We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schrödinger (NLS) equation with an isotropic harmonic oscillator potential. The dynamics in this regime is dominated by resonant interactions between quartets of linear normal modes, accurately captured by the corresponding resonant approximation. Within this approximation, we identify Fermi-Pasta-Ulam–like recurrence phenomena, whereby the normal-mode spectrum passes in close proximity of the initial configuration, and two-mode states with time-independent mode amplitude spectra that translate into long-lived breathers of the original NLS equation. We comment on possible implications of these findings for nonlinear optics and matter-wave dynamics in Bose-Einstein condensates.

中文翻译：

---

Fermi-Pasta-Ulam 现象和谐波陷阱中的持续呼吸

我们考虑由具有各向同性谐振子势的二维非线性薛定谔（NLS）方程控制的长期弱非线性演化。该状态下的动力学由线性正态模式四重奏之间的共振相互作用主导，由相应的共振近似准确捕获。在这个近似中，我们确定了类似费米-帕斯塔-乌拉姆的复发现象，其中正常模式频谱在初始配置的附近通过，以及具有时间无关模式振幅频谱的双模式状态转化为长寿命呼吸的原始 NLS 方程。我们评论了这些发现对玻色-爱因斯坦凝聚中的非线性光学和物质波动力学的可能影响。