# Transformation of the Peregrine Breather Into Gray Solitons on a Vertically Sheared Current

^{1}Department of Marine Environment and Engineering, National Sun Yat-sen University, Kaohsiung, Taiwan^{2}Aix 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.

## 1 Introduction

Within the framework of a fully nonlinear two-dimensional potential solver [4], 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 [7]) [3]. suggested that the Peregrine breather may provide a useful and simple model for rogue wave events [1]. presented the first experimental observation of the Peregrine breather in a water wave tank. More recently [2], 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 *k* is the carrier wavenumber and *h* the water depth. They found a good agreement between the experimental soliton and the dark soliton solution of the defocusing nonlinear Schrödinger equation. Dark solitons occur as envelope holes. Rogue waves are large-amplitude waves which occurs at the sea surface suddenly without warning. Such waves are accompanied by deep holes before and/or after the largest crest. Another possible mechanism of these holes in the ocean could be dark or gray soliton generation. There is an abundant literature on the interaction between surface water waves and spatially uniform currents. In the real ocean, currents are never uniform. Spatially varying currents may affect strongly the water wave behavior. Herein, we paid attention to the evolution of a Peregrine breather propagating at the surface of a vertically sheared current. In this Brief Research Report we propose, based on the NLS equation in infinite depth, a physical mechanism of gray soliton generation from a Peregrine breather evolving on slowly varying underlying water vorticity. Section 2 is devoted to the presentation of the vor-NLS equation in the presence of constant vorticity derived by [8]. The vor-NLS equation is self-focusing or defocusing according to the magnitude of the vorticity. Vorticity effect on the soliton solutions of the vor-NLS of self-focusing and defocusing types is displayed. A numerical simulation of the transformation of the Peregrine breather propagating at the free surface of a water flow of slowly varying vorticity is presented in section 3. A conclusion is given in section 4.

## 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

with *k* the carrier wavenumber and *g* the gravitational acceleration.

[5] 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

Note that

where *m* fixes the minimum of amplitude at the center of the soliton. For

**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 *κ* is a wavenumber in the Fourier space and *ν* the loss of the energy of the carrier wave is of the order of 1% over approximately 1,000 periods of time evolution of the carrier. The numerical simulation has been run in dimensionless units with *T* is the period of the background of the Peregrine breather. During its propagation along the ramp the width of the breather increases whereas its amplitude decreases. At

**FIGURE 3**. Time evolution of the tranformation of the Peregrine breather into a two-grey soliton with *T* is the period of the Peregrine breather background.

**FIGURE 4**. Gray soliton profiles at **(left)** and **(right)**. The solid lines correspond to the numerical simulation whereas the circles correspond the analytical solution of Eq. 4 for

## 4 Conclusion

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 [2] 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.

## Author Contributions

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.

## References

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6. Lo E, Mei CC. A numerical study of water-wave modulation based on a higher-order nonlinear schrödinger equation. *J Fluid Mech* (1985) 150:395–416. doi:10.1017/s0022112085000180

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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, AustraliaReviewed by:

Miguel Onorato, University of Turin, ItalyTon 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, kharif@irphe.univ-mrs.fr