Nonplanar dust-acoustic Gardner solitons in a four-component dusty plasma

The nonlinear propagation of Gardner solitons (GSs) in a nonplanar (cylindrical and spherical) four-component dusty plasma (composed of inertial positively and negatively dust, Boltzmann electrons, and ions) is studied by the reductive perturbation method. The modified Gardner equation is derived and numerically solved. It has been found that the basic characteristics of the dust-acoustic (DA) GSs, which are shown to exist for $\mu$ around its critical value ${\mu }_c$ [where $\mu =Z_{dp}{m}_{dn}/Z_{dn}{m}_{dp}$, $Z_{dn}$ $\left(Z_{dp}\right)$ is the number of electrons (protons) residing on a negative (positive) dust, $m_{dp}$ $\left(m_{dn}\right)$ is the mass of the positive (negative) dust, ${\mu }_c$ is the value of $\mu$ corresponding to the vanishing of the nonlinear coefficient of the Korteweg–de Vries (KdV) equation, e.g., ${\mu }_c\simeq 0.174$ for ${\mu }_e=n_{e0}/Z_{dn}{n}_{dn0}=0.2$, ${\mu }_i=n_{i0}/Z_{dn}{n}_{dn0}=0.4$, and $\sigma =T_i/T_e=0.1$, $n_{e0}$, $n_{i0}$, and $n_{dn0}$ are, respectively, electron, ion, and dust number densities, and $T_i$ ($T_e$) is the ion (electron) temperature], are different from those of the KdV solitons, which do not exist for $\mu$ around ${\mu }_c$. It has been also found that the propagation characteristics of nonplanar DA GSs significantly differ from those of planar ones.

Figure 1

How ${\mu }_c$ [obtained from $A(\mu ={\mu }_c)=0$] varies with $\sigma$ and ${\mu }_e$ for ${\mu }_i=0.4$.

Figure 2

How ${\mu }_c$ [obtained from $A(\mu ={\mu }_c)=0$] varies with $\sigma$ and ${\mu }_i$ for ${\mu }_e=0.2$.

Figure 3

Effects of cylindrical geometry on DA positive GSs for $\mu =0.18$, ${\mu }_e=0.2$, ${\mu }_i=0.4$, $\sigma =0.1$, and $U_0=0.05$.

Figure 4

Effects of cylindrical geometry on DA negative GSs for $\mu =0.16$, ${\mu }_e=0.2$, ${\mu }_i=0.4$, $\sigma =0.1$, and $U_0=0.05$.

Figure 5

Effects of spherical geometry on DA positive GSs for $\mu =0.18$, ${\mu }_e=0.2$, ${\mu }_i=0.4$, $\sigma =0.1$, and $U_0=0.05$.

Figure 6

Effects of spherical geometry on DA negative GSs for $\mu =0.16$, ${\mu }_e=0.2$, ${\mu }_i=0.4$, $\sigma =0.1$, and $U_0=0.05$.