We analytically study the impact of a short laser pulse onto an inhomogeneous cold diluted plasma at rest, in particular: the duration of the hydrodynamic regime; the formation and the features of plasma waves (PWs); their wave-breakings (WBs); the motion of test electrons injected in the PWs.
If the pulse is a plane wave travelling in the $z$-direction, and the initial plasma density (IPD) depends only on $z$, then suitable matched bounds on the maximum and relative variations of the IPD, as well as the intensity and duration of the pulse, ensure a strictly hydrodynamic evolution of the electron fluid during its whole interaction with the pulse, while ions can be regarded as immobile. This evolution is ruled by a family (parametrized by $Z\ge 0$) of decoupled systems of non-autonomous Hamilton equations with 1 degree of freedom, which determine how electrons initially located in the layer $Z\le z<Z+dZ$ move; $ξ=ct-z$ replaces time $t$ as the independent variable. This family of ODEs is obtained by reduction from the Lorentz-Maxwell and continuity PDEs for the electrons' fluid within the spacetime region where the change of the pulse is negligible. After the laser-plasma interaction the Jacobian of the map from Lagrangian to Eulerian coordinates is linear-quasi-periodic in $ξ$. We determine spacetime locations and features of the first wave-breakings of the wakefield PWs, the motion of test electrons (self-)injected in the PWs. The energy of those trapped in a single PW trough grows linearly with the distance gone, where the IPD is constant.
If the pulse has cylindrical symmetry and a not too small radius, the same conclusions hold for the part of the plasma enclosed within the causal cone swept by it.
This computationally light approach may help in a preliminary study of extreme acceleration mechanisms of electrons (LWFA, etc.), before 2D or 3D PIC simulations.
