Boundary value problem for plane nonlinear solitary waves in Hall MHD

The equations for plane nonlinear traveling waves in the Hall MHD are derived. The first integral of the “energy” of the equations of traveling waves is obtained and their Hamiltonian property in Lagrangian coordinates is proved. A classification of singular points of the equations of traveling waves is carried out. For an isothermal plasma resting at infinity, the boundary problem of finding plane solitary waves whose parameters have given values at infinity and propagating through space with a given phase velocity is also posed. The ranges of change in the phase velocity are analytically found for which the boundary value problem is solvable. It is shown that there are two families of solutions to the boundary value problem, which differ in the magnitude of the phase velocity – fast waves, the phase velocity of which is greater than the sonic one, and slow waves with a phase velocity less than the sonic one. The solitary waves found were verified by substituting them into the Hall MHD equations.