Bernstein-Greene-Kruskal approach for the quantum Vlasov equation

Abstract

The one-dimensional stationary quantum Vlasov equation is analyzed using the energy as one of the dynamical variables, similarly as in the solution of the Vlasov-Poisson system by means of the Bernstein-Greene-Kruskal method. In the semiclassical case where quantum tunneling effects are small, an infinite series solution is developed and shown to be immediately integrable up to a recursive chain of quadratures in position space only. As it stands, the treatment of the self-consistent, Wigner-Poisson system is beyond the scope of the method, which assumes a given smooth time-independent external potential. Accuracy tests for the series expansion are also provided. Examples of anharmonic potentials are worked out up to a high order on the quantum diffraction parameter.

Bernstein–Greene–Kruskal and Case–Van Kampen Modes for the Landau–Vlasov Equation

“…By construction, these so-called BGK modes are exact nonlinear plasma oscillations which do not present damping or growth. The BGK approach where the energy is the central dynamical variable can be adapted for the derivation of phase-space hole structures [14][15][16][17] and, to a more limited extent, to quantum plasmas [18].…”

3D BGK model for electron phase-space holes including polarization drift

“…Some theories have been developed for solitary waves in quantum plasma [19] based on a fluid model or a Bernstein-Greene-Kruskal (hereafter BGK) approach for quantum Vlasov equation (e.g. [20]). More generally, coherent structures appear in many fields of physics such as condensed matter or hydrodynamics.…”