Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping

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.

Stochastic Diffusion of Energetic Ions in Wendelstein-Type Stellarators

The collisionless stochastic diffusion of energetic ions in optimized stellarators of the Wendelstein type has been considered. The phenomenon concerned was predicted earlier in the framework of a simplified theory describing the separatrix crossing by ions. The jumps of the adiabatic invariant in magnetic configurations of a stellarator are calculated. The analysis of the results obtained confirms the importance of the stochastic diffusion and demonstrates that the diffusion coefficient can considerably exceed the available result.