  Development of the self-sustained quantum-electrodynamical (QED) cascade in a
single strong laser pulse is studied analytically and numerically. The
hydrodynamical approach is used to construct the analytical model of the
cascade evolution, which includes the key features of the cascade observed in
3D QED particle-in-cell (QED-PIC) simulations such as the magnetic field
predominance in the cascade plasma and laser energy absorption. The equations
of the model are derived in the closed form and are solved numerically. Direct
comparison between the solutions of the model equations and 3D QED-PIC
simulations shows that our model is able to describe the complex nonlinear
process of the cascade development qualitatively well. The various regimes of
the interaction based on the intensity of the laser pulse are revealed in both
the solutions of the model equations and the results of the QED-PIC
simulations.
