We study the nonlinear Schrödinger equation (NLS) with bounded initial data which does not vanish at infinity. Examples include periodic, quasi-periodic and random initial data. On the lattice we prove that solutions are polynomially bounded in time for any bounded data. In the continuum, local existence is proved for real analytic data by a Newton iteration scheme. Global existence for NLS with a regularized nonlinearity follows by analyzing a local energy norm.

中文翻译：

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具有有界初始数据的 Z 和 R 上的非线性薛定谔方程：示例和猜想

我们研究了非线性薛定谔方程 (NLS)，其初始数据在无穷远时不会消失。示例包括周期性、准周期性和随机初始数据。在格子上，我们证明对于任何有界数据，解在时间上都是多项式有界的。在连续统中，通过牛顿迭代方案证明了真实解析数据的局部存在性。具有正则化非线性的 NLS 的全局存在性遵循分析局部能量范数。