Fully kinetic simulation study of ion-acoustic solitons in the presence of trapped electrons

The nonlinear fluid theory developed by Schamel suggests a modified KdV equation to describe the temporal evolution of ion acoustic (IA) solitons in the presence of trapped electrons. The validity of this theory is studied here by verifying solitons' main characteristic, i.e., stability against successive mutual collisions. We have employed a kinetic model as a more comprehensive theory than the fluid one, and utilized a fully kinetic simulation approach (both ions and electrons are treated based on the Vlasov equation). In the simulation approach, these solitons are excited self-consistently by employing the nonlinear process of IA solitons formation from an initial density perturbation (IDP). The effect of the size of IDPs on the chain formation is proved by the simulation code as a benchmark test. It is shown that the IA solitons, in the presence of trapped electrons, can retain their features (both in spatial and velocity direction) after successive mutual collisions. The collisions here include encounters of IA solitons with the same trapping parameter, while differing in size. Kinetic simulation results reveal a complicated behavior during a collision between IA solitons in contrast to the fluid theory predictions and simulations. In the range of parameters considered here, two oppositely propagating solitons rotate around their collective center in the phase space during a collision, independent of their trapping parameters. Furthermore, they exchange some portions of their trapped populations.

Figure 1

The evolution of the distribution function versus velocity is shown for $\beta =0.2$ (a) and $\beta =-0.1$ (b). At $\tau =10$ the distribution function (black dotted line) is presented at $x=550$, middle of the DDP, which shows hump (hollow) for $\beta =0.2\left(\beta =-0.1\right)$. Blue solid line presents the distribution function at the middle of the first soliton just before its first collision. Filamented structures can be witness to grow inside the distribution function of trapped population.

Figure 2

The temporal evolution of electrons (a) and ions (b) number density, for the case of a small IDP ($\psi =0.05$, $\mathrm{\Delta }=10$) with $\beta =0$, is shown. The propagation of both Langmuir and IA wave packets can be witnessed. The number density are shown with color covering values from 1.0 to 1.005. Note that the colors are arranged based on a power-law distribution so the small amplitude Langmuir wave packet in the electron number density can be recognizable.

Figure 3

Profiles of electrons number density (a) and charge density (a) are shown for a large IDP ($\psi =0.2$, $\mathrm{\Delta }=500$) with $\beta =-0.1$. Four different times during disintegration process, namely, $\tau =0,50,100,150$, are presented with black (around $x=500$), blue (around $x=1000$), red (around $x=1500$), and green (around $x=2000$), respectively. The disintegration process of each of DDPs into three IA solitons can be seen in details.

Figure 4

The temporal evolution of electrons number density and charge density are shown for a large IDP ($\psi =0.2$, $\mathrm{\Delta }=500$) and two values of $\beta$. Panels (a) and (b) represent the electrons number density for $\beta =-0.1$, while (c) and (d) display the charge density for the same value of $\beta$. The electrons number density for $\beta =0.2$ is shown in (e) and (f), while (g) and (h) display the charge density. Twelve (eight) collisions are observed for each of six (four) IA solitons in case of $\beta =-0.1\left(\beta =0.2\right)$. Zoomed-in panels (b), (d), (f), and (h) on the right side of each figures display the details of three successive collisions around the time $550<\tau <750$. Since the trajectories of solitons don't change by this collisions, hence their velocities are conserved.

Figure 5

The temporal evolution of features such as amplitude (a), width (b), and velocity (c) of ions (blue solid line) and electrons (black dotted line) number densities are shown for the right-propagating first and dominant soliton in case of $\beta =-0.1,\psi =0.2$, and $\mathrm{\Delta }=500$. Anomalies take place in the measurement during collision times, i.e., $180<\tau <250$, $380<\tau <450$, $580<\tau <650$. The fluctuation of the values around the average for propagation times (excluding the collision intervals) are less than $1\%$, $5\%$, and $8\%$ for amplitude ($1.12<a_e<1.14,1.14<a_i<1.16$), width ($85<w_e<95$, $70<w_i<80$), and velocity ($11.0<v_i=v_e<9.0$), respectively.

Figure 6

For a large IDP ($\psi =0.2$, $\mathrm{\Delta }=500$) with $\beta =-0.1$, the number density profiles of the first and dominant IA soliton are presented for different times. The number density profiles of electrons (a) and ions (b) before and after the first ($\tau =180$, $\tau =250$) and the second ($\tau =380$, $\tau =450$) triple collision are shown. The same is presented for electrons (c) and ions (d) for the third ($\tau =580$, $\tau =650$) and the fourth ($\tau =780$, $\tau =900$) triple collisions.

Figure 7

For the case of large IDP ($\psi =0.2$ and $\mathrm{\Delta }=500$) and $\beta =-0.1$, the hollow in the electrons' distribution function accompanying the right-propagating first and dominant IA soliton are shown in the phase space. The phase space structure of the trapped population is presented before and after first ($\tau =180,250)$, second ($\tau =380,450$), third ($\tau =580,650$), and fourth ($\tau =780,900$) triple collisions, respectively, starting from the top left corner. The size and shape of hollows stay the same for all the figures, confirming the stability. However, the symmetry of the hollow (in the phase space) are distorted increasingly as the number of collisions increases. See Supplemental Material [24] for the early stage development of the hollow, i.e., $\tau <180$.

Figure 8

For the case of large IDP, i.e., $\psi =0.2$ and $\mathrm{\Delta }=500$, with $\beta =0.0\left(\beta =0.2\right)$, the electrons' distribution function are shown in the phase space in the two top (bottom) figures. These figures represent the trapping area associated with the right-propagating first and dominant IA soliton. The phase space structure of the plateau ($\beta =0.0$) and the hump ($\beta =0.2$) is presented before the first collision ($\tau =190,200$, respectively) and after the final collision ($\tau =950)$. Stability of IA solitons can be confirmed by comparing the size and shape of nonlinear structures. However, their symmetry is distorted due to the number of mutual collisions. Furthermore, plateau structure prove to be more robust than the other two shapes.

Figure 9

Collision of hollows; for the case of $\beta =-0.1$, the distribution functions of electron and ions are presented during a collision between right (R) and left (L) propagating IA solitons. The frame is moving alongside the right-propagating IA soliton and shows the times from $\tau =37$ up to $\tau =48$ (arranged from top left corner). The collision for the electrons appears as a rotation of both phase-space hollows around their collective center of mass. Furthermore two solitons exchange a portion of their trapped population during collision. However, for the ions, the collision simply consists of two displacements moving through each other. See Supplemental Material [24] for complete successive steps of this collision, i.e., $0<\tau <130$.

Figure 10

Collision of plateaus; A collision between two IA solitons for the case of $\beta =0.0$ is presented in the phase space of electrons from $\tau =41$ up to $\tau =52$ (starting from the top left corner). The collision takes place between right (R) and left (L) propagating solitons. The frame follows the right-propagating (R) soliton. The collision causes the two plateaus in the phase space to rotate once around their collective center and exchange some parts of their populations.

Figure 11

Collision of humps; for the case of $\beta =0.2$, a collision of two oppositely propagating IA solitons is shown in the phase space of electrons. The collision happens from $\tau =40$ up to $\tau =49$ (starting from the top left corner). The phase space structure accompanying the IA solitons are humps here, hence the red color. One rotation around the collective center and the exchange of trapped population takes place during collision between the two humps.