  The one-dimensional (1D) problem on bilatiral irradiation by proton beams of
the plane layer of condensed DT mixture with length $2H$ and density $\rho_0
\leqslant 100\rho_s$, where $\rho_s$ is the fuel solid-state density at
atmospheric pressure and temperature of 4 K, is considered. The proton kinetic
energy is 1 MeV, the beam intensity is $10^{19}$ W/cm$^2$ and duration is 50
ps. A mathematical model is based on the one-fluid two-temperature
hydrodynamics with a wide-range equation of state of the fuel, electron and ion
heat conduction, DT fusion reaction kinetics, self-radiation of plasma and
plasma heating by alpha-particles. If the ignition occurs, a plane detonation
wave, which is adjacent to the front of the rarefaction wave, appears. Upon
reflection of this detonation wave from the symmetry plane, the flow with the
linear velocity profile along the spatial variable $x$ and with a weak
dependence of the thermodynamic functions of $x$ occurs. An appropriate
solution of the equations of hydrodynamics is found analytically up to an
arbitrary constant, which can be chosen so that the analytical solution
describes with good accuracy the numerical one. The gain with respect to the
energy of neutrons $G\approx 200$ at $H\rho_0 \approx 1$ g/cm$^2$, and $G>2000$
at $H\rho_0 \approx 5$ g/cm$^2$. To evaluate the ignition energy
$E_{\mathrm{ig}}$ of cylindrical targets, the quasi-1D model, limiting
trajectories of $\alpha$-particles by a cylinder of a given radius, is
suggested. The model reproduces the known theoretical dependence
$E_{\mathrm{ig}} \sim \rho_0^{-2}$ and gives $E_{\mathrm{ig}} = 160$ kJ for
$\rho_0 = 100\rho_s \approx 22$ g/cm$^3$.
