# Bound Coherent Structures Propagating on the Free Surface of Deep Water

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## Abstract

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## 1. Introduction

## 2. Deep Water Wave Equations

## 3. Bound Structures in the Super Compact Dyachenko-Zakharov Equation

#### 3.1. Numerical Method for Obtaining Bound Structures. Experiments with Two Breathers

#### 3.2. Numerical Method for Obtaining Bound Structures. Experiments with Bi-Solitons of the NLSE

## 4. Bound Structures in the Full System of Nonlinear Equations

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

NLSE | nonlinear Schrödinger equation |

SCDZE | super compact Dyachenko-Zakharov equation |

RV (equations) | fully nonlinear equations written in conformal variables |

## References

- Korteweg, D.J.; De Vries, G.X.L.I. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1895**, 39, 422–443. [Google Scholar] [CrossRef] - Kadomtsev, B.B.; Petviashvili, V.I. On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl.
**1970**, 15, 539–541. [Google Scholar] - Boussinesq, J. Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl.
**1872**, 17, 55–108. [Google Scholar] - Davey, A.; Stewartson, K. On three-dimensional packets of surface waves. Proc. R. Soc. London. Math. Phys. Sci.
**1974**, 338, 101–110. [Google Scholar] - Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M. Method for solving the Korteweg-deVries equation. Phys. Rev. Lett.
**1967**, 19, 1095. [Google Scholar] [CrossRef] - Zakharov, V.E.; Shabat, A.B. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Funct. Anal. Its Appl.
**1974**, 8, 226–235. [Google Scholar] [CrossRef] - Zakharov, V.E.; Manakov, S. On the complete integrability of a nonlinear Schrödinger equation. Theor. Math. Phys.
**1974**, 19, 551–559. [Google Scholar] [CrossRef] - Peregrine, D.H. Water waves, nonlinear Schrödinger equations and their solutions. ANZIAM J.
**1983**, 25, 16–43. [Google Scholar] [CrossRef] [Green Version] - Akhmediev, N.; Eleonskii, V.; Kulagin, N. Generation of periodic trains of picosecond pulses in an optical fiber: Exact solutions. Sov. Phys. JETP
**1985**, 62, 894–899. [Google Scholar] - Kuznetsov, E.A. Solitons in a parametrically unstable plasma. DoSSR
**1977**, 236, 575–577. [Google Scholar] - Ma, Y.C. The perturbed plane-wave solutions of the cubic Schrödinger equation. Stud. Appl. Math.
**1979**, 60, 43–58. [Google Scholar] [CrossRef] - Dysthe, K.B. Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. London. Math. Phys. Sci.
**1979**, 369, 105–114. [Google Scholar] - Dyachenko, A.; Kachulin, D.; Zakharov, V.E. Super compact equation for water waves. J. Fluid Mech.
**2017**, 828, 661–679. [Google Scholar] [CrossRef] [Green Version] - Dyachenko, A.I.; Kuznetsov, E.A.; Spector, M.; Zakharov, V.E. Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A
**1996**, 221, 73–79. [Google Scholar] [CrossRef] - Dyachenko, A.I. On the Dynamics of an Ideal Fluid with a Free Surface. Dokl. Math.
**2001**, 63, 115–117. [Google Scholar] - Dyachenko, A.I.; Zakharov, V.E. On the formation of freak waves on the surface of deep water. JETP Lett.
**2008**, 88, 307. [Google Scholar] [CrossRef] - Slunyaev, A. Numerical simulation of “limiting” envelope solitons of gravity waves on deep water. J. Exp. Theor. Phys.
**2009**, 109, 676–686. [Google Scholar] [CrossRef] - Slunyaev, A.; Clauss, G.F.; Klein, M.; Onorato, M. Simulations and experiments of short intense envelope solitons of surface water waves. Phys. Fluids
**2013**, 25, 067105. [Google Scholar] [CrossRef] [Green Version] - Slunyaev, A.; Klein, M.; Clauss, G.F. Laboratory and numerical study of intense envelope solitons of water waves: Generation, reflection from a wall, and collisions. Phys. Fluids
**2017**, 29, 047103. [Google Scholar] [CrossRef] [Green Version] - Kachulin, D.; Dyachenko, A.; Gelash, A. Interactions of coherent structures on the surface of deep water. Fluids
**2019**, 4, 83. [Google Scholar] [CrossRef] [Green Version] - Dyachenko, A.I.; Kachulin, D.; Zakharov, V.E. On the nonintegrability of the free surface hydrodynamics. JETP Lett.
**2013**, 98, 43–47. [Google Scholar] [CrossRef] - Petviashvili, V.I. Equation of an extraordinary soliton. FizPl
**1976**, 2, 469–472. [Google Scholar] - Dyachenko, A.; Kachulin, D.; Zakharov, V.E. Envelope equation for water waves. J. Ocean. Eng. Mar. Energy
**2017**, 3, 409–415. [Google Scholar] [CrossRef] - Kachulin, D.; Gelash, A. On the phase dependence of the soliton collisions in the Dyachenko–Zakharov envelope equation. Nonlinear Process. Geophys.
**2018**, 25, 553–563. [Google Scholar] [CrossRef] [Green Version] - Kachulin, D.; Dyachenko, A.; Dremov, S. Multiple Soliton Interactions on the Surface of Deep Water. Fluids
**2020**, 5, 65. [Google Scholar] [CrossRef] - Shabat, A.; Zakharov, V. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP
**1972**, 34, 62. [Google Scholar] - Zakharov, V.E. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys.
**1968**, 9, 190–194. [Google Scholar] [CrossRef] - Korotkevich, A.; Pushkarev, A.; Resio, D.; Zakharov, V.E. Numerical verification of the weak turbulent model for swell evolution. Eur. J. Mech. B/Fluids
**2008**, 27, 361–387. [Google Scholar] [CrossRef] [Green Version] - Dyachenko, A.; Kachulin, D.; Zakharov, V.E. Collisions of two breathers at the surface of deep water. Nat. Hazards Earth Syst. Sci.
**2013**, 13, 3205–3210. [Google Scholar] [CrossRef] [Green Version] - Frigo, M.; Johnson, S.G. The design and implementation of FFTW3. Proc. IEEE
**2005**, 93, 216–231. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Profiles |c(x)| of bound structures formed by two single breathers with ${\mu}_{ini}=0.2$ and different $\kappa $ in normal (

**a**) and log (

**b**) scales. Profiles plotted after damping is switched off.

**Figure 2.**The examples of initial Nonlinear Schrödinger equation (NLSE) bi-solitons for different $\kappa $ and ${\mu}_{ini}$ = 0.1.

**Figure 3.**Profiles of |c(x)| for bound structures resulting from the initial NLSE bi-solitons with different $\kappa $ and ${\mu}_{ini}$ = 0.1 (

**a**), 0.2 (

**b**), 0.3 (

**c**) and 0.4 (

**d**). As ${\mu}_{ini}$ increases, the influence of $\kappa $ becomes insignificant.

**Figure 4.**The process of splitting the NLSE bi-soliton in two separate solitons for the case of ${\mu}_{ini}$ = 0.2 and $\kappa $ = 1.1: the initial state of bi-soliton (

**a**), the interaction of bi-soliton (

**b**), start of the splitting process of bound state (

**c**), two separate solitons (

**d**).

**Figure 6.**Time evolution of $\left|c\right(x\left)\right|$ in the super compact Dyachenko-Zakharov equation (SCDZE) (

**a**) and $\eta \left(x\right)$ in RV model (

**b**), ${\mu}_{ini}=0.2$, ${\mu}_{fin}\approx $ 0.175, and $\kappa $ = 1.5.

**Figure 7.**Profile of $\eta \left(x\right)$ (

**a**) and ${\eta}^{2}\left(x\right)$ in log scale (

**b**) for bound structure obtained in the model of RV equations with ${\mu}_{fin}\approx 0.25$.

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**MDPI and ACS Style**

Kachulin, D.; Dremov, S.; Dyachenko, A.
Bound Coherent Structures Propagating on the Free Surface of Deep Water. *Fluids* **2021**, *6*, 115.
https://doi.org/10.3390/fluids6030115

**AMA Style**

Kachulin D, Dremov S, Dyachenko A.
Bound Coherent Structures Propagating on the Free Surface of Deep Water. *Fluids*. 2021; 6(3):115.
https://doi.org/10.3390/fluids6030115

**Chicago/Turabian Style**

Kachulin, Dmitry, Sergey Dremov, and Alexander Dyachenko.
2021. "Bound Coherent Structures Propagating on the Free Surface of Deep Water" *Fluids* 6, no. 3: 115.
https://doi.org/10.3390/fluids6030115