Structure and evolution of magnetohydrodynamic solitary waves with Hall and finite Larmor radius effects

Nonlinear and low-frequency solitary waves are investigated in the framework of the one-dimensional Hall-magnetohydrodynamic model with finite Larmor effects and two different closure models for the pressures. For a double adiabatic pressure model, the organization of these localized structures in terms of the propagation angle with respect to the ambient magnetic field $\theta$ and the propagation velocity $C$ is discussed. There are three types of regions in the $\theta -C$ plane that correspond to domains where either solitary waves cannot exist, are organized in branches, or have a continuous spectrum. A numerical method valid for the two latter cases, which rigorously proves the existence of the waves, is presented and used to locate many waves, including bright and dark structures. Some of them belong to parametric domains where solitary waves were not found in previous works. The stability of the structures has been investigated by performing a linear analysis of the background plasma state and by means of numerical simulations. They show that the cores of some waves can be robust, but, for the parameters considered in the analysis, the tails are unstable. The substitution of the double adiabatic model by evolution equations for the plasma pressures appears to suppress the instability in some cases and to allow the propagation of the solitary waves during long times.

Figure 1

Characteristic velocities (left) and domains of stability of ${\mathbf{\xi }}_0$ in the $C/V_A-\theta$ plane (right). The regions are (1) saddle-center, (2) focus-focus, (3) saddle-saddle, (4) and center-center. The parameters used for panels (a) and (b) [(c) and (d)] correspond to case 1 (2) in Table 1. The green dotted lines in panels (a) and (c) are the velocities making $p_1=0$ in Eq. (19).

Figure 2

Panels (a) and (b) show $u_y\left({\widehat{x}}^{*}\right)$ versus $\theta$ for $C/V_A=1$, and $u_y\left({\widehat{x}}^{*}\right)$ versus $C/V_A$ for $\theta ={75}^{\circ }$, respectively. Other parameters correspond to case 1 in Table 1. Positive (negative) values of $u_y\left({\widehat{x}}^{*}\right)$ are denote with blue crosses (red dots).

Figure 3

Branches of solitary waves in the $C/V_A-\theta$ plane for case 1.

Figure 4

Solitary waves named (a) and (b) in Fig. 3.

Figure 5

Solitary waves named (c) and (d) in Fig. 3.

Figure 6

Solitary waves named (e) and (f) in Fig. 3.

Figure 7

$\mid u_y\left({\widehat{x}}^{*}\right)\mid$ versus $\phi$ diagram (a)] and example of solitary wave in the focus-focus domain (b)–(d).

Figure 8

Panels (a) and (b) show the maximum growth rate $\gamma$ in the $k-\theta$ plane for cases 1 and 2 in Table 1, respectively.

Figure 9

Evolutions of some solitary waves belonging to the saddle-center regime. Panel (a) shows an example of unstable core for $\theta ={80}^{\circ }$ and $C/v_A=0.745715$. Panel (b) shows an example of unstable background for $\theta =30.{415}^{\circ }$ and $C/v_A=0.9$. Panel (c) shows an example of robust solution, using Eqs. (27) and (28) for the pressures with the same parameters as panel (b).