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

Results of theoretical and numerical investigations concerning the space–time evolution in 1D of Coulombian and Newtonian systems with densities departing from a uniform and homogeneous background are reported here. In the Coulombian case, the model is called a One Component Plasma. In the Newtonian case, we have the cosmological models of Cold Dark Matter in expanding universes with expansion parameters depending or not upon the cosmological constant. A canonical Hamiltonian formulation is given for studying single-speed solutions of their Coulomb- or Jeans–Vlasov–Poisson descriptions. It is shown that using the Gel’fand mass coordinate the equations of motion are exactly integrable and that the corresponding equations of their characteristics are inhomogeneous, linear and second order ODE’s with variable coefficients for the cosmological models. It is furthermore shown that, using correlated initial conditions, Burgers’ type of implicit equations for the velocity fields are obtained. Comparison is made between this way to generate exact solutions for the characteristics of the models and that put forward by Zel’dovich. Two examples illustrate the resulting regular and singular mean-field dynamics of the models: a periodic initial excess density for the One Component Plasma and a local departure from homogeneity for the cosmological models.