We theoretically and experimentally examine the effect of forcing and damping on systems that can be described by the nonlinear Schrödinger equation (NLSE), by making use of the phase-space predictions of the three-wave truncation. In the latter, the spectrum is truncated to only the fundamental frequency and the upper and lower sidebands. Our experiments are performed on deep water waves, which are better described by the higher-order NLSE, the Dysthe equation. We therefore extend our analysis to this system. However, our conclusions are general for NLSE systems. By means of experimentally obtained phase-space trajectories, we demonstrate that forcing and damping cause a separatrix crossing during the evolution. When the system is damped, it is pulled outside the separatrix, which in the real space corresponds to a phase-shift of the envelope and therefore doubles the period of the Fermi–Pasta–Ulam–Tsingou recurrence cycle. When the system is forced by the wind, it is pulled inside the separatrix, lifting the phase-shift. Furthermore, we observe a growth and decay cycle for modulated plane waves that are conventionally considered stable. Finally, we give a theoretical demonstration that forcing the NLSE system can induce symmetry breaking during the evolution.

我们在理论上和实验上通过利用三波截断的相空间预测，研究了强迫和阻尼对可以由非线性Schrödinger方程（NLSE）描述的系统的影响。在后者中，频谱仅被截断为基频和上，下边带。我们的实验是在深水波上进行的，用高阶NLSE Dysthe方程可以更好地描述。因此，我们将分析扩展到该系统。但是，我们的结论对NLSE系统是通用的。通过实验获得的相空间轨迹，我们证明了力和阻尼在演化过程中引起了分离线的穿越。当系统受到阻尼时，它会被拉到分离线之外，在实际空间中，它对应于包络的相移，因此使费米－帕斯塔－乌拉姆－青果复发周期的周期加倍。当系统受到风的推动时，将其拉到分离线内部，抬起相移。此外，我们观察到通常认为稳定的调制平面波的增长和衰减周期。最后，我们给出了一个理论论证，即强迫NLSE系统可以在进化过程中引起对称性破坏。