Long Wavelength Limit for the Quantum Euler-Poisson Equation

In this paper, we consider the long wavelength limit for the quantum Euler--Poisson equation. Under the Gardner--Morikawa transform, we derive the quantum Korteweg--de Vries (KdV) equation by a reductive perturbation method. We show that the KdV dynamics can be seen at time interval of order $O(\epsilon^{-3/2})$. When the nondimensional quantum parameter $H=2$, it reduces to the inviscid Burgers equation.

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