Solitary and periodic waves in collisionless plasmas: The Adlam-Allen model revisited

We consider the Adlam-Allen (AA) system of partial differential equations, which, arguably, is the first model that was introduced to describe solitary waves in the context of propagation of hydrodynamic disturbances in collisionless plasmas. Here, we identify the solitary waves of the model by implementing a dynamical systems approach. The latter suggests that the model also possesses periodic wave solutions—which reduce to the solitary wave in the limiting case of an infinite period—as well as rational solutions that are obtained herein. In addition, employing a long-wave approximation via a relevant multiscale expansion method, we establish the asymptotic reduction of the AA system to the Korteweg–de Vries equation. Such a reduction is not only another justification for the above solitary wave dynamics, but may also offer additional insights for the emergence of other possible plasma waves. Direct numerical simulations are performed for the study of multiple solitary waves and their pairwise interactions. The stability of solitary waves is discussed in terms of potentially relevant criteria, while the robustness of spatially periodic wave solutions is touched upon via numerical experiments.

Figure 1

Top row: Panel (a) shows the effective potential $V\left(w\right)$ (27), along with some typical energy levels depicted by straight horizontal lines. Parameter values: $B_0=R_0=1$, $v=1.5$. The continuous (red) line corresponds to the energy $E_0=V\left(0\right)=0$ corresponding to the homoclinic orbit. The dashed lines depict energy values $E_0<0$, which correspond to closed, periodic orbits. Panel (b) shows orbits in the $(w,w^{\prime })$ phase-plane of Eq. (26), corresponding to the energy levels of panel (a). The continuous (red) curve depicts the homoclinic orbit, pertinent to the energy $E_0=V\left(0\right)=0$. The dashed closed curves are associated with the periodic solutions corresponding to energy values $E_0<0$. Middle row: Panel (c) shows the profile of the solitary wave for $R$, given by Eq. (32), at $t=0$. Shown also, as an inset, is the quantity $1/R$, which, physically, represents the density of the charged particles. Panel (d) shows the profile of the soliton solution for $B$ given by (33), at $t=0$. Bottom row: A spatially periodic solution corresponding to the energy $E_0=-0.015$. Panel (e) depicts the profile of the $R^p$ periodic component of the solution, at $t=0$, while panel (f) depicts the profile of its $B^p$ periodic component, at $t=0$.

Figure 2

Contour plots showing the spatiotemporal evolution of the solitary waves. Panel (a) [panel (b)] depicts the evolution of the $R$-component (the $B$-component) of the solution. Parameter values: velocity $v=1.5$, $B_0=R_0=1$ (i.e., $C=1$), and $L=150$.

Figure 3

Snapshots of the evolution of two colliding solitary waves. In each panel, the $B$-component of the solution is depicted. The initial positions and velocities for the solitary waves are $x_1=-110$, $v_1=1.8$ and $x_2=-70$, $v_2=1.2$. Parameter values are $B_0=1$, $R_0=1$, and $L=150$.

Figure 4

Contour plots showing the spatiotemporal evolution of the two-soliton initial conditions $(R^{ds}(x,0),B^{ds}(x,0))$ for velocities $v_1=1.8$ and $v_2=1.2$. The rest of the parameters are fixed as in Fig. 3. Panel (a) [panel (b)] shows the $R$-component ($B$-component) of the solution.

Figure 5

Snapshots of the evolution of the two solitary waves with velocities $v_1=1.8$ and $v_2=1.2$. A magnification around the tail (during and after the collision of the solitons) of the $B$-component of the solution is shown. The rest of the parameters are fixed as in Fig. 3.

Figure 6

Similar to Fig. 3, but now for solitons’ initial positions and velocities $x_1=-110$, $v_1=1.9$ and $x_2=-70$, $v_2=1.8$. Parameter values are $B_0=1$, $R_0=1$, and $L=150$.

Figure 7

Contour plots of the spatiotemporal evolution of the two solitary wave initial conditions $(R^{\mathrm{ds}}(x,0)$, $B^{\mathrm{ds}}(x,0))$ for $v_1=1.9$ and $v_2=1.8$. The rest of the parameter values are fixed as in Fig. 6. (a) The $R$-component of the solution is shown. (b) The same as before but for the $B$-component.

Figure 8

Snapshots of the evolution of the two wave initial condition $(R^{\mathrm{ds}}(x,0),B^{\mathrm{ds}}(x,0))$, with velocities $v_1=1.9$ and $v_2=1.8$. A magnification around the tail, during the repulsive interaction of the solitons, of the $B$-component of the solution is shown. The rest of the parameter values are fixed as in Fig. 6.

Figure 9

(a) The dependence of $N$ on the velocity $v$. Here, $N\left(v\right)$ has been calculated in the small velocity (KdV) limit, using Eq. (45) [dashed (blue) curve], and for all permitted velocities, using Eqs. (13) and (14) [solid (red) curve]. It is observed that, in both cases, $dN/dv>0$, which implies stability of the traveling solitary waves according to the KdV stability criterion (see the text). (b) The dependence of the momentum $P$ [Eq. (18)] on the velocity $v$. As in (a), the dashed (blue) curve and the solid (red) curve correspond to the KdV limit and the exact traveling wave [Eqs. (13) and (14)]. Observe that, in both cases, $dP/dv>0$, which also suggests stability, per the discussion in the text.

Figure 10

Contour plot of the spatiotemporal evolution of the perturbed spatially periodic solution $(R^p(x,0),B^p(x,0)+0.01sin\left(3Kx\right))$. (a) The $R$-component of the solution. (b) The $B$-component of the solution. Parameter values: $B_0=R_0=1$, $v=1.5$. The spatial interval is $[0,m\lambda ]$ with $m=5$; $\lambda$ is the spatial period of the solution, and $K=2\pi /\lambda$ is the wave number associated with the wavelength $\lambda$. The energy associated with the unperturbed initial conditions is $E_0=-0.03$.

Figure 11

Contour plot of the spatiotemporal evolution of the perturbed spatially periodic solution $(R^p(x,0)+0.01sin\left(10Kx\right),B^p(x,0))$ with $m=3$. (a) The $R$-component of the solution. (b) The $B$-component of the solution. Parameter values: $B_0=R_0=1$, $v=1.5$. Energy of the unperturbed initial conditions $E_0=-0.01$.

Figure 12

(a) The graph of the effective potential when $v=C$, the case of the degenerate ODE, Eq. (46). The (green) horizontal line defines the energy $V\left(0\right)=E_0=0$ of the homoclinic connection associated with the analytical rational solutions (48) and (49). (b) The (green) continuous curve is the homoclinic connection of energy $V\left(0\right)=E_0=0$. The closed (dashed) periodic orbits correspond to energy values $E_0<0$ [dashed horizontal lines in (a)]. (c) The profile of the rational solution $R^r(x,t)$, given by Eq. (48), at $t=0$. (d) The profile of the rational solution $B^r(x,t)$, given by Eq. (49), at $t=0$.