In this talk we will describe numerical and theoretical results obtained jointly with a number of collaborators: David Nicholls, Mariana Haragus, Yi Li, and Gerard Misolek.

We study solitary wave interactions in the Euler-Poisson equations modeling ion acoustic plasmas and their approximation by KdV n-solitons. While largely correct qualitatively, the n-soliton solutions do not accurately capture the scattering shifts observed in the plasma waves. Our analysis indicates that the scattering shifts are due to resonant interactions of the solitary waves, related to the null space of the linearized KdV equation.

A mathematical proof of the existence of solutions of the Euler-Poisson equations appears still to be an open question. The equations are in a certain sense hyperbolic conservation laws with dispersion rather than viscosity preventing the occurence of shocks. We discuss some recent pointwise estimates and their role in a priori exclusion of shock
discontinuities.