On a class of implicit solutions of the continuity and Euler's equations for 1D systems with long range interactions

Results of computer simulations and of theoretical analysis done to investigate and interpret the space-time evolution of the mass density and the velocity field of the inviscid self-gravitating (attractive) and (repulsive) Coulomb liquids in 1D with correlated initial conditions, namely proportionality between the mass density and the divergence of the velocity field are reported here. Numerical data gathered for both models in a collisionless regime reveal an evolution with a time-dependent proportionality factor. Feeding this result in the continuity and div-Euler equations leads to the introduction of another field which is shown to satisfy a Burgers type of implicit equation. A thorough description of regular implosion followed by singular collapses in the attractive case, and of regular explosion in the repulsive case is obtained. Time-inversion symmetry is investigated, energy conservation and stability properties are shown to apply in the regular regions of smooth solutions. The velocity potential satisfies a new local and inhomogeneous PDE.