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

TL;DR

This paper investigates how forcing and damping influence the dynamics of NLSE-like systems, revealing phenomena like separatrix crossing, symmetry breaking, and altered recurrence cycles through theoretical analysis and water wave experiments.

Contribution

It extends the understanding of NLSE systems by analyzing the effects of forcing and damping, including experimental validation with water waves and the Dysthe equation, highlighting phenomena beyond linear predictions.

Findings

Forcing causes the system to move inside the separatrix, damping pulls it outside.

Damping doubles the FPUT recurrence period due to phase shifts.

For modulation frequencies outside the MI-band, growth and decay cycles are observed.

Abstract

We theoretically and experimentally examine the effect of forcing and damping on systems that can be described by the nonlinear Schr\"odinger equation (NLSE), by making use of the phase-space predictions of the three-wave truncation of the spectrum. In the latter, only the fundamental frequency and the upper and lower sidebands are retained. Plane wave solutions to the NLSE exhibit modulation instability (MI) within a frequency band determined by a linear stability analysis. For modulation frequencies inside the MI-band, we experimentally demonstrate that forcing and damping cause a separatrix crossing during the evolution. 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. When the system is damped by the viscosity of the water, 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. Furthermore, for modulation frequencies outside the conventional MI-band, we experimentally demonstrate that contrary to the linear prediction, we do observe a growth and decay cycle of the plane-wave modulation. Finally, we give a theoretical demonstration that forcing the NLSE system can induce symmetry breaking during the evolution.