Long time behavior of the NLS-Szegő equation

We are interested in the influence of filtering the positive Fourier modes to the integrable non linear Schrödinger equation. Equivalently, we want to study the effect of dispersion added to the cubic Szegő equation, leading to the NLS-Szegő equation on the circle $\mathbb{S}^1$\begin{align}i \partial_{t} u + \epsilon^{\alpha} \partial^2_x u = \Pi ({\lvert u \rvert}^2 u) , \qquad 0 \lt \epsilon \lt 1 , \qquad \alpha \geq 0.\end{align}There are two sets of results in this paper. The first result concerns the long time Sobolev estimates for small data. The second set of results concerns the orbital stability of plane wave solutions. Some instability results are also obtained, leading to the wave turbulence phenomenon.

Keywords

cubic Schrödinger equation, Szegő projector, small dispersion, stability, wave turbulence, Birkhoff normal form

2010 Mathematics Subject Classification

Primary 37-xx, 70-xx, 76-xx, 92-xx. Secondary 34-xx, 35-xx, 80-xx, 82-xx.

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Received 25 April 2019

Published 30 August 2019