Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping
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ORIGINAL PAPER
Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping D. Eeltink
· A. Armaroli · C. Luneau · H. Branger · M. Brunetti · J. Kasparian
Received: 27 May 2020 / Accepted: 19 October 2020 © The Author(s) 2020
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 phasespace 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 Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11071-020-06043-1) contains supplementary material, which is available to authorized users. D. Eeltink (B) · A. Armaroli · M. Brunetti · J.Kasparian Group of Applied Physics and Institute for Environmental Sciences, University of Geneva, Geneva, Switzerland e-mail: [email protected] H. Branger IRPHE, AMU, CNRS, ECM, Marseille, France C. Luneau Institut Pytheas, AMU,CNRS,IRD, Marseille, France
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. Keywords Phase-shift · NLS · Gravity surface waves · Separatrix crossing · Symmetry breaking
1 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 [1–4]. Elementary solutions of the NLSE include plane waves, solitons and breathers. 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 therefore modulates the amplitude of the plane wave, generating a train of sharp pulses [6]. Remarkably, the MI can exhibit cyclic behavior, known as the Fermi–Pasta–Ulam– Tsingou (FPUT) recurrence [7]: Despite complex nonlinear dynamics, the system returns to its initial condition.
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We are interested in the effect of forcing and damping on t
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