Quantum ion-acoustic waves

The one-dimensional two-species quantum hydrodynamic model is considered in the limit of small mass ratio of the charge carriers. Closure is obtained by adopting an equation of state pertaining to a zero-temperature Fermi gas for the electrons and by disregarding pressure effects for the ions. By an appropriate rescaling of the variables, a nondimensional parameter H, proportional to quantum diffraction effects, is identified. The system is then shown to support linear waves, which in the limit of small H resemble the classical ion-acoustic waves. In the weakly nonlinear limit, the quantum plasma is shown to support waves described by a deformed Korteweg-de Vries equation which depends in a nontrivial way on the quantum parameter H. In the fully nonlinear regime, the system also admits traveling waves which can exhibit periodic patterns. The quasineutral limit of the system is also discussed.