The flowchart of PIC method is shown in Fig. 8.
In process of head-on collision between the KdV solitary wave and the envelope solitary waves, we assume that there are two opposite propagating solitary waves. the KdV solitary wave is propagating in the positive x direction and the envelop wave is in the negative x direction. The initial number density and the velocity of the dust particles are: