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arXiv · 2009.09855

The perturbational stability of the Schr$\ddot{o}$dinger equation

Xixia Ma
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Abstract

By using the Wigner transform, it is shown that the nonlinear Schr$\ddot{\textmd{o}}$dinger equation can be described, in phase space, by a kinetic theory similar to the Vlasov equation which is used for describing a classical collisionless plasma. In this paper we mainly show Landau damping in the quantum sense, namely,quantum Landau damping exists for the Wigner-Poisson system. At the same time, we also prove the existence and the stability of the nonlinear Schr$\ddot{\textmd{o}}$dinger equation under the quantum stability assumption.

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Xixia Ma. 2020-08-05. The perturbational stability of the Schr$\ddot{o}$dinger equation. https://arxiv.org/abs/2009.09855

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