Nonlinear Schrodinger Approximation for the Electron Euler-Poisson Equation

The nonlinear Schrodinger (NLS for short) equation plays an important role in describing slow modulations in time and space of an underlying spatially and temporarily oscillating wave packet. In this paper, the authors study the NLS approximation by providing rigorous error estimates in Sobolev spaces for the electron Euler-Poisson equation, an important model to describe Langmuir waves in a plasma. They derive an approximate wave packet-like solution to the evolution equations by the multiscale analysis, then they construct the modified energy functional based on the quadratic terms and use the rotating coordinate transform to obtain uniform estimates of the error between the true and approximate solutions.