  The theory of soliton gas had been previously developed for unidirectional
integrable dispersive hydrodynamics in which the soliton gas properties are
determined by the overtaking elastic pairwise interactions between solitons. In
this paper, we extend this theory to soliton gases in bidirectional integrable
Eulerian systems where both head-on and overtaking collisions of solitons take
place. We distinguish between two qualitatively different types of
bidirectional soliton gases: isotropic gases, in which the position shifts
accompanying the head-on and overtaking soliton collisions have the same sign,
and anisotropic gases, in which the position shifts for head-on and overtaking
collisions have opposite signs. We construct kinetic equations for both types
of bidirectional soliton gases and solve the respective shock-tube problems for
the collision of two "monochromatic" soliton beams consisting of solitons of
approximately the same amplitude and velocity. The corresponding weak solutions
of the kinetic equations consisting of differing uniform states separated by
contact discontinuities for the mean flow are constructed. Concrete examples of
bidirectional Eulerian soliton gases for the defocusing nonlinear Schr\"odinger
(NLS) equation and the resonant NLS equation are considered. The kinetic
equation of the resonant NLS soliton gas is shown to be equivalent to that of
the shallow-water bidirectional soliton gas described by the Kaup-Boussinesq
equations. The analytical results for shock-tube Riemann problems for
bidirectional soliton gases are shown to be in excellent agreement with direct
numerical simulations.
