Gross-Pitaevskii-Poisson model for an ultracold plasma: Density waves and solitons

We introduce one- A nd two-dimensional (1D and 2D) models of a degenerate bosonic gas composed of ions carrying positive and negative charges (cations and anions), under the condition of the electroneutrality. The system may exist in the mean-field condensate state, enabling the competition of the Coulomb coupling, contact repulsion, and kinetic energy of the particles, provided that their effective mass is reduced by means of a lattice potential. The respective model combines the Gross-Pitaevskii (GP) equations for the two-component wave function of the cations and anions, coupled to the Poisson equation for the electrostatic potential mediating the Coulomb interaction. In addition to its direct introduction, the contact interaction in the GP system can be derived, in the Thomas-Fermi approximation, from a system of three GP equations, which includes the wave function of heavy neutral (buffer) atoms. In the system with fully repulsive contact interactions, we construct stable spatially periodic patterns (density waves, akin to ionic crystals). The transition to the density wave is identified by analysis of the modulational instability of a uniformly mixed neutral state. The density-wave pattern, which represents the system's ground state (GS), is accurately predicted by a variational approximation. In the 2D case, a stable pattern is produced too, with a quasi-1D shape. The 1D system with contact self-attraction in each component produces bright solitons of three types: Neutral ones, with fully mixed components, dipoles, with the components separated by the interspecies contact repulsion, and quadrupoles, with a layer of one component sandwiched between side lobes formed by the other. The transition from the neutral solitons to dipoles is accurately modeled analytically. A chart of the GSs of the different types (neutral solitons, dipoles, or quadrupoles) is produced. Different soliton species do not coexist as stable states. Collisions between traveling solitons are studied too. Collisions are elastic for dipole-dipole pairs, while dipole-antidipole ones merge into stable quadrupoles via multiple collisions.