Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite gap tori for the 2D cubic NLS equation
Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite gap tori for the 2D cubic NLS equation
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DOI：
10.4171/jems/1200
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发表时间：
2018-10
影响因子：
2.6
通讯作者：
M. Guardia;Z. Hani;E. Haus;A. Maspero;M. Procesi
中科院分区：
文献类型：
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作者：
M. Guardia;Z. Hani;E. Haus;A. Maspero;M. Procesi
本文研究二维环面上的三次非线性Schr odinger方程（NLS）.该方程包含一类特殊的椭圆不变准周期环，称为有限间隙解。这些都是继承自可积的一维模型（立方NLS的圆圈），考虑解决方案，只依赖于一个变量。我们研究了2D NLS模型的这种不变环面的长时间稳定性，并表明，在一定的假设下，在足够长的时间尺度上，它们在Sobolev空间$H^s（\mathbb{T}^2）$（$0<s<1$）中表现出强形式的横向不稳定性。更准确地说，我们构造的二维三次NLS的解决方案，开始任意接近这样的不变环面在$H^s$拓扑和其$H^s$范数可以增长任何给定的因素。这项工作的部分动机的问题的无限能量级联的二维NLS，似乎是第一个实例（不稳定）长时间非线性动力学（线性稳定）准周期环面附近的研究和建设。
We consider the defocusing cubic nonlinear Schr\"odinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long timescales, they exhibit a strong form of transverse instability in Sobolev spaces $H^s(\mathbb{T}^2)$ ($0<s<1$). More precisely, we construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the $H^s$ topology and whose $H^s$ norm can grow by any given factor. This work is partly motivated by the problem of infinite energy cascade for 2D NLS, and seems to be the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed.