Singular traveling waves for the Euler-Poisson system

Abstract

We consider the Euler-Poisson system for ions where the electrons are given by a Maxwell-Boltzmann distribution, and we investigate the existence of one-dimensional periodic traveling waves. More precisely, we first establish the existence of a smooth global branch of bifurcation emanating from a constant equilibrium. We then construct a singular traveling wave emerging as the limiting profile at the end of the global curve of bifurcation. Our analysis accommodates a wide class of pressure laws and provides a comprehensive characterization of both smooth and singular traveling waves. A central difficulty in this model arises from the exponential nonlinearity, induced by the nonlocal Poisson-Boltzmann equation, which prevents any explicit representation of the electron field in terms of the ion density. This poses significant obstacles compared to previous studies on related models, where such explicit formulas were crucial for global bifurcation arguments.