In this article, we present, throughout two basic models of damped nonlinear Schrödinger (NLS)–type equations, a new idea to bound from above the fractal dimension of the global attractors for NLS‐type equations. This could answer the following open issue: consider, for instance, the classical one‐dimensional cubic nonlinear Schrödinger equation

$$u_t+i{u}_{xx}+i|u{|}^{2}u+\gamma u=f,f\in {𝕃}^2(ℝ).$$

“How can we bound the fractal dimension of the associate global attractor without the need to assume that the external forcing term f has some decay at infinity (that is belonging to some weighted Lebesgue space)?”

中文翻译：

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关于耗散非线性Schrödinger型方程整体吸引子的有限分形维数的注记

在本文中，我们在整个阻尼非线性Schrödinger（NLS）型方程的两个基本模型中，提出了一个新的想法，从上面约束NLS型方程的整体吸引子的分形维数。这可能会回答以下未解决的问题：例如，考虑经典的一维三次非线性薛定ding方程

$${ü}_{Ť}+\mathrm{一世}{ü}_{XX}+\mathrm{一世}|ü{|}^2ü+\gamma ü=F，F\in {𝕃}^2（ℝ）。$$

“我们如何在不假定外部强迫项f在无穷大处具有一定衰减（属于某个加权Lebesgue空间）的情况下，约束整体全局吸引子的分形维数？”