The KdV equation and the nonlinear Schrödinger (NLS) equation can be derived as formal approximation equations describing the long wavelength limit and the envelopes of slowly modulated spatially and temporarily oscillating wave packet-like solutions to the ion Euler-Poisson equation. In this talk, we rigorously justify such approximations by giving error estimates in Sobolev norms between exact solutions of the ion Euler-Poisson system and the formal approximation obtained via the KdV and the NLS equation.
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