1${1\over 2}$D time-dependent solution for the problem of the beam – barrier interaction was constructed, i.e., electromagnetic fields (${\bf E}=E_x{\bf \hat
  e}_x+E_y{\bf \hat e}_y$; ${\bf B}=B{\bf \hat e}_z$) and motion of the beam particles in the $x$ – $y$ plane were found. The structure of the fields and motion of the beam particles were treated self-consistently except the external current $J_y$ which was fixed in our simulations.

The main results of our simulations are the following ([@Smolsky96]; [@Usov98]).

1\. When the energy densities of the beam and the magnetic field, ${\bf B}_0$, of the barrier are comparable,

$$\alpha = 8\pi n_0m_pc^2(\Gamma_0-1)/B^2_0\sim 1\,, 
\label{alpha}$$

where $m_p$ is the proton mass, the process of the beam – barrier interaction is strongly nonstationary, and the density of protons after their reflection from the barrier is strongly non-uniform. The ratio of the maximum density of reflected protons and their minimum density is $\sim 10$.

2\. At $\alpha > \alpha _{\rm cr}\simeq 0.4$, the depth of the beam particle penetration into the barrier increases in time, $x_{\rm pen}\simeq v_{\rm pen} t$, where $v_{\rm pen}$ is the mean velocity of the penetration into the barrier. The value of $v_{\rm pen}$ is subrelativistic and varies from zero 