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 S i∂tu+ ǫ ∂ xu = Π(|u|u), 0 < ǫ < 1, α ≥ 0. 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.

中文翻译：

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NLS-Szegő 方程的长时间行为

我们感兴趣的是过滤正傅立叶模式对可积非线性薛定谔方程的影响。等效地，我们想研究色散添加到三次 Szegő 方程的影响，导致 NLS-Szegő 方程对圆 S i∂tu+ ǫ ∂ xu = Π(|u|u), 0 < ǫ < 1, α ≥ 0。本文有两组结果。第一个结果涉及 Sobolev 对小数据的长时间估计。第二组结果涉及平面波解的轨道稳定性。也得到了一些不稳定的结果，导致了波浪湍流现象。