Dear all,
We are pleased to invite you to the next WISE Zoominar on the topic of Modulational instability on 19/09/2024 at 17.00-18.00 Paris time. The schedule of the additional zoominars for 2024/2025 will be released later.
Join us via https://tudelft.zoom.us/my/wisezoominars on 19th September 17.00-18.00 Paris time!
First speaker: Raphael Stuhlmeier, University of Plymouth
Title: Phase coherence, steady states, and new solutions in the Benjamin-Feir instability: a dynamical systems approach
Abstract: The Benjamin-Feir instability is the most prominent special case of the four-wave interaction which governs energy exchange of waves on the surface of deep water. It has classically been treated via linear stability analysis, starting from the nonlinear Schrödinger equation (Zakharov, 1968) or the reduced Zakharov equation (Crawford et al, 1981). Rather than linearise, I will discuss how to treat the resonant interaction equations of the Zakharov formulation using simple dynamical systems techniques. These elucidate how energy exchange leads to stationarity of dynamic phase, clarify the role of steady-state solutions found recently by Liao and co-workers, and provide new, explicit solutions to the Zakharov equation which are discrete analogues of the Akhmediev breather. We will see how vestiges of energy exchange persist even when the system has stabilised, and that the largest growth rate does not lead to the most growth, facts which are not apparent from linear theory alone.
References:
Crawford, D.R., Lake, B.M., Saffman, P.G. & Yuen, H.C. 1981 Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech. 105, 177–191.
Andrade, D. & Stuhlmeier, R. 2023 The nonlinear Benjamin–Feir instability – Hamiltonian dynamics, discrete breathers and steady solutions. J. Fluid Mech. 958, A17.
Zakharov, V.E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.
Second speaker: Andrey A. Gelash, EPFL
Title: Bi-solitons on the surface of a deep fluid: an inverse scattering transform approach
Abstract: I will present results of our recent paper [A. Gelash et al, Bi-solitons on the surface of a deep fluid: an inverse scattering transform perspective based on perturbation theory. Phys. Rev.
Lett., 2024], where we investigate theoretically and numerically the dynamics of long-living
oscillating coherent structures – bi-solitons – in the exact and approximate models for waves
on the free surface of deep water. To elucidate the long-living bi-soliton complex nature we
apply an analytical-numerical approach based on the perturbation theory and the inverse
scattering transform (IST) for the one-dimensional focusing nonlinear Schrödinger equation
model. We observe a periodic energy and momentum exchange between solitons and
continuous spectrum radiation resulting in repetitive oscillations of the coherent structure.
We find that soliton eigenvalues oscillate on stable trajectories experiencing a slight drift on a
scale of hundreds of the structure oscillation periods so that the eigenvalue dynamics is in
good agreement with predictions of the IST perturbation theory. Based on the obtained
results, we conclude that the IST perturbation theory justifies the existence of the long-living
bi-solitons on the surface of deep water which emerge as a result of a balance between their
dominant solitonic part and a portion of continuous spectrum radiation.
Please note that WISE Zoominars including the Q&A will be recorded and posted on the WISE YouTube Channel afterwards (https://www.youtube.com/@wisezoominars). By participating, you consent to any information you share to be included in the recording and shared.
Best wishes,
Tripp, Alvise, Morteza and Ton
(The WISE Zoominar organizing committee)