Stability of the Hartree equation with time-dependent coefficients

Stability of the Hartree equation with time-dependent coefficients

Exact solutions of the mKdV equation with time-dependent coefficients

In this paper, we study the time variable coefficient modified Korteweg-de Vries (mKdV) equation from group-theoretic point of view. We obtain Lie point symmetries admitted by the mKdV equation for various forms for the time variable coefficients. We use the symmetries to construct the group-invariant solutions for each of the cases of the arbitrary variable coefficients. Finally, the solitary ...

On the Time Dependent Gross Pitaevskii- and Hartree Equation

We are interested in solutions Ψt of the Schrödinger equation of N interacting bosons under the influence of a time dependent external field, where the range and the coupling constant of the interaction scale with N in such a way, that the interaction energy per particle stays more or less constant. Let N0 be the particle number operator with respect to some φ0 ∈ L (R → C). Assume that the rela...

N-dimensional nonlinear Fokker-Planck equation with time-dependent coefficients.

An N-dimensional nonlinear Fokker-Planck equation is investigated here by considering the time dependence of the coefficients, where drift-controlled and source terms are present. We exhibit the exact solution based on the generalized Gaussian function related to the Tsallis statistics. Furthermore, we show that a rich class of diffusive processes, including normal and anomalous ones, can be ob...

A Microscopic Derivation of the Time-Dependent Hartree-Fock Equation with Coulomb Two-Body Interaction

We study the dynamics of a Fermi gas with a Coulomb interaction potential, and show that, in a mean-field limiting regime, the dynamics is described by the Hartree-Fock equation. This extends previous work of Bardos et al. [3] to the case of unbounded interaction potentials. We also express the mean-field limit as a “superhamiltonian” system, and state our main result in terms of a Heisenberg-p...