Coordinates at Small Energy and Refined Profiles for the Nonlinear Schr?dinger Equation
Coordinates at Small Energy and Refined Profiles for the Nonlinear Schr?dinger Equation
复制标题
非线性薛定谔方程的小能量坐标和精细轮廓
DOI：
10.1007/s40818-021-00105-2
复制
发表时间：
2021
期刊：
影响因子：
2.8
通讯作者：
Maeda Masaya
中科院分区：
文献类型：
--
作者：
Cuccagna Scipio;Maeda Masaya
本文给出了文献[6]中给出的非线性薛定谔方程(NLS)小能解驻波选择定理的一个新的简化证明。我们考虑一类具有多个本征值的薛定谔算子的非线性最小二乘问题，并证明了任一小能量解都收敛到时间周期解加散射项的轨道。新的思想是考虑“精化轮廓”，这是一个时间上的准周期函数，几乎求解NLS，并对解的离散模式进行编码。通过初等方法得到的精化轮廓，直接给出了一个最佳坐标系，避免了文献[6]中的规范形式争论，也使我们更好地理解了费米黄金规则。
In this paper we give a new and simplified proof of the theorem on selection of standing waves for small energy solutions of the nonlinear Schrödinger equations (NLS) that we gave in [6]. We consider a NLS with a Schrödinger operator with several eigenvalues, with corresponding families of small standing waves, and we show that any small energy solution converges to the orbit of a time periodic solution plus a scattering term. The novel idea is to consider the “refined profile”, a quasi–periodic function in time which almost solves the NLS and encodes the discrete modes of a solution. The refined profile, obtained by elementary means, gives us directly an optimal coordinate system, avoiding the normal form arguments in [6], giving us also a better understanding of the Fermi Golden Rule.