Interaction of solitary waves in longitudinal magnetic field in two-fluid MHD

The interaction of solitary waves in a model of two-fluid MHD is studied analytically and numerically in the most general case of waves in cold plasma in longitudinal magnetic field. The distinctive feature of this work is the use of “exact” equations rather than an approximate approach (a model equation). Numerical analysis of the solutions of this system of eight partial differential equations shows that the the interaction of solitary waves found in this case is the same (with great accuracy) as that of solitons, i.e., solitary waves that are solutions of various model equations. The solitary waves considered here transport plasmoids with velocities of the order of the Alfven velocity. The main finite-difference method used here for solving the said equations is a natural generalization of the classical two-step Lax-Wendorff scheme.


Introduction
The study of solitons is traditionally based on certain model equations [1], starting with the Kortewegde Vries equation, the nonlinear Schrödinger equation, the sine-Gordon equation. For plasma environments, this list has been considerably extended due to Zakharov equations, Kadomtsev-Petviashvili equation, the equations of nonlinear Alfven waves and magnetic-sound waves [2], etc.
In this paper, we study numerically the interaction of solitary waves of a speical type: the so-called wave packets, each having the form of nonlinear vibrations whose amplitude is modulated by a solitary wave. It is the longitudinal magnetic field that (being responsible for the rotation of plasma particles in the transverse plane) makes the transverse velocity components, as well as the magnetic and the electric fields, experience nonlinear vibrations which, under certain conditions, form a wave packet running along the magnetic field. In the case of cold plasma, these waves are presented in [3]. The principal distinction of the present work from other similar studies consists in avoiding the ideology of model equations, thereby making it possible to use the exact equations of two-fluid plasma hydrodynamics [4] (i.e., the equations expressing the fundamental laws of conservation of mass, energy, and momentum of electrons and ions, as well as the laws of electrodynamics) for both finding the solitary waves and studying their interaction.
It is shown here that, when colliding, the wave packets are similar to material particles: they preserve their shape, velocity, amplitude, etc., and the collision process has a finite duration.

Basic Equations
In the absence of dissipation, for fully ionized two-component quasi-neutral cold plasma the equations of two-fluid hydrodynamics of plasma [4] can be written in the following one-fluid form [5]: Where each vector is expanded along ( )  and across ( )  the direction of wave vector k , and 0 J  , 0 D  ,  q k are arbitrary constants of integration. Moreover, we always have ( ) const H    in the traveling wave. In a special case, the two-fluid form of system (3) is obtained in [6], and in the general case, in [3].
In [7], the authors study collisions of solitary waves represented by the solutions of system (3) of the form   In order to find a solution of system (4)

Solving Equations (3) for Solitary Waves in the Case
Here, , E  are arbitrary constants, and the potential function ( ) H  has the form The desired solutions of the solitary wave type correspond to the degenerate cases and exist only if Consider the boundary value problem of exciting a solitary wave (in resting plasma of density   with the magnetic field strength H H    ) traveling along the magnetic field. For the characteristic scales we then have Then, from (6) we obtain the following constraints on the plase velocity a 2 Moreover, it is not difficult to obtain formulas for the relative amplitude ( ) A a of the solitary wave The function ( ) A a is monotonically increasing in a from 0 to 2 2 In particular, for electron-ion plasma, the maximal amplitude is   , which, for instance, in the case of hydrogen plasma is 42  .

Numerical Modeling Method
Consider three problems regarding the interaction of the above waves. We are interested in three cases: Where the x axis is directed along the vector k and complex notation is used for the components of the vectors , , U H E : , , In dimensionless form, system (1)-(2) can be writes as In this paper, these effects have been discovered numerically on the basis of a complete system of equations of two-fluid hydrodynamics of cold plasma, i.e., the laws of conservation of mass and momentum, together with the Maxwell equations. It should be mentioned that some mathematical properties (the Painleve property) of traveling wave equations (8) were considered in [9]. The present work continues the attempts to construct new models of plasma for the investigation of processes in plasma accelerators and new types of magnetic traps. This work has been financially supported by the Russian Scientific Foundation (Project 16-11-10278).