Investigations of the kinetic ion-acoustic soliton by the Bernstein–Greene–Kruskal integral method

The solitary waves are investigated through the Bernstein–Greene–Kruskal integral method with the ion response. We consider two specific cases of ions, i.e., the single stream with the waterbag distribution and the two counter-propagating streams with the Maxwellian distribution. The trapped electron distributions are derived for both two cases. The results show that the trapped electron distribution can be either a hole or a hump in the phase space, depending on the competition between the contributions from the passing electron distribution, the potential profile, and the ion response. We obtain the boundary between the ion-acoustic soliton (IAS) and the electron hole in the parameter space. The effects of the potential amplitude, width, and the ion-to-electron mass ratio on the separatrices are discussed. The Vlasov simulations are conducted to verify the stability of the IAS constructed by the integral method.