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.
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