A Nonlinear Adiabatic Theorem for Coherent States

Abstract

We present a result obtained in collaboration with Rémi Carles on the propagation of coherent states for a 1-d cubic nonlinear Schrödinger equation in a semi-classical regime ($\epsilon \ll 1$):

$$i\epsilon {\partial }_{t}{\psi }^{\epsilon } + \frac{{\epsilon }^{2}} {2} {\partial }_{x}^{2}{\psi }^{\epsilon } = V (x){\psi }^{\epsilon } +\, {\epsilon }^{\alpha }\vert {\psi }^{\epsilon }{\vert }^{2}{\psi }^{\epsilon },\;\;{\psi }^{\epsilon } :{ \mathbf{R}}_{ t} \times {\mathbf{R}}_{x} \rightarrow {\mathbf{C}}^{N},\;\;\alpha > 0.$$