Abstract We study two resonant Hamiltonian systems on the phase space the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified scattering limit for an NLS equation with cigar shaped trap. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scales. We then pursue a thorough study of the dynamics of the resonant systems themselves. Our central finding is that these resonant equations fit into a larger class of Hamiltonian systems that have many striking dynamical features: non-trivial symmetries such as invariance under the Fourier transform and the flow of the linear Schrödinger equation with harmonic trapping, a robust well-posedness theory, including global well-posedness in L2 and all higher L2 Sobolev spaces, and an infinite family of orthogonal, explicit stationary wave solutions in the form of the Hermite functions.

中文翻译：

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与具有谐波俘获的一维非线性薛定谔方程相关的共振哈密顿系统

摘要 我们研究了相空间上的两个谐振哈密顿系统，即五次一维连续谐振方程，以及一个在文献中作为雪茄形陷阱的 NLS 方程的修正散射极限出现的三次谐振系统。我们证明这些系统在长时间尺度的小数据范围内近似于具有谐波捕获的五次和三次一维 NLS 的动力学。然后，我们深入研究共振系统本身的动力学。我们的核心发现是，这些共振方程适合更大类的哈密顿系统，这些系统具有许多显着的动力学特征：非平凡的对称性，例如傅立叶变换下的不变性和具有谐波捕获的线性薛定谔方程的流动，这是一个强大的井-姿势论，