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

D Eeltink,J Kasparian,H Branger,C Luneau,M Brunetti,A Armaroli

doi:10.1007/s11071-020-06043-1

Abstract

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.

Highlights

The nonlinear Schrödinger equation (NLSE) describes the propagation of the field envelope in many different systems, for instance in optical fibers, Bose– Einstein condensates, water waves, and Langmuir waves in hot plasmas [1,2,3,4]
1) Our main finding is the experimental demonstration that when wind forcing is sufficient to overcome the viscous damping, the system is attracted toward P1 solutions, inducing a separatrix crossing during the evolution
4) We experimentally show that while no growth is expected outside the modulation instability (MI)-band based on a linear stability analysis, there is a growth and decay pattern of P2 solutions here, confirming the theoretical findings in Ref. [15]

Summary

Introduction

The nonlinear Schrödinger equation (NLSE) describes the propagation of the field envelope in many different systems, for instance in optical fibers, Bose– Einstein condensates, water waves, and Langmuir waves in hot plasmas [1,2,3,4]. The plane wave solution is subject to modulation instability (MI) [5]: the linear stability analysis of the NLSE reveals that within a certain frequency bandwidth, a modulation—perturbation—to the plane wave will grow exponentially. It modulates the amplitude of the plane wave, generating a train of sharp pulses [6]. Many systems that can be described by the NLSE naturally undergo dissipation, not many allow to be forced [8]. Water waves can undergo both: While viscosity is a natural source of damping, wind can provide forcing

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Conclusion