Theory of Nonlinear Waves in Plasma with a Constant Magnetic Field

In the paper, for the first time, the analytic solution of the collisionless time-dependent Vlasov-Boltzmann kinetic equation in the presence of a constant magnetic field. By means of this new solution, the general electrodynamic equation, describing the behavior of waves, also in the nonlinear stage, is considered. The analysis is restricted to the consideration of the first nonlinear approximation, keeping the second power of the electric strength. As it is known, the plasma with a magnetic field is characterized by the abundance of different wave branches, describing which demands rather long and tedious calculations. We restrict ourselves to the consideration of perturbations propagating along the constant magnetic field. Obviously, any modification of the distribution function can lead to the essential change of wave-like behavior of plasma and the whole plasma electrodynamics. It is shown that in the nonlinear stage, the longitudinal and transversal waves cannot propagate independently – they are coupled with each other.

A large number of papers and textbooks are devoted to the theory of the waves in the magnetized plasma [1-3]. Possible branches of oscillations in the magnetized plasma allows to consider different approaches and interpretations for waves [4,5]. In the predominant part of published papers, the solution of the linearized Vlasov-Boltzmann kinetic equation is used.

After finding the zero-order solution of the main equation, the authors construct approximations for any higher order. In most parts of books on the kinetic theory of gases, also in original papers, one cannot observe attempts to immediately determine the solution of the Vlasov-Boltzmann kinetic equation [6,7]. In papers [8-10] a general solution of the Vlasov-Boltzmann kinetic equation is presented, but this solution is connected with the infinite chain of approximations, which should be cut off.

In the present paper, for the first time, the exact analytic solution of the time-dependent Vlasov-Boltzmann kinetic equation for plasma with a constant magnetic field. It is known that all wave-like behaviors (also the electromagnetic characteristics) of plasma are determined by the distribution function (DF) of charged particles. In the presence of the magnetic field, a high number of possibilities exists for the investigation of different branches of waves.

Using the time-dependent DF in the present paper, only waves propagating along the constant magnetic field as a first step are considered. The description of waves with other directions we plan for the future, as they are connected with tedious and long calculations. The nonlinearity of waves, expressed in the form of the second power of the electric field, is taken into account. It should be emphasized that the nonlinearity of waves in a magnetoactive plasma is rarely discussed in the literature [1,4,5].

We start from the collisionless Vlasov-Boltzmann equation for particles of α-type,

$\frac{\partial f_{\alpha }}{\partial t}+\overrightarrow{v}\frac{\partial f_{\alpha }}{\partial \overrightarrow{r}}+\frac{Z_{\alpha }e}{m_{\alpha }}\left\{\overrightarrow{E}\left(\overrightarrow{r},t\right)+\frac{1}{c}\left[\overrightarrow{v}\times \left(\overrightarrow{B}\left(\overrightarrow{r},t\right)+{\overrightarrow{B}}_0\right)\right]\right\}\frac{\partial f_{\alpha }}{\partial \overrightarrow{v}}=0$ , (1)

where ${\overrightarrow{B}}_0$ is the constant magnetic field, Zα is the number of elementary charges e possessed by the particle (for instance, for electrons Ze =-1). Using notions of vector $\overrightarrow{A}\left(\overrightarrow{r},t\right),{\overrightarrow{A}}_0\left(\overrightarrow{r}\right)$ and scalar $\varphi \left(\overrightarrow{r},t\right)$ potentials for the sum of fields in the Lorentz force, we obtain

$\overrightarrow{E}\left(\overrightarrow{r},t\right)+\frac{1}{c}\left[\overrightarrow{v}\times \left(\overrightarrow{B}\left(\overrightarrow{r},t\right)+{\overrightarrow{B}}_0\right)\right]=-\frac{1}{c}\frac{\partial }{\partial t}\overrightarrow{A}-\overrightarrow{\nabla }\varphi +\frac{1}{c}\left[\overrightarrow{v}\times \left[\overrightarrow{\nabla }\times \left(\overrightarrow{A}+{\overrightarrow{A}}_0\right)\right]\right]$ (2)

Using the relation, one can give the relation (2) the form [11]:

$\overrightarrow{E}\left(\overrightarrow{r},t\right)+\frac{1}{c}\left[\overrightarrow{v}\times \left(\overrightarrow{B}\left(\overrightarrow{r},t\right)+{\overrightarrow{B}}_0\right)\right]=-\frac{d}{dt}\left\{{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\left\{\frac{1}{c}\frac{\partial }{\partial t^{\prime }}\overrightarrow{A}\left(\overrightarrow{r},t^{\prime }\right)+\overrightarrow{\nabla }\varphi \left(\overrightarrow{r},t^{\prime }\right)\right\}}+\frac{1}{c}{\overrightarrow{A}}_0\left(\overrightarrow{r}\right)\right\}$ , $+v_i\cdot \overrightarrow{\nabla }\cdot \left\{{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\left\{\frac{1}{c}\frac{\partial }{\partial t^{\prime }}{A}_i\left(\overrightarrow{r},t^{\prime }\right)+{\nabla }_i\varphi \left(\overrightarrow{r},t^{\prime }\right)\right\}}+\frac{1}{c}{A}_{0_i}\left(\overrightarrow{r}\right)\right\}$ (4)

Lower limits of integrals will be determined by assumption that the electric field is switched on at the initial moment t₀:

${\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t\right)=\frac{Z_{\alpha }e}{m_{\alpha }}\left\{{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\cdot \overrightarrow{E}\left(\overrightarrow{r},t^{\prime }\right)-\frac{1}{c}}{\overrightarrow{A}}_0\left(\overrightarrow{r}\right)\right\}$

where $\overrightarrow{E}\left(\overrightarrow{r},t\right)=-\frac{1}{c}\frac{\partial }{\partial t}\overrightarrow{A}\left(\overrightarrow{r},t\right)-\overrightarrow{\nabla }\varphi \left(\overrightarrow{r},t\right)$ . The kinetic equation (1) acquires the form

$\frac{\partial f_{\alpha }}{\partial t}+\overrightarrow{v}\frac{\partial f_{\alpha }}{\partial \overrightarrow{r}}+\frac{d}{dt}{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t\right)\cdot \frac{\partial f_{\alpha }}{\partial \overrightarrow{v}}-\left(v_i\cdot \overrightarrow{\nabla }\cdot {\epsilon }_{\alpha }{}_i\left(\overrightarrow{r},t\right)\right)\cdot \frac{\partial f_{\alpha }}{\partial \overrightarrow{v}}=0$ . (5)

Afterwards, it is convenient to introduce a new velocity variable, $\overrightarrow{w}=\overrightarrow{v}-{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t\right)$ , ( $\overrightarrow{v}=\overrightarrow{w}+{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t\right)$ ), that implies the change of derivatives in the equation (5),

$\frac{\partial f_{\alpha }\left(t,\overrightarrow{r},\overrightarrow{v}\right)}{\partial t}\to \frac{\partial f_{\alpha }\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial t}-\frac{\partial }{\partial t}{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t\right)\cdot \frac{\partial f_{\alpha }\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}$ , (6)

$\frac{\partial f_{\alpha }\left(t,\overrightarrow{r},\overrightarrow{v}\right)}{\partial r_i}\to \frac{\partial f_{\alpha }\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial r_i}-\frac{\partial }{\partial r_i}{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t\right)\cdot \frac{\partial f_{\alpha }\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}$ , (7)

which results in the compact form of the kinetic equation (5)

$\frac{\partial f_{\alpha }}{\partial t}+\left[H\cdot f_{\alpha }\right]=0$ , (8)

where $\left[H\cdot f_{\alpha }\right]=\frac{\partial H\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}\cdot \frac{\partial f_{\alpha }}{\partial \overrightarrow{r}}-\frac{\partial H\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}\cdot \frac{\partial f_{\alpha }}{\partial \overrightarrow{w}}$ (9)

Is the Poisson’s brackets defined by means of the kinetic energy,

$H\left(t,\overrightarrow{r},\overrightarrow{w}\right)=\frac{1}{2}{\left\{\overrightarrow{w}+{\overrightarrow{\epsilon }}_{\alpha }\left(t,\overrightarrow{r}\right)\right\}}^2\equiv H\left(t\right)$ . (10)

In the following, for shortening, we sometimes use the notation H(t). The characteristic equations for Eq. (8) read

$\frac{d\overrightarrow{w}}{dt}=-\frac{\partial H\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}$ , (11)

$\frac{d\overrightarrow{r}}{dt}=\frac{\partial H\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}$ . (12)

Further calculations are carried out for the characteristic equation only for $\overrightarrow{w}$ . Results for the characteristic equation for the coordinate $\overrightarrow{r}$ (see equation (12)) can be obtained by a simple change of notations. From equation (11) we obtain

$\frac{d\overrightarrow{w}}{dt}=-\frac{\partial }{\partial t}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime },\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}}=-\frac{d}{dt}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime },\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}}+\left(\frac{d\overline{r}}{dt}\cdot \frac{\partial }{\partial \overrightarrow{r}}\right){\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime },\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}}+\left(\frac{d\overrightarrow{w}}{dt}\cdot \frac{\partial }{\partial \overrightarrow{w}}\right){\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime },\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}}=-\frac{d}{dt}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime }\right)}{\partial \overrightarrow{r}}}+\left(\frac{dH\left(t\right)}{d\overrightarrow{w}}\cdot \frac{\partial }{\partial \overrightarrow{r}}\right){\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime }\right)}{\partial \overrightarrow{r}}}-\left(\frac{dH\left(t\right)}{d\overrightarrow{r}}\cdot \frac{\partial }{\partial \overrightarrow{w}}\right){\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime }\right)}{\partial \overrightarrow{r}}}$ . (13)

Introducing the operator $\left\{\widehat{t}\right\}=\frac{d\overrightarrow{r}}{dt}\cdot \frac{\partial }{\partial \overrightarrow{r}}+\frac{d\overrightarrow{w}}{dt}\cdot \frac{\partial }{\partial \overrightarrow{r}}=\frac{\partial H\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}\cdot \frac{\partial }{\partial \overrightarrow{r}}-\frac{\partial H\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}\cdot \frac{\partial }{\partial \overrightarrow{r}}$ (14)

and continue the transformation, made for obtaining Eq. (13), we find

$\frac{d\overrightarrow{w}}{dt}=-\frac{d}{dt}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime }\right)}{\partial \overrightarrow{r}}}+\left\{\widehat{t}\right\}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime }\right)}{\partial \overrightarrow{r}}}=-\frac{d}{dt}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime }\right)}{\partial \overrightarrow{r}}}+\frac{d}{dt}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\cdot }\left\{{\widehat{t}}^{\prime }\right\}{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\frac{\partial H\left(t^{″}\right)}{\partial \overrightarrow{r}}-\left\{\widehat{t}\right\}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\cdot \left\{{\widehat{t}}^{\prime }\right\}{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\frac{\partial H\left(t^{″}\right)}{\partial \overrightarrow{r}}}}}$ (15)

Last term in the right-hand (rh) side of Eq.(15), by analogy to preceding terms, can be also transformed to the form with the full time derivative. The similar procedure can be realized for all subsequent terms. In consequence we obtain the expression for the first constant integral for Eq.(1):

$\overrightarrow{U}=\overrightarrow{w}+{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial H\left(t^{\prime }\right)}{\partial \overrightarrow{r}}}-{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\cdot \left\{{\widehat{t}}^{\prime }\right\}{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\frac{\partial H\left(t^{″}\right)}{\partial \overrightarrow{r}}+}}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\cdot \left\{{\widehat{t}}^{\prime }\right\}{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\left\{{\widehat{t}}^{″}\right\}{\displaystyle \underset{}{\overset{t^{″}}{\int }}d{t}^{‴}}\frac{\partial H\left(t^{‴}\right)}{\partial \overrightarrow{r}}-}}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\cdot \left\{{\widehat{t}}^{\prime }\right\}{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\left\{{\widehat{t}}^{″}\right\}{\displaystyle \underset{}{\overset{t^{″}}{\int }}d{t}^{‴}}\left\{{\widehat{t}}^{‴}\right\}{\displaystyle \underset{}{\overset{t^{‴}}{\int }}d{t}^{IV}}\frac{\partial H\left(t^{IV}\right)}{\partial \overrightarrow{r}}+}}$ (16)

After using the transformation of integrals,

${\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\mathrm{...}=}}{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{″}{\displaystyle \underset{t^{″}}{\overset{t}{\int }}d{t}^{\prime }\mathrm{....}}}$ ,(17)

the expression (16) acquires the form

$\overrightarrow{U}=\overrightarrow{w}+{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\left\{1-{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\cdot \left\{t^{″}\right\}+\frac{1}{2}{\left\{{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}}\cdot \left\{t^{″}\right\}\right\}}^2-\frac{1}{6}}{\left\{{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}}\cdot \left\{t^{″}\right\}\right\}}^3+\mathrm{....}\right\}}\cdot \frac{\partial H\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}$ (18)

For obtaining (18) and (19) see Appendix A Expression for the second constant of integration $\overrightarrow{R}$ can be obtained by means of a change of variables in equations (12), (13), and (18):

$\overrightarrow{R}=\overrightarrow{r}-{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\left\{1-{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\cdot \left\{t^{″}\right\}+\frac{1}{2}{\left\{{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}}\cdot \left\{t^{″}\right\}\right\}}^2-\frac{1}{6}}{\left\{{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}}\cdot \left\{t^{″}\right\}\right\}}^3+\mathrm{....}\right\}}\cdot \frac{\partial H\left(t,\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}$ . (19)

Expressions (18) and (19) represent expansions of the following relations

$\overrightarrow{U}=\overrightarrow{w}+{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}H\left\{t^{\prime },\overrightarrow{r}-{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}},}\overrightarrow{w}+{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}}\right\}}$ , (20)

$\overrightarrow{R}=\overrightarrow{r}-{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{w}}H\left\{t^{\prime },\overrightarrow{r}-{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}},\overrightarrow{w}+{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}}\right\}}$ . (21)

By direct substitution one can easily be convinced that the expression,

$f_{\alpha }=f_{\alpha }\left\{\overrightarrow{R}\left(t,\overrightarrow{r},\overrightarrow{w}\right),\overrightarrow{U}\left(t,\overrightarrow{r},\overrightarrow{w}\right)\right\}$ ,(22)

together with Eqs. (20) and (21) is the solutions of the Vlasov-Boltzmann equation (1). Indeed from Eqs.(8) and (9) we can construct the identical equation:

$\frac{\partial f_{\alpha }}{\partial \overrightarrow{R}}\left\{\frac{\partial \overrightarrow{R}}{\partial t}+\frac{\partial H}{\partial \overrightarrow{w}}\frac{\partial \overrightarrow{R}}{\partial \overrightarrow{r}}-\frac{\partial H}{\partial \overrightarrow{r}}\frac{\partial \overrightarrow{R}}{\partial \overrightarrow{w}}\right\}+\frac{\partial f_{\alpha }}{\partial \overrightarrow{U}}\left\{\frac{\partial \overrightarrow{U}}{\partial t}+\frac{\partial H}{\partial \overrightarrow{w}}\frac{\partial \overrightarrow{U}}{\partial \overrightarrow{r}}-\frac{\partial H}{\partial \overrightarrow{r}}\frac{\partial \overrightarrow{U}}{\partial \overrightarrow{w}}\right\}=0$ . (23)

Taking derivatives with time from equations (20) and (21) we find,

$\frac{\partial \overrightarrow{R}}{\partial t}=-\left\{\widehat{t}\right\}\cdot \overrightarrow{R}\text{ and }\frac{\partial \overrightarrow{U}}{\partial t}=-\left\{\widehat{t}\right\}\cdot \overrightarrow{U}$ (24)

Hence the distribution function of α-particles then should be presented in the form (22). For further analysis, we need to determine inverse solutions for $\overrightarrow{r}$ and $\overrightarrow{w}$ of Eqs.(20) and (21). From these equations, it follows that [11]:

$\overrightarrow{r}-{\displaystyle \underset{t}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}}=\overrightarrow{\rho }\left(t\right)=\overrightarrow{R}+{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{w}}H\left\{\overrightarrow{\rho }\left(t^{\prime }\right),\overrightarrow{\omega }\left(t^{\prime }\right)\right\}},$ , $\overrightarrow{r}-{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}}=\overrightarrow{\rho }\left(t^{\prime }\right)=\overrightarrow{R}+{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\frac{\partial }{\partial \overrightarrow{w}}H\left\{\overrightarrow{\rho }\left(t^{″}\right),\overrightarrow{\omega }\left(t^{″}\right)\right\}},$ , $\overrightarrow{r}-{\displaystyle \underset{t^{″}}{\overset{t^{\prime }}{\int }}d{t}^{‴}\frac{\partial H\left(t^{‴},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}}}=\overrightarrow{\rho }\left(t^{″}\right)=\overrightarrow{R}+{\displaystyle \underset{}{\overset{t^{″}}{\int }}d{t}^{‴}\frac{\partial }{\partial \overrightarrow{w}}H\left\{\overrightarrow{\rho }\left(t^{‴}\right),\overrightarrow{\omega }\left(t^{‴}\right)\right\}},$ (25)

$\overrightarrow{w}+{\displaystyle \underset{t}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}}=\overrightarrow{\omega }\left(t\right)=\overrightarrow{U}-{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}H\left\{\overrightarrow{\rho }\left(t^{\prime }\right),\overrightarrow{\omega }\left(t^{\prime }\right)\right\}},$ (26)

The chain of expressions (25, 25’,.....) and (26, 26’,......) can be continued which allows to present inverse solutions in the following form:

$\overrightarrow{r}=\overrightarrow{R}+{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }}\frac{\partial }{\partial \overrightarrow{U}}H\{t^{\prime },\overrightarrow{R}+{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\frac{\partial }{\partial \overrightarrow{U}}H\left\{t^{″},\overrightarrow{\rho }\left(t^{″}\right),\overrightarrow{\omega }\left(t^{″}\right)\right\},}\overrightarrow{U}-{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\frac{\partial }{\partial \overrightarrow{R}}H\left\{t^{″},\overrightarrow{\rho }\left(t^{″}\right),\overrightarrow{\omega }\left(t^{″}\right)\right\}}\}$ ,(27)

$\overrightarrow{w}=\overrightarrow{U}-{\displaystyle \underset{}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{R}}H\left\{t^{\prime },\overrightarrow{R}+{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\frac{\partial }{\partial \overrightarrow{U}}H\left\{t^{″},\overrightarrow{\rho }\left(t^{″}\right),\overrightarrow{\omega }\left(t^{″}\right)\right\},\overrightarrow{U}-{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\frac{\partial }{\partial \overrightarrow{R}}H\left\{t^{″},\overrightarrow{\rho }\left(t^{″}\right),\overrightarrow{\omega }\left(t^{″}\right)\right\}}}\right\}}$ . (28)

By successively substituting $\overrightarrow{\rho }\left(t^{″}\right),\overrightarrow{\rho }\left(t^{‴}\right),\mathrm{....}$ , and $\overrightarrow{\omega }\left(t^{″}\right),\overrightarrow{\omega }\left(t^{‴}\right),\mathrm{...}$ into solutions (27) and (28), these solutions take the form of an infinite series of the constants of integration $\overrightarrow{R}$ and $\overrightarrow{U}$ in the arguments of functions $H\left\{t^{″},\overrightarrow{\rho }\left(t^{″}\right),\overrightarrow{\omega }\left(t^{″}\right)\right\},H\left\{t^{‴},\overrightarrow{\rho }\left(t^{‴}\right),\overrightarrow{\omega }\left(t^{‴}\right)\right\},\mathrm{....}$ . Solutions (27) and (28) we’ll use below.

We define the distribution function (DF) assuming that at the initial moment, t₀, the DF of α particles depends only on the velocity

$f_{\alpha }\left\{\overrightarrow{R}\left(t_0,\overrightarrow{r},\overrightarrow{v}\right),\overrightarrow{U}\left(t_0,\overrightarrow{r},\overrightarrow{v}\right)\right\}={f_{0\alpha }\left(v\right)|}_{t_0}$ (29)

From relations at the initial time t₀ it follows:,

$\overrightarrow{R}\left(t_0,\overrightarrow{r},\overrightarrow{w}\right)=R\text{ and }\overrightarrow{U}\left(t_0,\overrightarrow{r},\overrightarrow{w}\right)=U$ (30)

where, $\overrightarrow{w}=\overrightarrow{v}-\overrightarrow{\epsilon }\left(\overrightarrow{r},t_0\right)$ and $\overrightarrow{\epsilon }\left(\overrightarrow{r},t\right)$ is determined by Eq. (4). Then by means of Eq. (28) we can find expression for the velocity $\overrightarrow{v}$ at the moment t₀ :

$\overrightarrow{v}=-\frac{Z_{\alpha }e}{m_{\alpha }}\frac{1}{c}{\overrightarrow{A}}_0++\overrightarrow{U}-{\displaystyle \underset{}{\overset{t_0}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{R}}H\left\{\overrightarrow{R}+{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\frac{\partial }{\partial \overrightarrow{U}}H\left\{\overrightarrow{R}+\mathrm{....},\overrightarrow{U}-\mathrm{....}\right\},\overrightarrow{U}-{\displaystyle \underset{}{\overset{t^{\prime }}{\int }}d{t}^{″}\frac{\partial }{\partial \overrightarrow{R}}H\left\{\overrightarrow{R}+\mathrm{....},\overrightarrow{U}-\mathrm{....}\right\}}}\right\}}$ (31)

(31) Substituting the relation (20) with $\overrightarrow{w}=\overrightarrow{v}-{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t\right)$ into the right-hand side of Eq.(31) we obtain an argument for the distribution function,

$\overrightarrow{v}-\frac{Z_{\alpha }e}{m_{\alpha }}{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\cdot \overrightarrow{E}\left(\overrightarrow{r},t\right)+}+{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}H\left\{t^{\prime },\overrightarrow{r}-{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}},}\overrightarrow{w}+{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}}\right\}}$ (32)

Satisfying the condition (29). Hence for the DF of α-particles we obtain

$f_{\alpha }=f_{0\alpha }\{\overrightarrow{v}-\frac{Z_{\alpha }e}{m_{\alpha }}{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\cdot \overrightarrow{E}\left(\overrightarrow{r},t\right)+}+{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}H\left\{t^{\prime },\overrightarrow{r}-{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}},}\overrightarrow{w}+{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}}\right\}}\}$ (33)

As $f_{0\alpha }\left(\overrightarrow{v}\right)$ we can choose the Maxwell distribution function with normalizing coefficient, ${\left(1/\sqrt{2\pi }\right)}^{3/2}\cdot \exp \left(-v^2/2\right)$ :

$f_{\alpha }=n_{0\alpha }{\left(\frac{m_{\alpha }}{2\pi T_{\alpha }}\right)}^{3/2}\exp \{-\frac{m_{\alpha }}{2T_{\alpha }}[\overrightarrow{v}-\frac{Z_{\alpha }e}{m_{\alpha }}{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\cdot \overrightarrow{E}\left(\overrightarrow{r},t\right)+}{+{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}H\left\{t^{\prime },\overrightarrow{r}-{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{w}},}\overrightarrow{w}+{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial H\left(t^{″},\overrightarrow{r},\overrightarrow{w}\right)}{\partial \overrightarrow{r}}}\right\}}]}^2\}$ (34)

In the following we’ll restrict ourselves taking into account the electric strength up to second power inclusive. Using expansions (18), and relations (10), (14) we find

$f_{\alpha }=n_{0\alpha }{\left(\frac{m_{\alpha }}{2\pi T_{\alpha }}\right)}^{3/2}\exp \{-\frac{m_{\alpha }}{2T_{\alpha }}[\overrightarrow{w}-\frac{Z_{\alpha }e}{m_{\alpha }}{\overrightarrow{A}}_0(\overrightarrow{r})+{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}}\frac{1}{2}{\left\{\overrightarrow{w}+{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t^{\prime }\right)\right\}}^2{-{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\cdot {\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\left\{\frac{\partial }{\partial \overrightarrow{w}}\frac{1}{2}{\left\{\overrightarrow{w}+{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t^{″}\right)\right\}}^2\cdot \frac{\partial }{\partial \overrightarrow{r}}-\frac{\partial }{\partial \overrightarrow{r}}\frac{1}{2}{\left\{\overrightarrow{w}+{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t^{″}\right)\right\}}^2\cdot \frac{\partial }{\partial \overrightarrow{w}}\right\}\frac{\partial }{\partial \overrightarrow{r}}\frac{1}{2}{\left\{\overrightarrow{w}+{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t^{\prime }\right)\right\}}^2}}]}^2\}$ (35)

It is convenient to introduce farther new variables instead of (4):

${\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t\right)={\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)-{\overrightarrow{a}}_{\alpha }\left(\overrightarrow{r}\right)$ , (36)

${\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)=\frac{Z_{\alpha }e}{m_{\alpha }}{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\cdot \overrightarrow{E}\left(\overrightarrow{r},t^{\prime }\right)},{\overrightarrow{a}}_{\alpha }\left(\overrightarrow{r}\right)=\frac{Z_{\alpha }e}{m_{\alpha }c}{\overrightarrow{A}}_0\left(\overrightarrow{r}\right)$ , (37)

Substituting relations $\overrightarrow{w}=\overrightarrow{v}-{\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t\right)$ and (36) into Eq.(35) (choosing ${\overrightarrow{\epsilon }}_{\alpha }\left(\overrightarrow{r},t\right)$ for different time-moments $t^{\prime },t^{″},\mathrm{...}$ ) we obtain

$f_{\alpha }\left(t,\overrightarrow{r},\overrightarrow{v}\right)=n_{0\alpha }{\left(\frac{m_{\alpha }}{2\pi T_{\alpha }}\right)}^{3/2}\exp \{-\frac{m_{\alpha }}{2T_{\alpha }}[\overrightarrow{v}-{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)+{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}}\frac{1}{2}{\left\{\overrightarrow{v}-{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)+{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t^{\prime }\right)\right\}}^2-$ ${\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}}\frac{1}{2}{\left\{\overrightarrow{v}-{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)+{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t^{\prime }\right)\right\}}^2-{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\{\frac{\partial }{\partial \overrightarrow{v}}\frac{1}{2}}}{\left(\overrightarrow{v}-{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)+{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t^{″}\right)\right)}^2\frac{\partial }{\partial \overrightarrow{r}}-$ $\frac{\partial }{\partial \overrightarrow{r}}\frac{1}{2}{\left(\overrightarrow{v}-{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)+{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t^{″}\right)\right)}^2\frac{\partial }{\partial \overrightarrow{v}}\}\cdot \frac{\partial }{\partial \overrightarrow{r}}{\frac{1}{2}{\left\{\overrightarrow{v}-{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)+{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t^{\prime }\right)\right\}}^2]}^2\}$ (38)

After squaring all terms indicated and expansion of the exponential function up to the second order of the electric strength, we obtain

$f_{\alpha }\left(t,\overrightarrow{r},\overrightarrow{u}+{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)\right)=n_{0\alpha }{\left(\frac{m_{\alpha }}{2\pi T_{\alpha }}\right)}^{3/2}\exp \left(-\frac{m_{\alpha }}{2T_{\alpha }}{u}^2\right)\{1-\frac{m_{\alpha }}{2T_{\alpha }}{u}_i{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}{\overrightarrow{\sigma }}_{\alpha i}\left(\overrightarrow{r},t^{\prime }\right).}{u}_k{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}{\overrightarrow{\sigma }}_{\alpha k}\left(\overrightarrow{r},t^{\prime }\right)-}$

$-\frac{m_{\alpha }}{T_{\alpha }}\overrightarrow{u}\cdot u_i({\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}{\sigma }_{\alpha i}\left(\overrightarrow{r},t^{\prime }\right)-{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial }{\partial r_i}.{\sigma }_{\alpha k}\left(\overrightarrow{r},t^{\prime }\right)\frac{\partial }{\partial \overrightarrow{r}}{\sigma }_{\alpha k}\left(\overrightarrow{r},t^{\prime }\right)-}}}$

$-{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}{\sigma }_{\alpha k}\left(\overrightarrow{r},t^{″}\right)\frac{\partial }{\partial r_k}\cdot \frac{\partial }{\partial \overrightarrow{r}}{\sigma }_{\alpha i}\left(\overrightarrow{r},t^{\prime }\right)}}+{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}\frac{\partial }{\partial r_k}{\sigma }_{\alpha i}\left(\overrightarrow{r},t^{″}\right)\cdot \frac{\partial }{\partial \overrightarrow{r}}{\sigma }_{\alpha k}\left(\overrightarrow{r},t^{\prime }\right)}})+$

$+\frac{m_{\alpha }}{T_{\alpha }}\overrightarrow{u}\cdot u_i\cdot u_k\frac{\partial }{\partial r_i}\frac{\partial }{\partial \overrightarrow{r}}{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}{\sigma }_{\alpha k}\left(\overrightarrow{r},t^{\prime }\right)}}-\frac{m_{\alpha }}{T_{\alpha }}\overrightarrow{u}{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }}{\sigma }_{\alpha k}\left(\overrightarrow{r},t^{\prime }\right)\cdot \frac{\partial }{\partial \overrightarrow{r}}{\sigma }_{\alpha k}\left(\overrightarrow{r},t^{\prime }\right)++\frac{1}{2}{\left(\frac{m_{\alpha }}{T_{\alpha }}\right)}^2\overrightarrow{u}\cdot u_i{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}{\overrightarrow{\sigma }}_{\alpha i}\left(\overrightarrow{r},t^{\prime }\right)}\cdot \overrightarrow{u}\cdot u_k{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\frac{\partial }{\partial \overrightarrow{r}}{\sigma }_{\alpha ki}\left(\overrightarrow{r},t^{\prime }\right)}\}.$ (39)

where the shifting velocity is introduced $\overrightarrow{u}=\overrightarrow{v}-{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)$ .

Analysis of waves we start from the main electrodynamic equation

$rot\cdot rot\cdot \overrightarrow{E}+\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}\overrightarrow{E}+\frac{4\pi }{c^2}{\displaystyle \sum _{\alpha }{Z}_{\alpha }e\frac{\partial }{\partial t}{\overrightarrow{J}}_{\alpha }\left(\overrightarrow{r},t\right)=0}$ , (40)

where the current ${\overrightarrow{J}}_{\alpha }\left(\overrightarrow{r},t\right)$ should be defined by means of Eq. (35),

${\overrightarrow{J}}_{\alpha }\left(\overrightarrow{r},t\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}d\overrightarrow{v}\cdot \overrightarrow{v}\cdot f_{0\alpha }\left(t,\overrightarrow{r},\overrightarrow{v}\right)}={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}d\overrightarrow{u}\cdot \left\{\overrightarrow{u}+{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)\right\}\cdot f_{0\alpha }\left(t,\overrightarrow{r},\overrightarrow{u}+{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)\right)}$ (41)

According to equations (38) and (39) one can be convinced that the constant magnetic field falls out of calculations. It means that the replacement of variables (realization of which is strictly justified for expressions under the integral) can eliminate the constant magnetic field from visual consideration, leaving all definitions obtained before that, valid. This result is apparently analogous to the situation, when in a system described by a “pure” Maxwell distribution function (DF), the force associated with the magnetic field disappears. In our case, the generalised expression of the Maxwell DF does not contain the constant magnetic field. Hence further investigation can be continued by means of the DF (38) and (39), since they are valid in the presence of a constant magnetic field. Obviously, one should only transform the argument of the DF as it is done in Eqs. (38) and (39). This leads to the necessity to change the interpretation of the initial equation. Equation (1) we rewrite in the form

$\frac{\partial f_{\alpha }}{\partial t}+\overrightarrow{v}\frac{\partial f_{\alpha }}{\partial \overrightarrow{r}}+\frac{Z_{\alpha }e}{m_{\alpha }}\left\{\overrightarrow{E}\left(\overrightarrow{r},t\right)+\frac{1}{c}\left[\overrightarrow{v}\times \left(\overrightarrow{B}\left(\overrightarrow{r},t\right)\right)\right]\right\}\frac{\partial f_{\alpha }}{\partial \overrightarrow{v}}+\left[\overrightarrow{v}\times {\overrightarrow{\Omega }}_{\alpha }\right]\frac{\partial f_{\alpha }}{\partial \overrightarrow{v}}=0$ , (42)

where ${\overrightarrow{\Omega }}_{\alpha }=\frac{Z_{\alpha }e}{m_{\alpha }c}{\overrightarrow{B}}_0$ . The expression (2), represented without the constant magnetic field ${\overrightarrow{B}}_0$ , after repeating calculations analogous to those used for obtaining Eqs. (3)-(5), acquires the form:

$\frac{Z_{\alpha }e}{m_{\alpha }}\left\{\overrightarrow{E}\left(\overrightarrow{r},t\right)+\frac{1}{c}\left[\overrightarrow{v}\times \overrightarrow{B}\left(\overrightarrow{r},t\right)\right]\right\}=\frac{d}{dt}{\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)+\overrightarrow{v}\cdot \overrightarrow{\nabla }\cdot {\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right),$ (43)

where ${\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)$ is defined by Eq.(37). Then, for the kinetic equation (42), we obtain

$\frac{\partial f_{\alpha }}{\partial t}+\overrightarrow{v}\frac{\partial f_{\alpha }}{\partial \overrightarrow{r}}+\frac{\partial {\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)}{\partial t}\frac{\partial f_{\alpha }}{\partial \overrightarrow{v}}-\left[\overrightarrow{v}\left[\overrightarrow{\nabla }\times {\overrightarrow{\sigma }}_{\alpha }\left(\overrightarrow{r},t\right)\right]\right]\frac{\partial f_{\alpha }}{\partial \overrightarrow{v}}+\left[\overrightarrow{v}\times {\overrightarrow{\Omega }}_{\alpha }\right]\frac{\partial f_{\alpha }}{\partial \overrightarrow{v}}=0$ . (44)

Using the relation (41) for the derivative of the current from Eq. (44), we find

$\frac{\partial }{\partial t}{\overrightarrow{J}}_{\alpha }=-\frac{\partial }{\partial r_k}{\displaystyle \underset{}{\overset{}{\int }}d\overrightarrow{v}\cdot \overrightarrow{v}\cdot v_k{f}_{0\alpha }+\frac{\partial {\overrightarrow{\sigma }}_{\alpha }}{\partial t}{\displaystyle \int d\overrightarrow{v}\cdot f_{0\alpha }}}-\left[{\overrightarrow{J}}_{\alpha }\left[\overrightarrow{\nabla }\times {\overrightarrow{\sigma }}_{\alpha }\right]\right]+\left[{\overrightarrow{J}}_{\alpha }\times {\overrightarrow{\Omega }}_{\alpha }\right]$ . (45)

Further detailed calculations are given in Appendix B. Substituting the relation (A5) from Appendix B into (45), we obtain the main equation for the analysis of oscillatory behaviour of plasma:

$rot\cdot rot\cdot \overrightarrow{E}+\frac{1}{c^2}\frac{{\partial }^2}{\partial t^2}\overrightarrow{E}+\frac{1}{c^2}{\displaystyle \sum _{\alpha }{\omega }_{p\alpha }^2}+\{\overrightarrow{E}\left(\overrightarrow{r},t\right)+V_{T_{\alpha }}^2{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }{\displaystyle \underset{t_0}{\overset{t^{\prime }}{\int }}d{t}^{″}}}\left(\Delta \overrightarrow{E}\left(\overrightarrow{r},t^{″}\right)+2\overrightarrow{\nabla }div\overrightarrow{E}\left(\overrightarrow{r},t^{″}\right)\right)$

$+\left[\left\{{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }}\overrightarrow{E}\left(\overrightarrow{r},t^{\prime }\right)+V_{T_{\alpha }}^2{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }}{\displaystyle \underset{t^{\prime }}{\overset{t}{\int }}d{t}^{″}{\displaystyle \underset{t_0}{\overset{t^{″}}{\int }}d{t}^{‴}}}\left(\Delta \overrightarrow{E}\left(\overrightarrow{r},t^{‴}\right)+2\overrightarrow{\nabla }div\overrightarrow{E}\left(\overrightarrow{r},t^{‴}\right)\right)\right\}\times {\overrightarrow{\Omega }}_{\alpha }\right]\}$

$-\frac{1}{c^2}{\displaystyle \sum _{\alpha }{\omega }_{p\alpha }^2\frac{Z_{\alpha }e}{m_{\alpha }}\{3\frac{\partial }{\partial r_k}\cdot {\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }}{E}_{\alpha k}\left(\overrightarrow{r},t^{\prime }\right){\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }}{\overrightarrow{E}}_{\alpha }\left(\overrightarrow{r},t^{\prime }\right)+\overrightarrow{E}\left(\overrightarrow{r},t\right){\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }{\displaystyle \underset{t_0}{\overset{t^{\prime }}{\int }}d{t}^{″}}div\overrightarrow{E}\left(\overrightarrow{r},t^{\prime }\right)}+}\left[{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }}\overrightarrow{E}\left(\overrightarrow{r},t^{\prime }\right)\left[\overrightarrow{\nabla }\times {\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }}\overrightarrow{E}\left(\overrightarrow{r},t^{\prime }\right)\right]\right]+$

$\left[\left\{{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }\cdot }{\displaystyle \underset{t_0}{\overset{t^{\prime }}{\int }}d{t}^{″}{E}_k\left(\overrightarrow{r},t^{″}\right)\frac{\partial }{\partial \overrightarrow{r}}{\displaystyle \underset{t_0}{\overset{t^{\prime }}{\int }}d{t}^{″}}{E}_k\left(\overrightarrow{r},t^{″}\right)+{\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }}E\left(\overrightarrow{r},t^{\prime }\right){\displaystyle \underset{t_0}{\overset{t}{\int }}d{t}^{\prime }{\displaystyle \underset{t_0}{\overset{t^{\prime }}{\int }}d{t}^{″}}div\overrightarrow{E}\left(\overrightarrow{r},t^{″}\right)}}\right\}\times {\overrightarrow{\Omega }}_{\alpha }\right]\}=0,$ (46)

Where is the Langmuir frequency of α-particles?

As it is mentioned above, plasma in a magnetic field is characterized by a great number of different types of waves [1,4,5]. Here, we consider longitudinal and transversal waves propagating along the magnetic field (z – direction).

(a) Longitudinal waves.

Obviously, for longitudinal waves we can assume that $\underset{\alpha }{\Sigma }{\omega }_{p\alpha }^2\approx {\omega }_{pe}^2$ , where is the electron Langmuir frequency. After taking the second derivative with time from Eq. (46) for $E_z\left(z,t\right)$ the component of the electric field, we find