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The one-dimensional (1D) problem on bilatiral irradiation by proton beams ofthe 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 atatmospheric pressure and temperature of 4 K, is considered. The proton kineticenergy is 1 MeV, the beam intensity is $10^{19}$ W/cm$^2$ and duration is 50ps. A mathematical model is based on the one-fluid two-temperaturehydrodynamics with a wide-range equation of state of the fuel, electron and ionheat conduction, DT fusion reaction kinetics, self-radiation of plasma andplasma heating by alpha-particles. If the ignition occurs, a plane detonationwave, which is adjacent to the front of the rarefaction wave, appears. Uponreflection of this detonation wave from the symmetry plane, the flow with thelinear velocity profile along the spatial variable $x$ and with a weakdependence of the thermodynamic functions of $x$ occurs. An appropriatesolution of the equations of hydrodynamics is found analytically up to anarbitrary constant, which can be chosen so that the analytical solutiondescribes with good accuracy the numerical one. The gain with respect to theenergy 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, limitingtrajectories of $\alpha$-particles by a cylinder of a given radius, issuggested. 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$^3$.

Plane thermonuclear detonation waves initiated by proton beams and quasi-one-dimensional model of fast ignition