BRIEF RESEARCH REPORT article
Sec. Interdisciplinary Physics
Volume 9 - 2021 | https://doi.org/10.3389/fphy.2021.631993
Transformation of the Peregrine Breather Into Gray Solitons on a Vertically Sheared Current
- 1Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan
- 2Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, F-13384, Marseille, France
In this Brief Research Report, we show, within the framework of the nonlinear Schrödinger equation in deep water and in the presence of vorticity (vor-NLS), that the Peregrine breather traveling at the free surface of a shear current of slowly varying vorticity may transform into gray solitons.
Within the framework of a fully nonlinear two-dimensional potential solver , computed the temporal evolution of a Stokes wavetrain with a small modulation. They found that the energy becomes focused, at the maximum of modulation, into a short wave packet of large amplitude they called a steep wave event (SWE). They showed that the Peregrine breather, which is an exact solution of the self-focusing nonlinear Schrödinger equation, was the most convenient approximation of the envelope of the SWE. Note that the Peregrine breather can be derived from the Kuznetsov-Ma breather and the Akhmediev breather in the limit of infinite temporal and spatial period (see ) . suggested that the Peregrine breather may provide a useful and simple model for rogue wave events . presented the first experimental observation of the Peregrine breather in a water wave tank. More recently , presented the first ever observation in a wave tank of dark solitons on the surface of water, so demonstrating the probable existence at the sea surface of dark solitons in finite depth for
2 The VOR-NLS
We choose an Eulerian frame
In the case of a wave train propagating at the surface of a deep water flow of constant vorticity
with k the carrier wavenumber and g the gravitational acceleration.
 have compared both the linear intrinsic phase velocities and total energies of gravity waves in the presence of constant vorticity in finite depth and deep water and came to the conclusion that linear gravity waves in finite depth propagating at the surface of a water flow of constant vorticity behave like waves in infinite depth if
FIGURE 1. Dimensionless group velocity deviation between finite and infinite depths as a function of the dispersive parameter for several values of the dimensionless vorticity. In dimensionless terms, the units of acceleration and length are the acceleration of gravity g and
2.1 Effect of Vorticity on the Peregrine Breather and Gray Soliton
The focusing vor-NLS equation admits the Peregrine breather as solution
FIGURE 2. Dimensionless profiles of the Peregrine breather (left) and the gray soliton (right) for several values of the vorticity with
3 Evolution of the Peregrine Breather on Slowly Varying Vortical Flow
Within the framework of Eq. 1, we have performed a numerical simulation of the transformation of a Peregrine breather traveling at the free surface of a vortical water flow whose vorticity varies very slowly.
The parameter σ is chosen such that the average temporal variation of the vorticity along the ramp is of
FIGURE 3. Time evolution of the tranformation of the Peregrine breather into a two-grey soliton with
FIGURE 4. Gray soliton profiles at
Within the framework of the vor-NLS equation in infinite depth we have shown numerically that a Peregrine breather propagating at the free surface of a slowly varying vortical flow may generate gray solitons. The present simulation confirms in a different context the result of  on the existence of dark solitons on the surface of shallow water (
Data Availability Statement
The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.
HH, Conceptualization of the project, Methodology, and formal analysis checking. MA, Formal analysis checking; Software design; Numerical simulations; Validation; Investigation; Visualization. YC, Conceptualization of the project and formal analysis checking. CK, Conceptualization; Methodology; Formal analysis; Investigation; Writing.
Conflict of Interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
2. Chabchoub A, Kimmoun O, Branger H, Hoffmann N, Proment D, Onorato M, et al. Experimental observation of dark solitons on the surface of water. Phys Rev Lett (2013) 110:124101. doi:10.1103/PhysRevLett.110.124101
4. Henderson KL, Peregrine DH, Dold JW. Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear schrödinger equation. Wave Motion (1999) 29:341–61. doi:10.1016/s0165-2125(98)00045-6
Keywords: nonlinear schrödinger equation, water waves, Solitons, breather, vorticity
Citation: Hsu HC, Abid M, Chen YY and Kharif C (2021) Transformation of the Peregrine Breather Into Gray Solitons on a Vertically Sheared Current. Front. Phys. 9:631993. doi: 10.3389/fphy.2021.631993
Received: 23 November 2020; Accepted: 26 January 2021;
Published: 17 March 2021.
Edited by:Amin Chabchoub, The University of Sydney, Australia
Reviewed by:Miguel Onorato, University of Turin, Italy
Ton Van Den Bremer, University of Oxford, United Kingdom
Copyright © 2021 Hsu, Abid, Chen and Kharif. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: C. Kharif, email@example.com