2. One-fluid modified Hall-MHD equations

In this section, we shall derive the modified one-fluid Hall-MHD model equations from the standard two-fluid equations for electrons and ions. The Hall-MHD is valid for the length scale $\lambda _e\ll \lambda \lesssim \lambda _i$ , where $\lambda$ is the scale length of the plasma. The basic equations for the model are Maxwell’s set of electromagnetic equations and two-fluid equations for electron and ion components (Bittencourt Reference Bittencourt2013). These equations are as follows.

Poisson’s equation

(2.1) \begin{equation} \boldsymbol{\nabla \boldsymbol{\cdot }E}=4\pi \rho ,\;\;\;\;\rho =\sum _{\alpha =i,e}\rho _{\alpha }, \end{equation}

where $\boldsymbol {E}$ is the electric field, $\rho$ is the total charge density and $\rho _{\alpha }=q_{\alpha }n_{\alpha }$ is the electric charge density of the $\alpha {\mbox{th}}$ species ( $e=\mbox{electron},\; i=\mbox{ion}$ ) particle with $q_e=-e$ and $q_i=+e$ .

The Faraday’s law is given by

(2.2) \begin{equation} \frac {\partial \boldsymbol {B}}{\partial t}=-c\boldsymbol{\nabla }\times \boldsymbol {E}, \end{equation}

where $\boldsymbol {B}$ is the ambient magnetic field.

The steadily moving charged object introduces an additional perturbed current which acts as a current source. To account for its effects, an additional current source $\boldsymbol {J}_{\!\boldsymbol {s}}$ is introduced in Ampère’s law, which then assumes the form

(2.3) \begin{equation} c\boldsymbol{\nabla }\times \boldsymbol {B}=4\pi \left (\boldsymbol {J}+\epsilon ^2\tilde {\boldsymbol {J}_{\!\boldsymbol {s}}}\right )+\frac {\partial \boldsymbol {E}}{\partial t}. \end{equation}

Here, $\boldsymbol {J}_{\!\boldsymbol {s}}=\epsilon ^2\tilde {\boldsymbol {J}_{\!\boldsymbol {s}}}$ with $\epsilon$ is a measure of the smallness of the perturbed amplitude, the second term in the right hand side is the displacement current density and $\boldsymbol {J}$ is the total conduction current density due to electrons and ions, given by

(2.4) \begin{equation} \boldsymbol {J}=\sum _{\alpha =i,e}\boldsymbol {J}_{\alpha }=\sum _{\alpha =i,e}\rho _{\alpha }\boldsymbol {v}_{\alpha }=e(n_{i}\boldsymbol {v}_{\boldsymbol {i}}-n_{e}\boldsymbol {v}_{\boldsymbol {e}}), \end{equation}

where $\boldsymbol {v}_{\alpha }$ and $\boldsymbol {J}_{\alpha }=\rho _{\alpha }\boldsymbol {v}_{\alpha }$ are the fluid velocity and conduction current density of the $\alpha {\mbox{th}}$ species particle, respectively.

It is worth noting that taking the divergence of (2.3) and using (2.1), we get

(2.5) \begin{equation} \frac {\partial \rho }{\partial t}+{\boldsymbol{\nabla }\boldsymbol{\cdot }}\left (\boldsymbol {J}+\epsilon ^2{\tilde {\boldsymbol {J}_{\!\boldsymbol {s}}}}\right )=0, \end{equation}

so that the total change density (including the perturbed current due to the moving charged obstacle) is conserved. All the above equations are coupled with $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol {B}=0$ .

The ion and electron continuity equations are given by

(2.6) \begin{align} \frac {\partial \rho _{\textit{mi}}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\rho _{\textit{mi}}\boldsymbol {v}_{\boldsymbol {i}}) &=0, \end{align}

(2.7) \begin{align} \frac {\partial \rho _{\textit{me}}}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\rho _{\textit{me}}\boldsymbol {v}_{\boldsymbol {e}}) &=\mathcal{S}_e. \end{align}

Here, $\boldsymbol {v}_{\boldsymbol {i}}(\boldsymbol {e})$ is the ion (electron) fluid velocity, $\rho _{mi(e)}=n_{i(e)}m_{i(e)}$ is the ion (electron) mass density, $m_{i(e)}$ is the ion (electron) mass and $n_{i(e)}$ is the ion (electron) density. The steadily moving objects are getting charged mostly due to the collisions with the electrons and ions. Also, the thermal velocity of the electron is larger than that of the ions so that the objects usually become negatively charged. As a consequence of the electron attachments on these objects, we introduce a sink term $\mathcal{S}_e$ in the electron continuity equation (2.7), where

(2.8) \begin{equation} \mathcal{S}_e=\epsilon ^2 m_e{\boldsymbol{\nabla }\boldsymbol{\cdot }}\left (\frac {\tilde {\boldsymbol {J}_{\!\boldsymbol {s}}}}{e}\right ). \end{equation}

The collective process is assumed to be sufficiently prevalent to keep the pressure tensor isotropic and as a consequence, we consider only the scalar pressure for ion(electron) fluid. Then, the ion and electron momentum density fluid equations are given by

(2.9) \begin{align} \rho _{\textit{mi}}\left (\frac {\partial }{\partial t}+\boldsymbol {v}_{\boldsymbol {i}}\boldsymbol{\cdot }\boldsymbol{\nabla }\right )\boldsymbol {v}_{\boldsymbol {i}} &=\rho _i\boldsymbol {E}+\frac {\boldsymbol {J}_{\boldsymbol {i}}\times {B}}{c}-\boldsymbol{\nabla }p_i, \end{align}

(2.10) \begin{align} \rho _{\textit{me}}\left (\frac {\partial }{\partial t}+\boldsymbol {v}_{\boldsymbol {e}}\boldsymbol{\cdot }\boldsymbol{\nabla }\right )\boldsymbol {v}_{\boldsymbol {e}} &=\rho _e\boldsymbol {E}+\frac {\boldsymbol {J}_{\boldsymbol {i}}\times {B}}{c}-\boldsymbol{\nabla }p_e-\mathcal{S}_e\boldsymbol {v}_{\boldsymbol {e}}. \end{align}

In (2.10), the electron momentum density loss due to the attachment of electrons is introduced. Additionally, the ion (electron) fluid is assumed to be isothermal so that the scalar pressure $p_{i(e)}$ is given by

(2.11) \begin{equation} p_{i(e)}=\rho _{mi(e)}T_{i(e)}/m_{i(e)}=n_{i(e)}T_{i(e)}, \end{equation}

where $T_{i(e)}$ is the ion (electron) temperature.

The following assumptions are made for the one-fluid Hall-MHD equations.

1. (i) We consider the low-frequency disturbances with the characteristic frequency $\ll \omega _{ce}\ll \omega _{\textit{pe}}$ , where $\omega _{ce}$ ( $=eB_0/(cm_e)$ is the electron cyclotron frequency. Under these conditions, Poisson’s equation (2.1) can be approximated as (Berezin & Karpman Reference Berezin and Karpman1964)

   (2.12) \begin{equation} \rho \approx 0\;\;\Rightarrow \;\;n_e\approx n_i=n\;\;\;(\mbox{say}), \end{equation}
   and thus, the MHD plasma becomes effectively quasi-neutral.

2. (ii) In Ampere’s law (see (2.3)),

   (2.13) \begin{align} \mid \boldsymbol {J}\mid \sim \sigma _0 E\quad \mbox{and}\quad \mid \partial \boldsymbol {E}/\partial t\mid \sim E/\tau _0, \end{align}
   where $\sigma _0$ and $\tau _0$ represent the characteristic conductivity and time. So that the estimated value of the ratio
   (2.14) \begin{align} \bigg |\frac {\partial \boldsymbol {E}/\partial t}{4\pi \boldsymbol {J}}\bigg |\sim \frac {1}{4\pi \sigma _0\tau _{0}}\ll 1, \end{align}
   as for the AW, $\tau _0^{-1}\sim (0.1-10)\,\mbox{Hz}$ is low and $\sigma _0\sim (n_0e^2/(m_e\nu _{eff}))\,\mbox{sec}^{-1}$ ( $\nu _{eff}$ is the effective electron–ion collision frequency) in MHD plasma is very high ( $\to \infty$ ) (Bittencourt Reference Bittencourt2013). Thus, the Ampere’s law (2.3) can be approximated as
   (2.15) \begin{equation} c\boldsymbol{\nabla }\times \boldsymbol {B}=4\pi \left (\boldsymbol {J}+\epsilon ^2\tilde {\boldsymbol {J}_{\!\boldsymbol {s}}}\right ). \end{equation}

3. (iii) The MHD variables; centre of mass density ( $\rho _m$ ) and ( $\boldsymbol {v}$ ) are defined as

   (2.16) \begin{eqnarray} \rho _m=\rho _{\textit{me}}+\rho _{\textit{mi}}=n(m_e+m_i)=nm_i(1+m_e/m_i)\approx nm_i=\rho _{\textit{mi}},\nonumber \\ {\boldsymbol {v}}=\frac {\rho _{\textit{mi}}\boldsymbol {v}_{\boldsymbol {i}}+\rho _{\textit{me}}\boldsymbol {v}_{\boldsymbol {e}}}{\rho _m}=\frac {\rho _{\textit{mi}}\left [\boldsymbol {v}_{\boldsymbol {i}}+(m_e/m_i)\boldsymbol {v}_{\boldsymbol {e}}\right ]}{\rho _m}\approx \boldsymbol {v}_{\boldsymbol {i}}. \end{eqnarray}
   Here, we have used the quasineutrality condition (2.12). Also, for the length scales $\lambda _e/\lambda _i\ll 1\Rightarrow \sqrt {m_e/m_i}\ll 1$ , so that the terms $O(m_e/m_i)$ are neglected.

Summation of (2.6) and (2.7) together with (2.8) leads to

(2.17) \begin{equation} \frac {\partial \rho _m}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(\rho _m\boldsymbol {v})=\mathcal{S}_e. \end{equation}

Using the Hall-MHD approximation (see (2.16)), (2.17) can be recast as

(2.18) \begin{equation} \frac {\partial n}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(n\boldsymbol {v})=\frac {\mathcal{S}_e}{m_i}, \end{equation}

where

(2.19) \begin{equation} \frac {\mathcal{S}_e}{m_i}=\epsilon ^2\left (\frac {m_e}{m_i}\right )\boldsymbol{\nabla \boldsymbol{\cdot }}\left (\frac {\tilde {\boldsymbol {J}_{\!\boldsymbol {s}}}}{e}\right )=\epsilon ^2\left (\frac {\lambda _e}{\lambda _i}\right )^2{\boldsymbol {\nabla }\boldsymbol{\cdot }}\left (\frac {\tilde {\boldsymbol {J}_{\!\boldsymbol {s}}}}{e}\right ). \end{equation}

On the Hall-MHD length scales $\lambda _e\ll \lambda _i\Rightarrow \mathcal{S}_e/m_i\to 0$ . Thus, the right-hand side of (2.18) is vanishingly small and the Hall-MHD continuity equation effectively becomes

(2.20) \begin{equation} \frac {\partial n}{\partial t}+\boldsymbol{\nabla }\boldsymbol{\cdot }(n\boldsymbol {v})\approx 0. \end{equation}

Summation of (2.9) and (2.10) together with (2.7), (2.16) and (2.17) yields

(2.21) \begin{equation} \rho _m\frac {{\textrm{d}}\boldsymbol {v}}{{\textrm{d}}t}=\frac {\boldsymbol {J}\times \boldsymbol {B}}{c}-\boldsymbol{\nabla }p-\mathcal{S}_e\boldsymbol {v},\quad \frac {\textrm{d}}{{\textrm{d}}t}=\frac {\partial }{\partial t}+\boldsymbol {v}\boldsymbol{\cdot }\boldsymbol{\nabla }. \end{equation}

Here, ${\textrm{d}}/{\textrm{d}}t$ is the total time derivative operator and $p=p_i+p_e=n(T_i+T_e)$ is the total MHD plasma scalar pressure. Using (2.12), Hall-MHD approximations (2.16) and neglecting the term $O (\mathcal{S}_e/m_i )$ as before, the momentum equation effectively becomes

(2.22) \begin{equation} \frac {{\textrm{d}}\boldsymbol {v}}{{\textrm{d}}t}\approx \frac {\boldsymbol {J}\times \boldsymbol {B}}{cnm_i}-\frac {\boldsymbol{\nabla }p}{nm_i}. \end{equation}

Using Ampere’s law (2.15), the Hall-MHD momentum equation (2.22) becomes

(2.23) \begin{equation} \frac {{\textrm{d}}\boldsymbol {v}}{{\textrm{d}}t}=\frac {\left [(\boldsymbol{\nabla }\times \boldsymbol {B})\times \boldsymbol {B}-\epsilon ^2(4\pi /c)\tilde {\boldsymbol {J}_{\!\boldsymbol {s}}}\times \boldsymbol {B}\right ]}{4\pi nm_i}-\frac {\boldsymbol{\nabla } p}{nm_i}. \end{equation}

To obtain an equation for the magnetic field $\boldsymbol {B}$ , we must eliminate $\boldsymbol {E}$ from (2.2). The MHD approximation (2.16) clearly shows that the Hall-MHD flow velocity $\boldsymbol { v}\approx \boldsymbol {v}_{\boldsymbol {i}}$ (the ion fluid velocity) and the quasi-neutrality condition (2.12) provides $\rho _i\approx ne$ and current density $\boldsymbol {J}\approx \boldsymbol {J}_{\boldsymbol {i}}\approx en\boldsymbol { v}$ . The ion responses are important for the low-frequency disturbance and also on the ion inertial length scale ( $\lambda _i\gt \lambda$ ), $\boldsymbol {E}$ is expressed from the ion momentum (2.9) (Berezin & Karpman Reference Berezin and Karpman1964; Hassam & Huba Reference Hassam and Huba1988) as

(2.24) \begin{equation} \boldsymbol {E}\approx \frac {m_i}{e}\frac {{\textrm{d}}\boldsymbol {v}}{{\textrm{d}}t}-\frac {\boldsymbol {v}\times \boldsymbol { B}}{c}+\frac {{\boldsymbol{\nabla }}p_i}{ne}. \end{equation}

By virtute of (2.22), (2.24) can be recast as

(2.25) \begin{equation} \boldsymbol {E}\approx \frac {\boldsymbol {J}\times \boldsymbol {B}}{cne}-\frac {\boldsymbol {v}\times \boldsymbol {B}}{c}-\frac {\boldsymbol{\nabla } p_e}{ne}, \end{equation}

which is the ‘generalised Ohm’s law’. The term $\propto {\boldsymbol {J}\times \boldsymbol {B}}$ is the Hall term which decouples the electron and ion motion on the ion inertial length scale. Substitution of this expression for $\boldsymbol {E}$ in Faraday’s law (2.2) leads to the following equation for $\boldsymbol {B}$ :

(2.26) \begin{equation} \frac {\partial \boldsymbol {B}}{\partial t}=\boldsymbol{\nabla }\times (\boldsymbol {v}\times \boldsymbol {B})-\frac {m_ic}{e}\boldsymbol{\nabla }\times \frac {{\textrm{d}}\boldsymbol { v}}{{\textrm{d}}t}. \end{equation}

Finally, (2.20), (2.23) and (2.26) constitute our full set of Hall-MHD model equations.

The equilibrium uniform magnetic field $\boldsymbol {B}_0$ is assumed to be in the $x$ -direction so that $\boldsymbol { B}_0=B_0\hat {e}_x$ . We are concerned with the plane wave propagating in the $x$ -direction so that all dynamical variables are assumed to be functions of $x$ (space) and $t$ (time) only. The charged density object is assumed to be moved in the $x$ -direction so that the additional perturbed current source term can be written as

(2.27) \begin{equation} \tilde {\boldsymbol {J}_{\!\boldsymbol {s}}}=\tilde {J}(x, t)\hat {e}_x \end{equation}

and thus, consider only the perturbations with $x$ variation. The geometry of the problem is shown in figure 1.

Figure 1.

The obstacle (source) is moving in the direction of the AW ( $x$ -direction) with constant speed $\tilde {v_d}$ . The perturbed magnetic field $\tilde {\boldsymbol{B}}$ is in the $y$ and $z$ directions with $\hat {\boldsymbol {b}}=\tilde {\boldsymbol {B}}/B_0$ , so that $b_y=\tilde {B}_y/B_0$ and $b_z=\tilde {B}_z/B_0$ . All the perturbed variables are assumed to be a function of $(x, t)$ .

The dimensionless variables

(2.28) \begin{align} \hat {t}=tv_A/\lambda _i, \!\quad\hat {x}=x/\lambda _i\;\hat {n}=n/n_0, \!\quad\hat {\boldsymbol {b}}=\boldsymbol {B}/B_0, \!\quad\hat {\boldsymbol {v}}=\boldsymbol {v}/v_A \!\quad\text{and} \!\quad\hat {J}=(4\pi \lambda _i/c) \tilde {\boldsymbol {J}_{\!\boldsymbol {s}}}/B_0,\\[-18pt]\nonumber \end{align}

are introduced to write the basic equations in dimensionless form. The normalised magnetic field ( $\hat {\boldsymbol {b}}$ ) and Hall-MHD fluid flow velocity ( $\hat {\boldsymbol {v}}$ ) are as follows:

(2.29) \begin{equation} \hat {\boldsymbol {b}}=b_y(x, t)\hat {e}_y+b_z(x, t)\hat {e}_z\quad\mbox{and}\quad\hat {\boldsymbol {v}}=v_x(x, t)\hat {e}_x+v_y(x, t)\hat {e}_y+v_z(x, t)\hat {e}_z, \end{equation}

where the $x$ -component of the perturbed magnetic field $b_x=0$ as the equilibrium magnetic $B_0\hat {e}_x$ is constant. Hereafter, we will be using these new variables and remove all the hats for simplicity of notation.

Finally, the (dimensionless) Hall-MHD equations (2.20), (2.23) and (2.26) in component form can be written as

(2.30) \begin{align} \left (\frac {\partial }{\partial t}+v_x\frac {\partial }{\partial x}\right )n &+n\frac {\partial v_x}{\partial x}=0, \end{align}

(2.31) \begin{align} \left (\frac {\partial }{\partial t}+v_x\frac {\partial }{\partial x}\right )v_x &+\frac {1}{n}\frac {\partial }{\partial x}\bigg (\frac {b_y^2+b_z^2}{2}\bigg )+\frac {\beta }{n}\frac {\partial n}{\partial x}=0, \end{align}

(2.32) \begin{align} \left (\frac {\partial }{\partial t}+v_x\frac {\partial }{\partial x}\right )v_y &-\frac {1}{n}\frac {\partial b_y}{\partial x}-\epsilon ^2\frac {\tilde {J} b_z}{n}=0, \end{align}

(2.33) \begin{align} \left (\frac {\partial }{\partial t}+v_x\frac {\partial }{\partial x}\right )v_z &-\frac {1}{n}\frac {\partial b_z}{\partial x}+\epsilon ^2\frac {\tilde {J} b_y}{n}=0, \end{align}

(2.34) \begin{align} \left (\frac {\partial }{\partial t}+v_x\frac {\partial }{\partial x}\right )b_y+b_y\frac {\partial v_x}{\partial x} &-\frac {\partial v_y}{\partial x}-\frac {\partial }{\partial x}\left (\frac {1}{n}\frac {\partial b_z}{\partial x}-\epsilon ^2\frac {\tilde {J} b_y}{n}\right )=0, \end{align}

(2.35) \begin{align} \left (\frac {\partial }{\partial t}+v_x\frac {\partial }{\partial x}\right )b_z+b_z\frac {\partial v_x}{\partial x} &-\frac {\partial v_z}{\partial x}+\frac {\partial }{\partial x}\left (\frac {1}{n}\frac {\partial b_y}{\partial x}+\epsilon ^2\frac {\tilde {J} b_z}{n}\right )=0. \end{align}

Here, $C_s=\sqrt {(T_i+T_e)/m_i}$ is the sound speed in MHD plasma. Also, $\beta =C_s^2/v_A^2=n_0T_e/(B_0^2/4\pi )$ defines the ratio of the plasma pressure to the magnetic pressure representing the plasma $\beta$ (Bittencourt Reference Bittencourt2013).

3. Linear analysis

The equilibrium values of the dynamical variables in the above set of Hall-MHD equations (2.30)–(2.35) are $n=1$ , $(v_x,\;v_y,\;v_z)=(0,\;0,\;0)$ , $(b_y,\;b_z)=(0,\;0)$ and $\tilde {J}=0$ . To study the dynamics of the linear collective mode in Hall-MHD, we linearise the set of basic equations (2.30)–(2.35) by assuming

(3.1) \begin{align} n=1+\epsilon \tilde {n},\quad (v_x, v_y, v_z)=(0, 0, 0)+\epsilon (\tilde {v_x}, \tilde {v_y}, \tilde {v_z}),\quad (b_y, b_z)=(0, 0)+\epsilon (\tilde {b_y}, \tilde {b_z}). \end{align}

Eliminating $\tilde {v_x}$ from the linearised equations (2.30) and (2.31) ( $O(\epsilon )$ ), we get the following equation:

(3.2) \begin{equation} \frac {\partial ^2 \tilde {n}}{\partial t^2}-\beta \frac {\partial ^2\tilde {n}}{\partial x^2}=0. \end{equation}

Again eliminating $\tilde {v_y}$ and $\tilde {v_z}$ from the linearised equations (2.32)–(2.35), we obtain the following coupled linear equations:

(3.3) \begin{equation} \frac {\partial ^2\tilde {b_y}}{\partial t^2}-\frac {\partial ^2\tilde {b_y}}{\partial x^2}=\frac {\partial ^3\tilde {b_z}}{\partial t\partial x^2},\;\;\;\;\; \frac {\partial ^2\tilde {b_z}}{\partial t^2}-\frac {\partial ^2 \tilde {b_z}}{\partial x^2}=-\frac {\partial ^3 \tilde {b_y}}{\partial t\partial x^2}. \end{equation}

Combining these two equations, we get a single equation for $\tilde {b_y}$ (or $\tilde {b_z}$ ) as follows:

(3.4) \begin{equation} \left (\frac {\partial ^2}{\partial t^2}-\frac {\partial ^2}{\partial x^2}\right )^2\tilde {b_y}(\tilde {b_z})=-\frac {\partial ^3}{\partial t\partial x^2}\left (\frac {\partial ^2}{\partial t^2}-\frac {\partial ^2}{\partial x^2}\right )\tilde {b_y}(\tilde {b_z}). \end{equation}

Finally, supposing that all perturbed variables are proportional to $\sim \exp i(\tilde {k}x-\tilde {\omega } t)$ (where $\tilde {k}$ and $\tilde {\omega }$ are the normalised wave number and frequency, respectively), from (3.2), we obtain the dispersion relation as

(3.5) \begin{equation} \tilde {\omega }^2=\beta \tilde {k}^2. \end{equation}

In the dimensional form, (3.5) reads as $\omega =kC_s$ , which is the usual longitudinal sound wave in MHD. No electric field, electric current density or magnetic field are associated with this wave mode.

In a similar way, from (3.4), we obtain another dispersion relation as

(3.6) \begin{equation} \left (\tilde {\omega }^2-\tilde {k}^2\right )^2=\tilde {\omega }^2\tilde {k}^4. \end{equation}

This dispersion relation (3.6) provides the solutions as

(3.7) \begin{equation} \tilde {\omega }=\pm \frac {\tilde {k}^2}{2}\pm \tilde {k}\;\left [1+ \left (\frac {\tilde {k}}{2}\right )^2\right ]^{1/2}. \end{equation}

In the long wavelength approximation $(\tilde {k}\ll 1)$ , the physically feasible solutions can be expressed as

(3.8) \begin{equation} \tilde {\omega }\simeq \tilde {k}\pm \frac {\tilde {k}^2}{2}+\frac {\tilde {k}^3}{8}+\ldots \;\Rightarrow \;\omega \simeq kv_A\left (1\pm \frac {k\lambda _i}{2}+O(k^2\lambda _i^2)\right ), \end{equation}

in dimensional form. This mode corresponds to the usual AW with weakly dispersive correction ( $\propto k\lambda _i$ ). This $k\lambda _i$ term is the ion inertia contribution which provides the essential dispersion of the mode. The $\pm$ sign in (3.8) corresponds to the state of polarisation of the AW, i.e. right circularly polarised (RCP) and left circularly polarised (LCP) waves, respectively. In the next section, we investigate the nonlinear excitation of this AW through a steadily moving charged object.

4. Interaction dynamics of nonlinear compressional Alfvén wave and moving charged obstacle

In this section, we shall derive the nonlinear transport equation that governs the dynamics of the weakly nonlinear AW excited by a steadily moving charged object. The linear mode analysis (dispersion relation in (3.8)) provides that the phase of the AW is given by

(4.1) \begin{equation} \tilde {k}x-\tilde {\omega }t=\tilde {k}(x-t)\mp \frac {\tilde {k}^2}{2}t+O(\tilde {k}^3). \end{equation}

To investigate the weakly nonlinear dynamics of the AW in the long wavelength ( $\tilde {k}\ll 1$ ) limit, the above phase relation (4.1) suggests us to introduce a coordinate stretching defined by (Ichikawa et al. Reference Ichikawa, Konno, Wadati and Sanuki1980)

(4.2) \begin{equation} \xi =\epsilon (x-t), \;\; \tau =\epsilon ^2t, \end{equation}

where $\tilde {k}$ is proportional to $\epsilon$ . The dependent variables $(n,\;v_x,\;v_{y(z)},\;b_{y(z)})$ are expanded around the uniform state in powers of $\epsilon$ as

(4.3) \begin{eqnarray} & \left [ n,\;\; v_x \right ] =\left [ 1,\;\; 0 \right ]+ \sum \limits _{j=1}^\infty \epsilon ^j \left [ n^{(j)}(\xi ,\tau ),\;\; v_x^{(j)}(\xi ,\tau ) \right ],\nonumber \\ & \left [ v_{y(z)},\;\; b_{y(z)} \right ] =\left [ 0,\;\; 0 \right ]+\sum _{j=0}^{\infty }\epsilon ^{j+\frac {1}{2}} \left [ v_{y(z)}^{(j+1)}(\xi ,\tau ),\;\; b_{y(z)}^{(j+1)}(\xi ,\tau ) \right ]. \end{eqnarray}

To study the effects of the additional perturbed current $\tilde {J}(x, t)$  (2.27) on the dynamics of the weakly nonlinear AW, we consider the implicit form $\tilde {J}(x, t)=\tilde {J} (\tilde {k}(x-v_d t) )$ , where $v_d$ is the normalised (by $v_A$ ) constant speed of the charged object moving in the $x$ -direction. Since $\tilde {k}\sim \epsilon$ , in the ( $\xi , \tau$ ) frame,

(4.4) \begin{equation} \tilde {J}(\xi , \tau )=\tilde {J}(\xi -\mathcal{M}\tau ), \end{equation}

where $\epsilon \mathcal{M}=v_d-1\Rightarrow v_d=1+\epsilon \mathcal{M}$ . The parameter $\mathcal{M}$ represents the relative Mach number of the charged object which measures how close the moving object velocity is to the Alfvèn velocity with $\mathcal{M}=0$ corresponding to exact resonance. As $\mathcal{M}$ increases from zero, the velocity eventually becomes supercritical, while it eventually becomes subcritical as $\mathcal{M}$ decreases from zero.

Using the new independent coordinates (4.2) and the perturbation expansion (4.3), we transform the set of basic equations (2.30)–(2.35) into a set of equations in the form of a series in $\epsilon$ . The resulting system can be written as

(4.5) \begin{align} & \epsilon ^{2}\frac {\partial }{\partial \xi }\left (v_x^{(1)}-n^{(1)}\right ) +\epsilon ^{3}\left [\frac {\partial n^{(1)}}{\partial \tau } +\frac {\partial }{\partial \xi }\big (n^{(1)}v_x^{(1)}+v_x^{(2)}-n^{(2)}\big )\right ]+O(\epsilon ^4)=0, \end{align}

(4.6) \begin{align} & \epsilon ^{2}\frac {\partial }{\partial \xi }\left [\beta n^{(1)}-v_x^{(1)}+\left (\frac {{b_y^{(1)}}^2+{b_z^{(1)}}^2}{2}\right )\right ] +\epsilon ^{3}\Bigg [\frac {\partial v_x^{(1)}}{\partial \tau }+\frac {\partial }{\partial \xi }\big (\beta n^{(2)}-v_x^{(2)}\big )+\frac {1}{2}\frac {\partial {v_x^{(1)}}^2}{\partial \xi }\nonumber \\ & \quad -\frac {\beta }{2}\frac {\partial {n^{(1)}}^2}{\partial \xi }+ \frac {\partial }{\partial \xi }\left (b_y^{(1)}b_z^{(2)}+b_y^{(2)}b_z^{(1)}\right )-n^{(1)}\frac {\partial }{\partial \xi }\left (\frac {{b_y^{(1)}}^2+{b_z^{(1)}}^2}{2}\right ) \Bigg ]+O(\epsilon ^{4})=0, \end{align}

(4.7) \begin{align} & -\epsilon ^{\frac {3}{2}}\frac {\partial }{\partial \xi }\left (v_y^{(1)}+b_y^{(1)}\right ) +\epsilon ^{\frac {5}{2}}\bigg [\frac {\partial v_y^{(1)}}{\partial \tau }-\frac {\partial }{\partial \xi }\left (v_y^{(2)}+b_y^{(2)}\right ) +v_x^{(1)}\frac {\partial v_y^{(1)}}{\partial \xi }+n^{(1)}\frac {\partial b_y^{(1)}}{\partial \xi } -\tilde {J} b_z^{(1)}\bigg ]\nonumber \\ & \quad +\, O\Big(\epsilon ^\frac {7}{2}\Big)=0, \end{align}

(4.8) \begin{align} & -\epsilon ^{\frac {3}{2}}\frac {\partial }{\partial \xi }\left (v_z^{(1)}+b_z^{(1)}\right ) +\epsilon ^{\frac {5}{2}}\bigg [\frac {\partial v_z^{(1)}}{\partial \tau }-\frac {\partial }{\partial \xi }\left (v_z^{(2)}+b_z^{(2)}\right ) +v_x^{(1)}\frac {\partial v_z^{(1)}}{\partial \xi }+n^{(1)}\frac {\partial b_z^{(1)}}{\partial \xi } +\tilde {J} b_y^{(1)}\bigg ]\nonumber \\ &\quad +\, O\Big(\epsilon ^\frac {7}{2}\Big)=0, \end{align}

(4.9) \begin{align} & \epsilon ^{\frac {3}{2}}\frac {\partial }{\partial \xi }\left (-v_y^{(1)}-b_y^{(1)}\right ) +\epsilon ^{\frac {5}{2}}\left [\frac {\partial b_y^{(1)}}{\partial \tau }-\frac {\partial }{\partial \xi }\left (v_y^{(2)}+b_y^{(2)}\right ) +\frac {\partial (v_x^{(1)}b_y^{(1)})}{\partial \xi }-\frac {\partial ^2 b_z^{(1)}}{\partial \xi ^2}\right ] \nonumber \\ &\quad +\, O\Big(\epsilon ^\frac {7}{2}\Big)=0, \end{align}

(4.10) \begin{align} & \epsilon ^{\frac {3}{2}}\frac {\partial }{\partial \xi }\left (-v_z^{(1)}-b_z^{(1)}\right ) +\epsilon ^{\frac {5}{2}}\left [\frac {\partial b_z^{(1)}}{\partial \tau }-\frac {\partial }{\partial \xi }\left (v_z^{(2)}+b_z^{(2)}\right ) +\frac {\partial (v_x^{(1)}b_z^{(1)})}{\partial \xi }+\frac {\partial ^2 b_y^{(1)}}{\partial \xi ^2}\right ]\nonumber \\ &\quad +\, O\Big(\epsilon ^\frac {7}{2}\Big)=0. \end{align}

The above equations are to be satisfied to all orders in $\epsilon$ . The lowest orders ( $O(\epsilon ^{3/2})$ and $O(\epsilon ^{2})$ ) term subject to the boundary conditions $n^{(1)},\;v_{x,y,z}^{(1)},\;b_{y,z}^{(1)}\to 0$ as $\xi \to -\infty$ provides the following relations:

(4.11) \begin{equation} v_y^{(1)}=-b_y^{(1)},\;\;v_z^{(1)}=-b_z^{(1)},\;\;n^{(1)}=v_x^{(1)},\;\;v_x^{(1)}=\beta n^{(1)}+\frac {{b_y^{(1)}}^2+{b_z^{(1)}}^2}{2} .\end{equation}

Equating the next higher order terms ( $O(\epsilon ^{5/2})$ ), we get the following nonlinear equations:

(4.12) \begin{align} & \frac {\partial v_y^{(2)}}{\partial \xi }+\frac {\partial b_y^{(2)}}{\partial \xi }=\frac {\partial v_y^{(1)}}{\partial \tau }-\tilde {J} b_z^{(1)},\nonumber \\ & \frac {\partial v_z^{(2)}}{\partial \xi }+\frac {\partial b_z^{(2)}}{\partial \xi }=\frac {\partial v_z^{(1)}}{\partial \tau }+\tilde {J} b_y^{(1)}, \end{align}

(4.13) \begin{align} & \frac {\partial v_y^{(2)}}{\partial \xi }+\frac {\partial b_y^{(2)}}{\partial \xi }=\frac {\partial b_y^{(1)}}{\partial \tau }-\frac {\partial ^2b_z^{(1)}}{\partial \xi ^2}+\frac {\partial }{\partial \xi }\big (v_x^{(1)}b_y^{(1)}\big ),\nonumber \\ & \frac {\partial v_z^{(2)}}{\partial \xi }+\frac {\partial b_z^{(2)}}{\partial \xi }=\frac {\partial b_z^{(1)}}{\partial \tau }-\frac {\partial ^2b_y^{(1)}}{\partial \xi ^2}+\frac {\partial }{\partial \xi }\big (v_x^{(1)}b_z^{(1)}\big ). \end{align}

Eliminating all the second order terms from the above set of equations and using linear relations provided in (4.11), we obtain the following coupled equations:

(4.14) \begin{align} \frac {\partial b_y^{(1)}}{\partial \tau }+\frac {1}{4(1-\beta )}\frac {\partial }{\partial \xi }\left [b_y^{(1)}\left (b_y^{(1)2}+b_z^{(1)2}\right )\right ]-\frac {1}{2}\frac {\partial ^2 b_z^{(1)}}{\partial \xi ^2} &=-\frac {\tilde {J} b_z^{(1)}}{2},\\[-12pt]\nonumber \end{align}

(4.15) \begin{align} \frac {\partial b_z^{(1)}}{\partial \tau }+\frac {1}{4(1-\beta )}\frac {\partial }{\partial \xi }\left [b_z^{(1)}\left (b_y^{(1)2}+b_z^{(1)2}\right )\right ]+\frac {1}{2}\frac {\partial ^2 b_y^{(1)}}{\partial \xi ^2} &=\frac {\tilde {J} b_y^{(1)}}{2}. \end{align}

Equations (4.14) and (4.15) describe the excitation dynamics of the weakly nonlinear AW induced by a steadily moving charged density object.

To incorporate the polarisation effects of the AW, we introduce a new variable $\phi =b_y^{(1)}\mp ib_z^{(1)}$ and combine (4.14) and (4.15), and finally obtain the following fDNLSE equation:

(4.16) \begin{eqnarray} i\frac {\partial \phi }{\partial \tau }+i\alpha \frac {\partial }{\partial \xi }\bigg (|\phi |^2\phi \bigg )+\frac {\ell }{2}\frac {\partial ^2 \phi }{\partial \xi ^2}=S(\xi , \tau )\phi ,\nonumber \\ \mbox{where}\quad \alpha =\frac {1}{4(1-\beta )},\;S(\xi , \tau )=S(\xi -\mathcal{M}\tau )=\frac {\tilde {J}(\xi -\mathcal{M}\tau )}{2}. \end{eqnarray}

Here, $\ell (=\pm 1)$ , before the dispersive term corresponds to the RCP (LCP) wave which is consistent with the linear dispersion relation (3.8). The coefficient of nonlinearity $(\alpha )$ changes its sign depending on the values of plasma $\beta$ according as $\beta \gtrless 1\Rightarrow \alpha \gtrless 0$ and thus, $\beta$ significantly modifies the nonlinearity in the system. The values $\beta \gt 1$ and $\beta \lt 1$ correspond to the high and low $\beta$ plasmas. The forcing term $S(\xi , \tau )$ is a real valued function and proportional to the current $\tilde {J}(\xi , \tau )$ produced by the moving charged object.

5. Exact solution: variational approach

In this section, we shall find the exact solution of the derived fDNLSE (4.16) using variational principle. In absence of the source term $S(\xi , \tau )=0$ , (4.16) is the well known Kaup–Newell equation (KNE)

(5.1) \begin{equation} i\frac {\partial \phi }{\partial \tau }+i\alpha \frac {\partial }{\partial \xi }\bigg (|\phi |^2\phi \bigg )+\frac {\ell }{2}\frac {\partial ^2 \phi }{\partial \xi ^2}=0, \end{equation}

which is solvable using inverse scattering method (Kaup & Newell Reference Kaup and Newell1978). Unfortunately, the KNE (5.1) cannot be expressed in a Hamiltonian form as

(5.2) \begin{equation} i\frac {\partial \phi }{\partial \tau }=\frac {\delta \mathcal{H}}{\delta \phi ^*},\quad i\frac {\partial \phi ^*}{\partial \tau }=-\frac {\delta \mathcal{H}}{\delta \phi }, \end{equation}

where $\delta$ denotes the variational derivative, the asterisk $(*)$ stands for the complex conjugate and $\mathcal{H}(\phi , \phi ^*)$ is the Hamiltonian (Hayashi & Ozawa Reference Hayashi and Ozawa1992; Guo & Wu Reference Guo and Wu1995). As a result the fDNLSE (4.16) (i.e. KNE with $S(\xi , \tau )\neq 0$ ) also can not be expressed as (5.2).

However, KNE (5.1) can be transformed to the above Hamiltonian form with the help of the gauge transformation (Hayashi & Ozawa Reference Hayashi and Ozawa1992; Colin & Ohta Reference Colin and Ohta2006),

(5.3) \begin{equation} u(\xi ,\tau )=\phi (\xi , \tau )\exp \left (\frac {i\alpha }{2}\int _{-\infty }^{\xi }\mid \phi (y, \tau )\mid ^2{\textrm{d}}y\right ). \end{equation}

Therefore, we use the gauge transformation (5.3) and transform the fDNLSE (4.16) to the equation

(5.4) \begin{equation} i\frac {\partial u}{\partial \tau }+i\alpha |u|^2\frac {\partial u}{\partial \xi }+\frac {\ell }{2}\frac {\partial ^2 u}{\partial \xi ^2}=S(\xi , \tau )u. \end{equation}

In absence of source term ( $S(\xi , \tau )=0$ ), (5.4) is the fully integrable Chen–Lee–Liu equation (CLLE) which obeys a number of conservation laws including the Hamiltonian (Chen et al. Reference Chen, Lee and Liu1979, Reference Chen, Lee and Lin1987). We call this (5.4) as the forced CLLE (fCLLE) which can be recast as the following Hamiltonian form:

(5.5) \begin{equation} i\frac {\partial u}{\partial \tau }=\frac {\delta \mathcal{H}}{\delta u^*},\;\;\;i\frac {\partial u^*}{\partial \tau }=-\frac {\delta \mathcal{H}}{\delta u}. \end{equation}

The Hamiltonian $\mathcal{H}(u, u^*)$ of the fCLLE is given by

(5.6) \begin{equation} \mathcal{H}=\int _{-\infty }^{\infty }\left [\frac {\ell }{2}\mid u_{\xi }\mid ^2-\frac {i\alpha }{4}\left (u^*u_{\xi }-u^*_{\xi }u\right )\mid u\mid ^2+S(\xi , \tau )\mid u\mid ^2\right ]{\textrm{d}}\xi , \end{equation}

where the integration is performed either over an infinite range for $\xi$ with $u(\xi , \tau )$ and its derivatives vanishing sufficiently fast as $\mid \xi \mid \to \infty$ or over a finite range with periodic boundary conditions.

In addtion to $\mathcal{H}$ , the fCLLE preserves the mass ( $\mathcal{N}$ ) and momentum ( $\mathcal{P}$ ) given by

(5.7) \begin{equation} \mathcal{N}=\int _{-\infty }^{\infty }\mid u\mid ^2\,{\textrm{d}}\xi \quad\mbox{and}\quad\mathcal{P}=\frac {i}{2}\int _{-\infty }^{\infty }\left (u^*u_{\xi }-u^*_{\xi }u\right )\,{\textrm{d}}\xi . \end{equation}

In proceeding subsection, we shall derive the stationary solutions of fCLLE (5.4) adopting the variational approach. These solutions provide us the stationary solutions of the derived fDNLSE (4.16) through the inverse transformation of (5.3) as

(5.8) \begin{equation} \phi (\xi , \tau )=u(\xi , \tau )\exp \left (-\frac {i\alpha }{2}\int _{-\infty }^{\xi }\mid u(y, \tau )\mid ^2{\textrm{d}}y\right )\;\Rightarrow \;\mid \phi (\xi , \tau )\mid ^2=\mid u(\xi , \tau )\mid ^2. \end{equation}

5.1. General theory: stationary solution

To derive the stationary solutions of the (5.4) (and hence, (4.16)), we consider the frame ( $\zeta$ ) of the charged object moving with the relative Mach number $\mathcal{M}$ ,

(5.9) \begin{equation} \zeta =\xi -\mathcal{M}\tau , \end{equation}

as the stationary frame. In this $\zeta$ frame, we consider the variational problem

(5.10) \begin{equation} {\delta }\left (\mathcal{H}+\omega \mathcal{N}+\mathcal{M}\mathcal{P}\right )\bigg |_{\zeta -\mbox{frame}}=0, \end{equation}

illustrating the fact that all the stationary solutions of (5.4) which decreases as $\mid \zeta \mid \to \infty$ are the stationary points of the functional $\mathcal{H}$ for fixed $\omega$ and $\mathcal{M}$ . Here, the real parameters $\omega$ and $\mathcal{M}$ have the meaning of Lagrangian multipliers. This variational problem yields the following nonlinear ordinary differential equation:

(5.11) \begin{equation} \frac {\ell }{2}\frac {d^2u}{{\textrm{d}}\zeta ^2}+i\alpha |u|^2\frac {{\textrm{d}} u}{{\textrm{d}}\zeta }-\left (\omega +S(\zeta )\right )u-i\mathcal{M}\frac {{\textrm{d}} u}{{\textrm{d}}\zeta }=0. \end{equation}

Equation (5.11) suggests the following stationary solution of (5.4) in the $\zeta$ -frame:

(5.12) \begin{equation} u(\xi ,\tau )=\varrho (\zeta )\exp (i\omega \tau ), \end{equation}

where $\varrho (\zeta )$ is a complex valued function. Substitution of (5.12) in (5.4) provides

(5.13) \begin{equation} \frac {\ell }{2}\frac {d^2\varrho }{{\textrm{d}}\zeta ^2}+i\alpha |\varrho |^2\frac {{\textrm{d}}\varrho }{{\textrm{d}}\zeta }-\left (\omega +S(\zeta )\right )\varrho -i\mathcal{M}\frac {{\textrm{d}}\varrho }{{\textrm{d}}\zeta }=0, \end{equation}

which is equivalent to (5.11) derived from the variational problem (6.16). Since $\varrho (\zeta )$ is complex, we assume

(5.14) \begin{equation} \varrho (\zeta )=\psi (\zeta )\exp (i\theta ),\quad\mbox{where}\quad\;\theta =\int _0^{\zeta }\kappa (s)\,{\textrm{d}}s \end{equation}

such that $\partial _{\zeta }\theta =\kappa (\zeta )$ is real representing the wave number and $\psi (\zeta )$ is a real-valued function which satisfies the localised boundary conditions, $\lim _{\mid \zeta \mid \to \infty }\psi (\zeta )=0$ .

Thus, the general solution of (5.4) can be written in the following form:

(5.15) \begin{equation} u(\xi ,\tau )=\psi (\zeta )\exp \left (i\varphi (\zeta )\right ), \end{equation}

with the phase of the solution ( $\varphi$ ) as

(5.16) \begin{equation} \varphi (\zeta )=\theta +\omega \tau =\int _0^{\zeta }\kappa (s)\,{\textrm{d}}s+\omega \tau . \end{equation}

Accordingly, with the help of the inverse transformation (5.8), the stationary solution of the derived fDNLSE (4.16) is given by

(5.17) \begin{equation} \phi (\xi ,\tau )=\phi (\zeta =\xi -\mathcal{M}\tau )=u(\zeta )\exp \left (i\varTheta (\zeta )\right ), \end{equation}

where the phase of the wave field ( $\varTheta$ ) is given by

(5.18) \begin{equation} \varTheta (\zeta )=\varphi (\zeta )-\frac {\alpha }{2}\int _{-\infty }^{\zeta }\mid u(y)\mid ^2{\textrm{d}}y. \end{equation}

Substituting (5.14) into (5.13) and separating the imaginary and real parts, we get

(5.19) \begin{equation} \ell \kappa =\mathcal{M}-\frac {\alpha }{2}\psi ^2 \end{equation}

for wave number and

(5.20) \begin{equation} \frac {1}{2}\frac {{\textrm{d}}^2\psi }{{\textrm{d}}\zeta ^2}-\left [\frac {\kappa ^2}{2}+\ell \left (\omega +S(\zeta )\right )-\ell \kappa \mathcal{M}\right ]\psi -\ell \alpha \kappa \psi ^3=0 \end{equation}

for the amplitude evolution with the help of the localised boundary condition. Again substituting (5.19) into (5.20), we finally obtain the following nonlinear ordinary differential equation with variable coefficients analogous to an autonomous quintic oscillator:

(5.21) \begin{equation} \frac {{\textrm{d}}^2\psi }{{\textrm{d}}\zeta ^2}+\left (\mathcal{M}^2-2\ell \omega \right )\psi -2\alpha \mathcal{M}\psi ^3+\frac {3\alpha ^2}{4}\psi ^5=2\ell S(\zeta )\psi . \end{equation}

This equation of motion for $\psi$ is reducible to quadrature and the exact solution for a specific analytic form of the source function $S(\zeta )$ can be expressed in terms of elliptic functions. Indeed, we may rewrite

(5.22) \begin{equation} \frac {{\textrm{d}}^2\psi }{{\textrm{d}}\zeta ^2}=-\frac {\textrm{d}}{{\textrm{d}}\psi }\left [\left (\frac {\mathcal{M}^2}{2}-\ell \omega \right )\psi ^2- 2\ell \int _{-\infty }^{\psi }S(\zeta )\psi \,{\textrm{d}}\psi -\frac {\alpha \mathcal{M}}{2}\psi ^4+\frac {3\alpha ^2}{8}\psi ^6\right ], \end{equation}

as the pseudo-potential. The integrand in the potential clearly suggests that integration is possible either for $S(\zeta )=\mbox{constant}$ or $S(\zeta )=S (\psi (\zeta ) )$ . In the proceeding subsection, we shall solve (5.4) (and hence, (4.16)) analytically for a particular localised form of $S(\zeta )$ for two situations of physical interest: (i) non-resonance ( $\mathcal{M}\neq 0$ ) and (ii) resonance ( $\mathcal{M}=0$ ).

5.2. Hyperbolic secant source

We consider the hyperbolic secant form of the localised source function $S(\zeta )$ as

(5.23) \begin{equation} S(\zeta )=f_0\;{\mbox{sech$\,$}}^l(\varDelta \zeta ),\;\;\;l\in \mathbb{R}, \end{equation}

where $f_0$ and $\varDelta$ are the real constants represent the amplitude and inverse of the width of the source. In view of the analytic form of $S(\zeta )$  (5.23), the solution of (5.21) is assumed to be of the form

(5.24) \begin{equation} \psi (\zeta )=\psi _0\;{\mbox{sech$\,$}}^{j}(\varDelta \zeta ),\;\;\;j\in \mathbb{R}, \end{equation}

where the real constant $\psi _0$ represents the amplitude of the wave field. Substituting (5.23) and (5.24) in (5.21), we obtain

(5.25) \begin{eqnarray} & \psi _0\left (j^2\varDelta ^2+\mathcal{M}^2-2\ell \omega \right )\;{\mbox{sech$\,$}}^{j}(\varDelta \zeta )-\psi _0\varDelta ^2j(j+1)\;{\mbox{sech$\,$}}^{(j+2)}(\varDelta \zeta )\nonumber \\ & -2\alpha \mathcal{M}\psi _0^3\;{\mbox{sech$\,$}}^{3j}(\varDelta \zeta )+\left (\frac {3\alpha ^2\psi _0^5}{4}\right )\;{\mbox{sech$\,$}}^{5j}(\varDelta \zeta )=2\ell \psi _0f_0\;{\mbox{sech$\,$}}^{(j+l)}(\varDelta \zeta ). \end{eqnarray}

We now derive the exact solitary wave solutions of both the LCP and RCP nonlinear compressible AW for both the nonresonance ( $\mathcal{M}\neq 0$ ) and resonance ( $\mathcal{M}=0$ ) situations.

First, we consider the nonresonance situation where the velocity is either in supercritical or in subcritical range. Then, (5.25) is solvable only if (i) $l+j=5j,\; j+2=3j$ or (ii) $j+2=5j,\; l+j=3j$ .

Condition (i) yields $l=4,\;j=1$ providing the following source function and solution:

(5.26) \begin{equation} S_A(\zeta )=f_0\;{\mbox{sech}}^{4}(\varDelta \zeta )\quad\mbox{and}\quad\psi (\zeta )=\psi _0\;{\mbox{sech}}(\varDelta \zeta ). \end{equation}

Similarly, condition (ii) yields $l=1,\;j= {1}/{2}$ providing the following source function and solution:

(5.27) \begin{equation} S_B(\zeta )=f_0\;{\mbox{sech}}(\varDelta \zeta )\quad\mbox{and}\quad\psi (\zeta )=\psi _0\;{\mbox{sech}}^{\frac {1}{2}}(\varDelta \zeta ). \end{equation}

5.2.1. Exact solution for source function: $S_A(\zeta )=f_0\;{\mbox{sech$\,$}}^{4}(\varDelta \zeta )$

For this case, substituting $l=4,\;j=1$ in (5.25) and equating the coefficients of ${\mbox{sech$\,$}}(\varDelta \zeta )$ , ${\mbox{sech$\,$}}^{3}(\varDelta \zeta )$ and ${\mbox{sech$\,$}}^{5}(\varDelta \zeta )$ , we get

(5.28) \begin{equation} \ell \omega =\frac {\varDelta ^2+\mathcal{M}^2}{2},\quad \psi _0^2=-\frac {\varDelta ^2}{\alpha \mathcal{M}}\quad\text{and}\quad f_0=\frac {3\ell \alpha ^2\psi _0^4}{8}=\frac {3\ell \varDelta ^4}{8\mathcal{M}^2}. \end{equation}

To ensure the existence of a localised solution, the width ( $\varDelta$ ) and amplitude ( $\psi _0$ ) must be real so that:

1. (i) $\ell \omega \gt 0$ and the expression of $\ell \omega$ in (5.28) provides

   (5.29) \begin{equation} 2\ell \omega -\mathcal{M}^2\gt 0\Rightarrow \ell \omega \gt \mathcal{M}^2/2; \end{equation}

2. (ii) $\psi _0^2\gt 0\Rightarrow -\alpha \mathcal{M}\gt 0\Rightarrow \alpha \mathcal{M}\lt 0$ ( $\alpha$ and $\mathcal{M}$ must have opposite sign).

Condition (ii) (with the help of the expression of $\alpha$  (4.16)) yields

(5.30) \begin{equation} \frac {\mathcal{M}}{4(1-\beta )}\lt 0\;\;\Rightarrow \;\; \mathcal{M}\gtrless 0\quad \mbox{for}\;\beta \gtrless 1. \end{equation}

Thus, for the existence of a localised solution, relative Mach number $\mathcal{M}$ possesses two values: positive for high $\beta$ ( $\beta \gt 1$ ) and negative for low $\beta$ ( $\beta \lt 1$ ). The solution (5.24) then becomes

(5.31) \begin{equation} \frac {\mathcal{M}}{4(1-\beta )}\lt 0\;\;\Rightarrow \;\; \mathcal{M}\gtrless 0\quad \mbox{for}\;\beta \gtrless 1. \end{equation}

The wave number $\kappa$ is determined from (5.19) as

(5.32) \begin{equation} \kappa =\ell \left [\mathcal{M}+\left (\frac {\varDelta ^2}{2\mathcal{M}}\right ){\mbox{sech$\,$}}^{2}(\varDelta \zeta )\right ]. \end{equation}

Accordingly, we get

(5.33) \begin{equation} \theta =\int _0^{\zeta }\kappa (s)\,{\textrm{d}}s=\ell \left [\mathcal{M}\zeta +\left (\frac {\varDelta }{2\mathcal{M}}\right )\tanh (\varDelta \zeta )\right ] \end{equation}

and the phase ( $\varphi$ )

(5.34) \begin{equation} \varphi =\ell \left [\mathcal{M}\zeta +\left (\frac {\varDelta }{2\mathcal{M}}\right )\tanh (\varDelta \zeta )+\ell \omega \tau \right ]. \end{equation}

A: solution of RCP Alfvén wave

For the RCP waves $\ell =+1$ , (5.28) provides $f_0\gt 0$ and $\omega \gt 0$ for the localised solution. Equation (5.15) then provides the following solution of the RCP waves:

(5.35) \begin{equation} u_R(\xi ,\tau )=\psi _0\;{\mbox{sech$\,$}}(\varDelta \zeta )\exp (i\varphi _R), \end{equation}

where the phase ( $\varphi _R$ ) of the wave field is given by

(5.36) \begin{equation} \varphi _R=\mathcal{M}\zeta +\left (\frac {\varDelta }{2\mathcal{M}}\right )\tanh (\varDelta \zeta )+\omega \tau . \end{equation}

B: solution of LCP Alfvén wave

For the LCP waves $\ell =-1$ , (5.28) demands $f_0\lt 0$ and $\omega \lt 0$ (as $\ell \omega \gt 0$ ) for the existence of a localised solution. As before, (5.15) provides the following solution of the LCP waves:

(5.37) \begin{equation} u_L(\xi ,\tau )=\psi _0\;{\mbox{sech$\,$}}(\varDelta \zeta )\exp (-i\varphi _L), \end{equation}

where the phase of the wave field ( $\varphi _L$ ) is given by

(5.38) \begin{equation} \varphi _L=\mathcal{M}\zeta +\left (\frac {\varDelta }{2\mathcal{M}}\right )\tanh (\varDelta \zeta )+\mid \omega \mid \tau . \end{equation}

Finally, the solution (5.17) of the derived fDNLSE (4.16) for the RCP (LCP) nonlinear AW is given by

(5.39) \begin{equation} \phi (\xi , \tau )=\pm \left (\frac {\varDelta }{\sqrt {-\alpha \mathcal{M}}}\right )\;{\mbox{sech$\,$}}(\varDelta \zeta )\exp (i\varTheta ). \end{equation}

The phase of the wave field ( $\varTheta$ ) (5.18) for the RCP and LCP AWs are respectively

(5.40) \begin{equation} \varTheta _R=\mathcal{M}\zeta +\left (\frac {\varDelta }{2\mathcal{M}}\right )\left [1+2\;\tanh (\varDelta \zeta )\right ]+\omega \tau \end{equation}

and

(5.41) \begin{equation} \varTheta _L=-\mathcal{M}\zeta -\mid \omega \mid \tau +\frac {\varDelta }{2\mathcal{M}}. \end{equation}

The corresponding source functions for the RCP and LCP AWs are respectively

(5.42) \begin{equation} S_A^R=\left (\frac {3\varDelta ^4}{8\mathcal{M}^2}\right )\;{\mbox{sech$\,$}}^{4}(\varDelta \zeta )\quad\mbox{and}\quad S_A^L=-\left (\frac {3\varDelta ^4}{8\mathcal{M}^2}\right )\;{\mbox{sech$\,$}}^{4}(\varDelta \zeta ). \end{equation}

The solution (5.39) corresponding to the source function $S_A$  (5.42) is valid for both the low ( $\beta \lt 1$ ) and high ( $\beta \gt 1$ ) $\beta$ plasmas with the proper algebraic signs of $\mathcal{M}$ according to condition (5.30).

5.2.2 Exact solutions for the source function: $S_B(\zeta )=f_0\;{\mbox{sech$\,$}}(\varDelta \zeta )$

For this case, substituting $l=1,\;j=1/2$ in (5.25) and equating the coefficients of ${\mbox{sech$\,$}}^{1/2}(\varDelta \zeta )$ , ${\mbox{sech$\,$}}^{3/2}(\varDelta \zeta )$ and ${\mbox{sech$\,$}}^{5/2}(\varDelta \zeta )$ , we get

(5.43) \begin{equation} \ell \omega =\frac {\varDelta ^2+4\mathcal{M}^2}{8},\quad \psi _0^4=\frac {\varDelta ^2}{\alpha ^2}\quad \text{and}\quad f_0=-\alpha \ell \mathcal{M}\psi _0^2=-\ell \mathcal{M}\varDelta . \end{equation}

Again, as before, to ensure the existence of a localised solution, the width ( $\varDelta$ ) and amplitude ( $\psi _0$ ) must be real so that:

1. (i) $\ell \omega \gt 0$ and the expression of $\ell \omega$ in (5.43) provides

   (5.44) \begin{equation} 8\ell \omega -4\mathcal{M}^2\gt 0\Rightarrow \ell \omega \gt \mathcal{M}^2/2; \end{equation}

2. (ii) $\psi _0^2\gt 0\Rightarrow \alpha \varDelta \gt 0$ ( $\alpha$ and $\varDelta$ must have same sign).

Condition (ii) can be expressed (using the expression of $\alpha$  (4.16)) as

(5.45) \begin{equation} \frac {\varDelta }{4(1-\beta )}\gt 0\Rightarrow \varDelta \gtrless 0\quad \mbox{for}\;\beta \lessgtr 1. \end{equation}

Thus, $\varDelta \gt 0$ ( $\varDelta \lt 0$ ) for low (high) $\beta$ . Then, the solution (5.24) can be written as

(5.46) \begin{equation} \psi (\zeta )=\psi _0\;{\mbox{sech$\,$}}^{\frac {1}{2}}(\varDelta \zeta ),\quad \psi _0=\pm \sqrt {\frac {\varDelta }{\alpha }},\quad \varDelta =\pm 2\sqrt {2\ell \omega -\mathcal{M}^2}. \end{equation}

The wave number $\kappa$ is determined from (5.19) as

(5.47) \begin{equation} \kappa =\ell \left [\mathcal{M}-\left (\frac {\varDelta }{2}\right )\;{\mbox{sech}}(\varDelta \zeta )\right ]. \end{equation}

Accordingly, we get

(5.48) \begin{equation} \theta =\int _0^{\zeta }\kappa (s)\,{\textrm{d}}s=\ell \left [\mathcal{M}\zeta -\frac {1}{2}\;\tan ^{-1}\left (\sinh (\varDelta \zeta )\right )\right ] \end{equation}

and the phase ( $\varphi$ )

(5.49) \begin{equation} \varphi =\ell \left [\mathcal{M}\zeta -\frac {1}{2}\;\tan ^{-1}\left (\sinh (\varDelta \zeta )\right )+\ell \omega \tau \right ]. \end{equation}

A: solution of RCP Alfvén wave

For the RCP waves $\ell =+1$ , (5.43) provides $f_0=-M\varDelta$ and $\omega \gt 0$ for the localised solution. Equation (5.15) then provides the following solution of the RCP wave:

(5.50) \begin{equation} u_R(\xi ,\tau )=\psi _0\;{\mbox{sech$\,$}}^{\frac {1}{2}}(\varDelta \zeta )\exp (i\varphi _R), \end{equation}

where the phase of the wave field ( $\varphi _R$ ) is given by

(5.51) \begin{equation} \varphi _R=\mathcal{M}\zeta -\frac {1}{2}\;\tan ^{-1}\left (\sinh (\varDelta \zeta )\right )+\omega \tau . \end{equation}

B: solution of LCP Alfvén wave

For the LCP waves $\ell =-1$ , (5.43) demands $f_0=M\varDelta$ and $\omega \lt 0$ (as $\ell \omega \gt 0$ ) for the existence of the localised solution. Equation (5.15) then provides the solution of the LCP wave,

(5.52) \begin{equation} u_L(\xi ,\tau )=\psi _0\;{\mbox{sech$\,$}}^{\frac {1}{2}}(\varDelta \zeta )\exp (-i\varphi _L), \end{equation}

where the phase of the wave field ( $\varphi _R$ ) is given by

(5.53) \begin{equation} \varphi _L=\mathcal{M}\zeta -\frac {1}{2}\;\tan ^{-1}\left (\sinh (\varDelta \zeta )\right )+\mid \omega \mid \tau . \end{equation}

Finally, the solution (5.17) of the derived fDNLSE (4.16) for the RCP (LCP) nonlinear AW is given by

(5.54) \begin{equation} \phi (\xi , \tau )=\pm \sqrt {\frac {\varDelta }{\alpha }}\;{\mbox{sech$\,$}}^{\frac {1}{2}}(\varDelta \zeta )\exp (i\varTheta ). \end{equation}

The phases of the RCP and LCP nonlinear AW field ( $\varTheta$ ) (5.18) are respectively

(5.55) \begin{equation} \varTheta _R=\mathcal{M}\zeta -\frac {1}{2}\left [\frac {\pi }{2}+2\;\tan ^{-1}\left (\sinh (\varDelta \zeta )\right )\right ]+\omega \tau \end{equation}

and

(5.56) \begin{equation} \varTheta _L=-\mathcal{M}\zeta -\frac {\pi }{4}-\mid \omega \mid \tau . \end{equation}

The corresponding source functions for the RCP and LCP nonlinear AW are respectively

(5.57) \begin{equation} S_B^R=-\mathcal{M}\varDelta \;{\mbox{sech$\,$}}(\varDelta \zeta )\quad \mbox{and}\quad S_B^L=\mathcal{M}\varDelta \;{\mbox{sech$\,$}}(\varDelta \zeta ). \end{equation}

The solution (5.54) corresponding to the source function $S_B$  (5.57) is valid for both the low ( $\beta \lt 1$ ) and high ( $\beta \gt 1$ ) $\beta$ plasmas with proper algebraic sign of $\varDelta$ according to the condition (5.45). The above results clearly show that the soliton solutions exists only for $\ell \omega \gt \mathcal{M}^2/2$ (condition (i) of both (5.29) and (5.44)) . Also, the solutions derived in this subsection are valid for the transcritical ( $\mathcal{M}\approx 0$ , but $\neq 0$ ) range of the source velocity.

In the next subsection, we consider the physical situation where the source velocity is resonant i.e. $v_d=1$ .

5.2.3. Exact solution for exact resonance: $\mathcal{M}=0$ ( $v_d=1$ )

At this resonance ( $\mathcal{M}=0$ ), (5.25) is solvable only if $l=2$ and $j=1/2$ providing the following source function and solution:

(5.58) \begin{equation} S(\xi )=f_0\;{\mbox{sech$\,$}}^2(\varDelta \xi )\quad \mbox{and}\quad \psi (\xi )=\psi _0\;{\mbox{sech$\,$}}^{\frac {1}{2}}(\varDelta \xi ). \end{equation}

The width ( $\varDelta$ ) and amplitude ( $\psi _0$ ) are then determined as

(5.59) \begin{equation} \varDelta ^2=8\ell \omega \quad \mbox{and}\quad \psi _0^4=\frac {8\ell (f_0+3\omega )}{3\alpha ^2}. \end{equation}

The solution (5.58) then becomes

(5.60) \begin{equation} \psi (\xi )=\pm \left (\frac {8\ell (f_0+3\omega )}{3\alpha ^2}\right )^{\frac {1}{4}}\;{\mbox{sech$\,$}}^{\frac {1}{2}}(2\sqrt {2\ell \omega }\;\xi ). \end{equation}

The wave number $\kappa$ is determined from (5.19) by putting $M=0$ as

(5.61) \begin{equation} \kappa =-\ell \sqrt {\frac {2\ell (f_0+3\omega )}{3}}\;{\mbox{sech$\,$}}(2\sqrt {2\ell \omega }\;\xi ). \end{equation}

Accordingly, we get

(5.62) \begin{equation} \theta =\int _0^{\xi }\kappa (s){\textrm{d}}s=-\frac {\ell }{2}\sqrt {\frac {f_0+3\omega }{3\omega }}\;\tan ^{-1}\left (\sinh (2\sqrt {2\ell \omega }\;\xi )\right ). \end{equation}

A: solution of RCP Alfvén wave

In this case ( $\ell =+1$ ), $\omega \gt 0$ and for the existence of a localised solution, we must have $\psi _0^2\gt 0$ as both $\varDelta$ and $\psi _0$ are real quantities. Equation (5.60) then provides the following solution of the RCP wave for $f_0\gt 0$ :

(5.63) \begin{equation} u_R(\xi )=\pm \left (\frac {8(f_0+3\omega )}{3\alpha ^2}\right )^{\frac {1}{4}}\;{\mbox{sech$\,$}}^{\frac {1}{2}}(2\sqrt {2\omega }\;\xi )\exp (i\varphi _R). \end{equation}

The phase of the wave field ( $\varphi _R$ ) is given by

(5.64) \begin{equation} \varphi _R=-\frac {1}{2}\sqrt {\frac {f_0+3\omega }{3\omega }}\;\tan ^{-1}\left (\sinh (2\sqrt {2\omega }\;\xi )\right )+\omega \tau . \end{equation}

However, in the case of $f_0\lt 0$ , the above solution is also valid provided $\omega \gt {f_0}/{3}$ holds and the solution will be

(5.65) \begin{equation} u_R(\xi )=\pm \left (\frac {8(3\omega -f_0)}{3\alpha ^2}\right )^{\frac {1}{4}}\;{\mbox{sech$\,$}}^{\frac {1}{2}}(2\sqrt {2\omega }\;\xi )\exp (i\varphi _R), \end{equation}

with the phase

(5.66) \begin{equation} \varphi _R=-\frac {1}{2}\sqrt {\frac {3\omega -f_0}{3\omega }}\;\tan ^{-1}\left (\sinh (2\sqrt {2\omega }\;\xi )\right )+\omega \tau . \end{equation}

B: solution of LCP Alfvén wave

In this case ( $\ell =-1$ ), (5.59) clearly shows that for the existence of localised solution, we must have:

1. (i) $\omega \lt 0$ as $\varDelta$ is real;

2. (ii) $\mid \omega \mid \gt {f_0}/{3}\Rightarrow \psi _0^4= {8(3\mid \omega \mid -f_0)}/{3\alpha ^2}$ provided $f_0\gt 0$ as $\psi _0$ is real.

Thus, in this case, the solution (5.60) can be written as

(5.67) \begin{equation} u_L(\xi )=\pm \left (\frac {8(3\mid \omega \mid -f_0)}{3\alpha ^2}\right )^{\frac {1}{4}}\;{\mbox{sech$\,$}}^{\frac {1}{2}}(2\sqrt {2\mid \omega \mid }\;\xi )\exp (-i\varphi _L). \end{equation}

The phase of the wave field ( $\varphi _L$ ) is given by

(5.68) \begin{equation} \varphi _L=-\frac {1}{2}\sqrt {\frac {3\mid \omega \mid -f_0}{3\mid \omega \mid }}\;\tan ^{-1}\left (\sinh (2\sqrt {2\mid \omega \mid }\;\xi )\right )+\mid \omega \mid \tau . \end{equation}

However, in case of $f_0\lt 0$ , the above solution is valid for all values of $\mid \omega \mid$ and the solution will be

(5.69) \begin{equation} u_L(\xi )=\pm \left (\frac {8(3\mid \omega \mid +f_0)}{3\alpha ^2}\right )^{\frac {1}{4}}\;{\mbox{sech$\,$}}^{\frac {1}{2}}(2\sqrt {2\mid \omega \mid }\;\xi )\exp (-i\varphi _L), \end{equation}

with the phase ( $\varphi _L$ ) given by

(5.70) \begin{equation} \varphi _L=-\frac {1}{2}\sqrt {\frac {3\mid \omega \mid +f_0}{3\mid \omega \mid }}\;\tan ^{-1}\left (\sinh (2\sqrt {2\mid \omega \mid }\;\xi )\right )+\mid \omega \mid \tau . \end{equation}

Finally, the solution (5.17) of the derived fDNLSE (4.16) for RCP nonlinear AW for $f_0\gtrless 0$ is given by

(5.71) \begin{equation} \phi _R(\xi , \tau )=\pm \left (\frac {8(3\omega \pm f_0)}{3\alpha ^2}\right )^{\frac {1}{4}}\;{\mbox{sech$\,$}}^{\frac {1}{2}}(2\sqrt {2\omega }\;\xi )\exp (i\varTheta _R). \end{equation}

The phase ( $\varTheta$ ) (5.18) of the RCP nonlinear AW field is given by

(5.72) \begin{equation} \varTheta _R=-\frac {1}{2}\sqrt {\frac {3\omega \pm f_0}{3\omega }}\left [\frac {\pi }{2}+2\;\tan ^{-1}\left (\sinh (2\sqrt {2\omega }\;\xi )\right )\right ]+\omega \tau . \end{equation}

Similarly, the solution (5.17) of the derived fDNLSE (4.16) for LCP nonlinear AWs for $f_0\gtrless 0$ is given by

(5.73) \begin{equation} \phi _L(\xi , \tau )=\pm \left (\frac {8(3\mid \omega \mid \mp f_0)}{3\alpha ^2}\right )^{\frac {1}{4}}\;{\mbox{sech$\,$}}^{\frac {1}{2}}(2\sqrt {2\mid \omega \mid }\;\xi )\exp (i\varTheta _L). \end{equation}

The phase ( $\varTheta$ ) (5.18) of the LCP nonlinear AW field is given by

(5.74) \begin{equation} \varTheta _L=-\frac {\pi }{4}\sqrt {\frac {3\mid \omega \mid \mp f_0}{3\mid \omega \mid }}-\mid \omega \mid \tau . \end{equation}

6. Stability of the solutions

In this section, we shall make some general statements about the stability of the derived solitary wave solutions using variational approach (Derrick Reference Derrick1964; Benjamin Reference Benjamin1972; Grillakis et al. Reference Grillakis, Shatah and Strauss1987, Reference Grillakis, Shatah and Strauss1990; Colin & Ohta Reference Colin and Ohta2006; Khare, Cooper & Dawson Reference Khare, Cooper and Dawson2018).

6.1. Stability of the solutions for nonresonant velocity

The exact solutions $u(\zeta )$ derived in the earlier section depend on the two Lagrangian parameters $\omega$ and $\mathcal{M}$ . In terms of these parameters, the conserved quantities $\mathcal{N}$ , $\mathcal{P}$ and $\mathcal{H}$ for (RCP and LCP) AW can be written as

(6.1) \begin{align} \mathcal{N} & =\int _{-\infty }^{\infty }\mid u(\zeta )\mid ^2{\textrm{d}}\zeta ,\;\;\;\;\mathcal{P}=\frac {i}{2}\int _{-\infty }^{\infty }\left [u^*(\zeta )u^{\prime }(\zeta )-u(\zeta )u^{*'}(\zeta )\right ]{\textrm{d}}\zeta , \end{align}

(6.2) \begin{align} \mathcal{H} & =\int _{-\infty }^{\infty }\left [\ell \mid u^{\prime }(\zeta )\mid ^2-\frac {i\alpha }{4}\{u^*(\zeta ) u^{\prime }(\zeta )-u(\zeta )u^{*\prime }(\zeta )\}\mid u(\zeta )\mid ^2+S(\zeta )\mid u(\zeta )\mid ^2\right ]{\textrm{d}}\zeta . \end{align}

Here, the prime notation represents the derivative with respect to $\zeta$ . Following Colin & Ohta (Reference Colin and Ohta2006) and Khare et al. (Reference Khare, Cooper and Dawson2018), we construct the following function:

(6.3) \begin{equation} d\left (\omega , M\right )=\mathcal{H}(u_{\omega , \mathcal{M}})+\omega \mathcal{N}(u_{\omega , \mathcal{M}})+\mathcal{M}\mathcal{P}(u_{\omega , \mathcal{M}}), \end{equation}

where $u_{\omega , \mathcal{M}}$ is the exact solitary wave solution. According to the theory (Colin & Ohta Reference Colin and Ohta2006; Khare et al. Reference Khare, Cooper and Dawson2018), the orbital stability (instability) of the solitary wave solutions depend on the negative (positive) sign of the determinant,

(6.4) \begin{equation} \mathcal{D}(\omega ,\mathcal{M} )=\text{det}[d^{\prime \prime }(\omega , \mathcal{M})]=\left | \begin{array}{ccccc} \dfrac {\partial \mathcal{P}}{\partial \mathcal{M}} && \dfrac {\partial \mathcal{P}}{\partial \omega }\\[10pt] \dfrac {\partial \mathcal{N}}{\partial \mathcal{M}} && \dfrac {\partial \mathcal{N}}{\partial \omega } \\ \end{array} \right | \lessgtr 0, \end{equation}

where $\mathcal{P}$ and $\mathcal{N}$ are evaluated for the $u_{\omega , \mathcal{M}}$ stationary solutions. The result for $\mathcal{D}(\omega , \mathcal{M})=0$ is inconclusive. The values of $\mathcal{N}$ (for LCP (RCP) wave) for the source functions $S_A$ and $S_B$ are respectively,

(6.5) \begin{align}\mathcal{N}_A &=\int _{-\infty }^{\infty }\mid u_{L(R)}(\zeta )\mid ^2\,{\textrm{d}}\zeta =\psi _0^2\int _{-\infty }^{\infty }{\mbox{sech}}^2(\varDelta \zeta )\,{\textrm{d}}\zeta =\mp \frac {2\;\sqrt {2\ell \omega -\mathcal{M}^2}}{\alpha \mathcal{M}},\end{align}

(6.6) \begin{align}\mathcal{N}_B &=\int _{-\infty }^{\infty }\mid u_{L(R)}(\zeta )\mid ^2\,{\textrm{d}}\zeta =\psi _0^2\int _{-\infty }^{\infty }{\mbox{sech}}(\varDelta \zeta )\,{\textrm{d}}\zeta =\frac {\pi }{\alpha }.\end{align}

Similarly, the values of $\mathcal{P}$ for $S_A$ and $S_B$ are respectively,

(6.7) \begin{align} \mathcal{P}_A &=-\ell \psi _0^2\int _{-\infty }^{\infty }\left [\mathcal{M}+\left (\frac {\varDelta ^2}{2\mathcal{M}}\right )\;{\mbox{sech}}^2(\varDelta \zeta )\right ]{\mbox{sech}}^2(\varDelta \zeta )\,{\textrm{d}}\zeta \nonumber\\ &\quad =\pm \frac {4\ell (\ell \omega +M^2)\sqrt {2\ell \omega -\mathcal{M}^2}}{3\alpha \mathcal{M}^2}, \end{align}

(6.8) \begin{align} \mathcal{P}_B &=-\ell \psi _0^2\int _{-\infty }^{\infty }\left [\mathcal{M}+\left (\frac {\varDelta }{2}\right ){\mbox{sech}}(\varDelta \zeta )\right ] {\mbox{sech}}(\varDelta \zeta )\,{\textrm{d}}\zeta =-\frac {\ell }{\alpha }\left [\mathcal{M}\pi \pm 2\sqrt {2\ell \omega -\mathcal{M}^2}\right ]. \end{align}

The determinant $\mathcal{D}(\omega , \mathcal{M})$ for the solutions corresponding to $S_A$ is given by

(6.9) \begin{equation} \mathcal{D}_A(\omega , \mathcal{M})=\left | \begin{array}{ccccc} \mp \frac {4\ell (m^4+4\omega ^2-\ell \omega \mathcal{M}^2)}{3\alpha \mathcal{M}^3\sqrt {2\ell \omega -\mathcal{M}^2}} && \pm \frac {4\ell \omega }{\alpha \mathcal{M}^2\sqrt {2\ell \omega -\mathcal{M}^2}}\\ \pm \frac {4\ell \omega }{\alpha \mathcal{M}^2\sqrt {2\ell \omega -\mathcal{M}^2}} && \mp \frac {2\ell }{\alpha \mathcal{M}\sqrt {2\ell \omega -\mathcal{M}^2}} \\ \end{array} \right |=-\frac {8(\mathcal{M}^2+\ell \omega )}{3\alpha ^2\mathcal{M}^4},\;(\ell ^2=1). \end{equation}

This clearly shows that the stability of the derived solitary wave solutions for both the RCP ( $\ell =+1$ ) and LCP ( $\ell =-1$ ) nonlinear compressive AWs depends on the values of $\omega$ in the parameter space $(\omega , \mathcal{M})$ . Also, for both the RCP and LCP AW, $\mathcal{D}_A(\omega , \mathcal{M})\lt 0$ for all real values of $\mathcal{M}$ and $\omega$ as $\ell \omega$ is always positive ( $\ell \omega \gt 0$ ). Moreover, we have already seen that for the existence of solitary wave solution, one must have $\ell \omega \gt \mathcal{M}^2/2$ (see (5.29) and (5.44)). Thus, the solitary wave solutions are orbitally stable for all real values of $\omega$ and $\mathcal{M}$ for which $\ell \omega \gt \mathcal{M}^2/2$ corresponding to the source function $S_A(\zeta )$ . However, for the source function $S_B(\zeta )$ , the result is inconclusive as $\mathcal{N}_B$ is independent of both $\omega$ and $\mathcal{M}$ (see (6.6)) implying $\mathcal{D}_B(\omega , \mathcal{M})=0$ .

6.2. Stability of the solutions at resonant velocity

Interestingly, the derived solitary wave solutions at resonance ( $\mathcal{M}=0$ ) can be treated as the static solutions at $\tau =0$ . Also, (5.4) is a Hamiltonian dynamical system and therefore we can apply Derrick’s theorem (Derrick Reference Derrick1964) to study the stability of the derived exact solutions to the particular scale transformation for the static situation (Khare et al. Reference Khare, Cooper and Dawson2018). The cornerstone of the theorem is to optimise the functional $\mathcal{H}$ as a function of a parameter $\mu$ for the scale transformation, $\xi \to \mu \xi$ , provided $\mathcal{N}$ (mass) is fixed. Practically, $\mathcal{H}$ as a function of $\mu$ has to be an extremum for the exact solution. Then, an exact solution is unstable if $\mathcal{H}$ decreases with $\mu$ . To apply this theorem, we put

(6.10) \begin{equation} u_{\mu }(\xi )=\sqrt {\mu }\;u(\mu \xi ), \end{equation}

where the static ( $\tau =0$ ) solution $u(\mu \xi )$ for both the RCP and LCP waves can be rewritten (with the help of (5.60), (5.61) and (5.62)) as

(6.11) \begin{align} u_{\mu }(\xi ) & =\pm \sqrt {\mu }\;\eta _{\textit{am}}\;{\mbox{sech}}^{\frac {1}{2}}(\mu \varDelta \xi )\exp [i\varphi (\mu \xi )],\quad \eta _{\textit{am}}=\left (\frac {\ell (3\ell \varDelta ^2+8f_0)}{3\alpha ^2}\right )^{\frac {1}{4}}, \end{align}

(6.12) \begin{align} \mbox{where}\quad \varphi (\mu \xi ) &=-\eta _{\textit{ph}}\;\tan ^{-1}\left (\sinh (\mu \varDelta \xi )\right ),\quad \eta _{\textit{ph}}=\frac {\ell }{2}\sqrt {\frac {3\ell \varDelta ^2+8f_0}{3\ell \varDelta ^2}}. \end{align}

The above scaling (6.10) conserves the mass ( $\mathcal{N}$ ),

(6.13) \begin{equation} \mathcal{N}=\int _{-\infty }^{\infty }\mid u(\xi )\mid ^2 {\textrm{d}}\xi =\frac {\pi \eta _{\textit{am}}^2}{\varDelta }. \end{equation}

The Hamiltonian ( $\mathcal{H}$ ) (5.6) can be expressed as

(6.14) \begin{equation} \mathcal{H}(\mu )=\frac {\ell }{2}\mathcal{H}_1(\mu )-\frac {i\alpha }{4}\mathcal{H}_2(\mu )+\mathcal{H}_3(\mu ), \end{equation}

where

(6.15) \begin{align} \mathcal{H}_1(\mu ) & =\int _{-\infty }^{\infty }\bigg |\frac {{\textrm{d}}u_{\mu }(\xi )}{{\textrm{d}}\xi }\bigg |^2 \,{\textrm{d}}\xi =\mu ^2\int _{-\infty }^{\infty }\bigg |\frac {{\textrm{d}}u(\xi )}{{\textrm{d}}\xi }\bigg |^2\, {\textrm{d}}\xi , \end{align}

(6.16) \begin{align} \mathcal{H}_2(\mu ) & =\int _{-\infty }^{\infty }\left [u_{\mu }^*(\xi )\frac {{\textrm{d}}u_{\mu }(\xi )}{{\textrm{d}}\xi }-\frac {{\textrm{d}}u_{\mu }^*(\xi )}{{\textrm{d}}\xi }u_{\mu }(\xi )\right ]\mid u_{\mu }(\xi )\mid ^2 \,{\textrm{d}}\xi \nonumber \\ & =\mu ^2\int _{-\infty }^{\infty }\left [u^*(\xi )\frac {{\textrm{d}}u(\xi )}{{\textrm{d}}\xi }-\frac {{\textrm{d}}u^*(\xi )}{{\textrm{d}}\xi }u(\xi )\right ]\mid u(\xi )\mid ^2 \,{\textrm{d}}\xi , \end{align}

(6.17) \begin{align} \mathcal{H}_3(\mu ) &=\int _{-\infty }^{\infty }S(\xi /\mu )\mid u(\xi )\mid ^2\,{\textrm{d}}\xi . \end{align}

The static solution provides

(6.18) \begin{eqnarray} && \mathcal{H}_1(\mu ) =\frac {\mathcal{N}\varDelta ^2}{8}\left (2+\frac {8f_0}{3\ell \varDelta ^2}\right ),\quad \mathcal{H}_2(\mu )=-i\varDelta \eta _{\textit{am}}^2\eta _{\textit{ph}}\mathcal{N},\nonumber \\ && \mathcal{H}_3(\mu ) =\frac {f_0\mathcal{N}}{\pi }\int _{-\infty }^{\infty }{\mbox{sech}}^2(\xi /\mu ){\mbox{sech}}(\xi )\,{\textrm{d}}\xi . \end{eqnarray}

Finally, with the help of (6.14), we get

(6.19) \begin{equation} \mathcal{H}_{\textit{new}}=\frac {\mathcal{H}(\mu )}{\mathcal{N}}=\left [\frac {\ell \varDelta ^2}{16}\left (2+\frac {8f_0}{3\ell \varDelta ^2}\right )\!-\!\frac {\alpha \varDelta \eta _{\textit{am}}^2\eta _{\textit{ph}}}{4}\right ]\mu ^2 \!+\!\frac {f_0}{\pi }\int _{-\infty }^{\infty }\!\!{\mbox{sech}}^2(\xi /\mu ){\mbox{sech}}(\xi )\,{\textrm{d}}\xi , \end{equation}

which provides

(6.20) \begin{eqnarray} \frac {{\textrm{d}}\mathcal{H}_{\textit{new}}}{{\textrm{d}}\mu }=\frac {\mu }{2}\left [\frac {\ell \varDelta ^2}{4}\left (2+\frac {8f_0}{3\ell \varDelta ^2}\right )-\alpha \varDelta \eta _{\textit{am}}^2\eta _{\textit{ph}}\right ]+\frac {f_0}{3\mu ^2},\nonumber \\ \frac {{\textrm{d}}^2\mathcal{H}_{\textit{new}}}{{\textrm{d}}\mu ^2}=\frac {1}{2}\left [\frac {\ell \varDelta ^2}{4}\left (2+\frac {8f_0}{3\ell \varDelta ^2}\right )-\alpha \varDelta \eta _{\textit{am}}^2\eta _{\textit{ph}}\right ]-\frac {2f_0}{3\mu ^3}, \end{eqnarray}

where, with the help of (6.11) and (6.12),

(6.21) \begin{equation} \alpha \varDelta \eta _{\textit{am}}^2\eta _{\textit{ph}}=\frac {\ell (3\ell \varDelta ^2+8f_0)}{6}. \end{equation}

According to Derrick’s theorem, $\mu =1$ is a stationary point of the Hamiltonian $\mathcal{H}_{\textit{new}}$ . After a little bit algebra (using (6.20) and (6.21)), we find that indeed for RCP ( $\ell =+1$ ) AWs,

(6.22) \begin{equation} \frac {{\textrm{d}}\mathcal{H}_{\textit{new}}}{{\textrm{d}}\mu }\bigg |_{\mu =1}=0\quad \text{and}\quad \frac {{\textrm{d}}^2\mathcal{H}_{\textit{new}}}{{\textrm{d}}\mu ^2}\bigg |_{\mu =1}=-f_0, \end{equation}

while for LCP ( $\ell =-1$ ) AWs,

(6.23) \begin{equation} \frac {{\textrm{d}}\mathcal{H}_{\textit{new}}}{{\textrm{d}}\mu }\bigg |_{\mu =1}=0\quad \text{for}\quad\varDelta ^2=\frac {8f_0}{3}\quad \text{and with this value},\;\frac {{\textrm{d}}^2\mathcal{H}_{\textit{new}}}{{\textrm{d}}\mu ^2}\bigg |_{\mu =1}=-f_0. \end{equation}

Equation (6.22) clearly shows that the static solitary wave solution of RCP AWs is stable (unstable) for $f_0\lt 0$ ( $f_0\gt 0$ ). While (6.23) shows that the static solitary wave solution of LCP AWs is always unstable as $f_0\gt 0$ (essential for real $\varDelta$ ).

7. Numerical results

In this section, we numerically solve the derived fDNLSE (4.16) using the split-step Fourier method (Fornberg Reference Fornberg1998; Agrawal Reference Agrawal2000; Boyd Reference Boyd2001). The numerical code is written on MATLAB and also benchmarked by reproducing earlier results of DNLSE (Spangler, Sheerin & Payne Reference Spangler, Sheerin and Payne1985; Ponce Dawson & Ferro Fontán Reference Ponce Dawson and Ferro Fontán1988; Dawson & Fontán Reference Dawson and Fontán1988). In the computation, the time step length ${\textrm{d}}\tau =10^{-3}$ , the number of grid points $N=2^{16}$ and the space length $L=1600$ , so that the space step length ${\textrm{d}}\xi =2L/N\approx 0.05$ . The nonlinear RCP AWs are excited by the current source function $S_A(\zeta )$ and $S_B(\zeta )$ with the initial conditions:

1. (i) $\phi (\xi , 0)=({\varDelta }/{\sqrt {-\alpha \mathcal{M}}})\; {\mbox{sech}\,}(\varDelta \xi ) \exp [i (\mathcal{M}\xi +({\varDelta }/{2\mathcal{M}})\{1+2\;\tanh (\varDelta \xi )\} ) ];$

2. (ii) $\phi (\xi , 0)=(\sqrt {{\varDelta }/{\alpha }})\;{\mbox{sech$\,$}}^{ {1}/{2}}(\varDelta \xi ) \exp [i (\mathcal{M}\xi -({1}/{2})\{({\pi }/{2})+2\;\tan ^{-1} (\sinh (\varDelta \xi ) )\} ) ]$ .

The numerical results are shown graphically in figures 2 and 3 for $S_A(\zeta )$ and $S_B(\zeta )$ , respectively. The analytical results in the previous section reveal that the solitary wave solution corresponding to $S_A(\zeta )$ exists only for $\mathcal{M}\lt 0$ (subcritical and transcritical velocity ranges) in the low $\beta$ ( $\alpha \gt 0$ ) and $\mathcal{M}\gt 0$ (supercritical and transcritical velocity ranges) in the high $\beta$ ( $\alpha \lt 0$ ) regions (irrespective of the sign of $\varDelta$ ). However, the solitary wave solution corresponding to $S_B(\zeta )$ exists only for $\varDelta \gt 0$ in the low $\beta$ and $\varDelta \lt 0$ in the high $\beta$ regions irrespective of the sign of $\mathcal{M}$ i.e. for subcritical, transcritical and supercritical velocity ranges. The parameter values are chosen accordingly in the computation and the results are shown in figures 2 and 3.

Figure 2.

Pinned soliton: numerical solution of fDNLSE (4.16) for RCP the waves corresponding to $S_A(\zeta )$ . The solid (blue) and dotted (red) curves are the analytical and numerical solutions, respectively. The source term (black) is multiplied by $10^2$ (in panels a, b, d) and $10$ (in panel c) to make them visible.

Figure 3.

Pinned soliton: numerical solution of fDNLSE (4.16) for RCP the waves corresponding to $S_B(\zeta )$ . The solid (blue) and dotted (red) curves are the analytical and numerical solutions, respectively. The source term (black) is multiplied by $10$ in panels (a), (b) and (c) to make them visible.

Figure 2(a, b) and 2(c, d) represent the numerical solutions for $\beta =10^{-3},\, \mathcal{M}=-0.1,\;-1$ and $\beta =1.2,\, \mathcal{M}=0.1,\;1$ , respectively. The results demonstrate that at time $\tau =5$ , the numerical (red dotted) results well agree with the analytical (blue solid).

Figure 3(a, b, c) and 3(d, e, f) represent the numerical solutions for $\beta =10^{-3},\, \varDelta =8\times 10^{-2},\, \mathcal{M}=-1,\;0.1,\;1$ and $\beta =1.2,\, \varDelta =-8\times 10^{-2}, \, \mathcal{M}=-1.\;0.1,\;1$ , respectively. The results show qualitatively similar results as illustrated in figure 2. Both the numerical (red dotted) and analytical (blue solid) solutions are in good agreement at $\tau =3$ for super and sub critical velocity, and $\tau =5$ for transcritical velocity. All the numerical solutions demonstrate that the source and the soliton moves in the same direction resembling the situation of pinned soliton. The numerical solutions of the LCP AWs show qualitatively similar behaviour as that of the RCP AWs.

8. Possible applications in astrophysics

The nonlinear coherent structures consisting of large amplitude density fluctuations or magnetic field perturbations naturally exist in the ionosphere from nonlinear saturation of instabilities of various plasma modes (Galperin et al. Reference Galperin1985; Pokhotelov et al. Reference Pokhotelov, Pokhotelov, Gokhberg, Feygin, Stenflo and Shukla1996; Petviashvili & Pohkotelov Reference Petviashvili and Pohkotelov2016; Bhattacharyya Reference Bhattacharyya2022). Recently, the compressional AWs launched by the charged space obstacle are observed in LEO region at an altitude $h\sim 600\;\text{km}$ characterised by the typical length scales $\lambda _i\sim 2\;{\textrm{km}}$ , $\lambda _e\sim 10\; {\textrm{m}}$ and $\lambda _D\sim 1\;{\textrm{cm}}$ (Bernhardt et al. Reference Bernhardt, Scott, Howarth and Morales2023). The estimated value of $\lambda _e/\lambda _i\sim 5\times 10^{-3}$ justifies the neglect of the term $O(\lambda _e/\lambda _i)^2$ in the derivation of the Hall-MHD one-fluid model in the present work. The charged obstacle (dimensional) speed $\tilde {v_d}=(GM_E/(h+R_E))^{1/2}$ (where $G$ is the gravitational constant and $M_E(R_E)$ is the mass (radius) of the earth) so that $\tilde {v_d}\sim 8\;\text{km}\;{\textrm{s}}^{-1}$ . In this region, the $O^{+}$ are the dominant ions so that $m_i\sim 16m_p$ , and the other parameters are $n_0\sim 10^6\;\mbox{cm}^{-3}$ , $T_i\sim 600\;\text{K}$ , $T_e\sim 2200\;\text{K}$ and $B_0\sim (0.3{-}0.5)\mbox{G}$ (Gurevich Reference Gurevich1976; Kelley Reference Kelley2009). Then, the Alfvén velocity $v_A\sim (160{-}320)\;\text{km}\; {\textrm{s}}^{-1}$ . These values estimate $v_d=\tilde {v_d}/v_A\sim (0.025{-}0.05)$ and $\mathcal{M}=v_d-1\sim -(0.95{-}0.98)$ and thereby the obstacle velocity is in subcritical range. Also, the plasma $\beta$ in this region is of the order of $10^{-4}$ and hence, has a negligible effect on the dynamics. Thus, the lower LEO region can be characterised as the low- $\beta$ region where the obstacle velocity is in the subcritical range. Accordingly, the results exhibiting pinned solitons in the subcritical range for low $\beta$ may be useful for tracking the debris (charged obstacle) objects in the lower LEO region as they travel with the same speed as the object and remain pinned to the obstacle.