Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb {T}^d$ with a multiplicative potential
Quasi-periodic solutions with Sobolev regularity of NLS on $\mathbb {T}^d$ with a multiplicative potential
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DOI：
10.4171/jems/361
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发表时间：
2013
影响因子：
2.6
通讯作者：
M. Berti;P. Bolle
中科院分区：
文献类型：
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作者：
M. Berti;P. Bolle
我们证明了具有T，d≥1上的乘势、有限可微的非线性项和沿预定方向约束的切向频率的薛定谔方程的拟周期解的存在性。解在时间和空间上都只具有Sobolev正则性。如果非线性和势为C∞，则解为C∞。证明是基于改进的Nash-Moser迭代格式，该迭代格式假定逆线性化算子(“格林函数”)沿Sobolev空间的尺度是最弱的Tame估计。通过一种新的多尺度归纳分析，证明了格林函数的关键非对角衰减估计。主要的新奇之处在于衡量标准和“复杂性”估计。
We prove the existence of quasi-periodic solutions for Schrodinger equations with a multiplicative potential on T, d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are C∞ then the solutions are C∞. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators (“Green functions”) along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and “complexity” estimates.