# Mathematical Modeling of Rogue Waves: A Survey of Recent and Emerging Mathematical Methods and Solutions

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## Abstract

**:**

## 1. Introduction

## 2. The Non-Linear Schrödinger Equation in the Prediction of Rogue Waves

#### The Solutions to the NLSE

## 3. The Korteweg–de Vries Equation

## 4. The Extended Dysthe Equation

## 5. The MMT Model

## 6. The Hirota Equation

## 7. The Ablowitz–Musslimani Models: Non-Local Rogue Waves

## 8. Conclusions

## Funding

## Conflicts of Interest

## Abbreviations

MMT | Majda–McLaughlin–Tabak |

AKNS | Ablowitz–Kaup–Newell–Segur |

## References

- Lehner, S.; Schulz-Stellenfleth, J.; Niedermeier, A.; Horstmann, J.; Rosenthal, W. Extreme waves detected by satellite borne synthetic aperture radar. In Proceedings of the ASME 2002 21st International Conference on Offshore Mechanics and Arctic Engineering, Oslo, Norway, 23–28 June 2002; American Society of Mechanical Engineers: New York, NY, USA, 2002; pp. 251–256. [Google Scholar]
- Rosenthal, W.; Lehner, S. Rogue waves: Results of the MaxWave project. J. Offshore Mech. Arct. Eng.
**2008**, 130, 021006. [Google Scholar] [CrossRef] - Didenkulova, I.; Slunyaev, A.; Pelinovsky, E.; Kharif, C. Freak waves in 2005. Nat. Hazards Earth Syst. Sci.
**2006**, 6, 1007–1015. [Google Scholar] [CrossRef] [Green Version] - Haver, S. A possible freak wave event measured at the Draupner Jacket January 1 1995. In Proceedings of the 2004 Rogue Waves, Brest, France, 20–22 October 2004; pp. 1–8. [Google Scholar]
- Stansell, P. Distributions of freak wave heights measured in the North Sea. Appl. Ocean Res.
**2004**, 26, 35–48. [Google Scholar] [CrossRef] - Dysthe, K.; Krogstad, H.E.; Müller, P. Oceanic rogue waves. Annu. Rev. Fluid Mech.
**2008**, 40, 287–310. [Google Scholar] [CrossRef] - Weisse, R. Marine Climate and Climate Change: Storms, Wind Waves and Storm Surges; Springer Science & Business Media: Berlin, Germany, 2010. [Google Scholar]
- Solli, D.; Ropers, C.; Koonath, P.; Jalali, B. Optical rogue waves. Nature
**2007**, 450, 1054. [Google Scholar] [CrossRef] [PubMed] - Stenflo, L.; Marklund, M. Rogue waves in the atmosphere. J. Plasma Phys.
**2010**, 76, 293–295. [Google Scholar] [CrossRef] - Moslem, W.; Shukla, P.; Eliasson, B. Surface plasma rogue waves. Europhys. Lett.
**2011**, 96, 25002. [Google Scholar] [CrossRef] - Tlidi, M.; Gandica, Y.; Sonnino, G.; Averlant, E.; Panajotov, K. Self-Replicating spots in the brusselator model and extreme events in the one-dimensional case with delay. Entropy
**2016**, 18, 64. [Google Scholar] [CrossRef] - Kibler, B.; Fatome, J.; Finot, C.; Millot, G.; Dias, F.; Genty, G.; Akhmediev, N.; Dudley, J.M. The Peregrine soliton in nonlinear fibre optics. Nat. Phys.
**2010**, 6, 790–795. [Google Scholar] [CrossRef] [Green Version] - Levi-Civita, T. Determination rigoureuse des ondes permanentes d’ampleur finie. Math. Ann.
**1925**, 93, 264–314. [Google Scholar] [CrossRef] - Nekrasov, A. On waves of permanent type. Izv. Ivanovo-Voznesensk. Politekhn. Inst.
**1921**, 3, 52–65. [Google Scholar] - Smith, R. Giant waves. J. Fluid Mech.
**1976**, 77, 417–431. [Google Scholar] [CrossRef] - Zakharov, V.E. Collapse of Langmuir waves. Sov. Phys. JETP
**1972**, 35, 908–914. [Google Scholar] - Dai, C.Q.; Wang, Y.Y.; Tian, Q.; Zhang, J.F. The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation. Ann. Phys.
**2012**, 327, 512–521. [Google Scholar] [CrossRef] - Akhmediev, N.; Soto-Crespo, J.M.; Ankiewicz, A. Extreme waves that appear from nowhere: on the nature of rogue waves. Phys. Lett. A
**2009**, 373, 2137–2145. [Google Scholar] [CrossRef] - Akhmediev, N.; Ankiewicz, A.; Soto-Crespo, J. Rogue waves and rational solutions of the nonlinear Schrödinger equation. Phys. Rev. E
**2009**, 80, 026601. [Google Scholar] [CrossRef] [PubMed] - Akhmediev, N.; Ankiewicz, A.; Soto-Crespo, J.; Dudley, J.M. Rogue wave early warning through spectral measurements? Phys. Lett. A
**2011**, 375, 541–544. [Google Scholar] [CrossRef] [Green Version] - Chabchoub, A.; Hoffmann, N.; Branger, H.; Kharif, C.; Akhmediev, N. Experiments on wind-perturbed rogue wave hydrodynamics using the Peregrine breather model. Phys. Fluids
**2013**, 25, 101704. [Google Scholar] [CrossRef] [Green Version] - Cousins, W.; Sapsis, T.P. Quantification and prediction of extreme events in a one-dimensional nonlinear dispersive wave model. Phys. D Nonlinear Phenom.
**2014**, 280, 48–58. [Google Scholar] [CrossRef] [Green Version] - Cousins, W.; Sapsis, T.P. Unsteady evolution of localized unidirectional deep-water wave groups. Phys. Rev. E
**2015**, 91, 063204. [Google Scholar] [CrossRef] [PubMed] - Cousins, W.; Sapsis, T.P. Reduced-order precursors of rare events in unidirectional nonlinear water waves. J. Fluid Mech.
**2016**, 790, 368–388. [Google Scholar] [CrossRef] [Green Version] - Tlidi, M.; Panajotov, K. Two-dimensional dissipative rogue waves due to time-delayed feedback in cavity nonlinear optics. Chaos Interdiscip. J. Nonlinear Sci.
**2017**, 27, 013119. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lugiato, L.A.; Lefever, R. Spatial dissipative structures in passive optical systems. Phys. Rev. Lett.
**1987**, 58, 2209. [Google Scholar] [CrossRef] [PubMed] - Panajotov, K.; Clerc, M.G.; Tlidi, M. Spatiotemporal chaos and two-dimensional dissipative rogue waves in Lugiato-Lefever model. Eur. Phys. J. D
**2017**, 71, 176. [Google Scholar] [CrossRef] - Akhmediev, N.; Kibler, B.; Baronio, F.; Belić, M.; Zhong, W.P.; Zhang, Y.; Chang, W.; Soto-Crespo, J.M.; Vouzas, P.; Grelu, P.; et al. Roadmap on optical rogue waves and extreme events. J. Opt.
**2016**, 18, 063001. [Google Scholar] [CrossRef] [Green Version] - Dai, C.; Wang, Y.; Yan, C. Chirped and chirp-free self-similar cnoidal and solitary wave solutions of the cubic-quintic nonlinear Schrödinger equation with distributed coefficients. Opt. Commun.
**2010**, 283, 1489–1494. [Google Scholar] [CrossRef] - Haghgoo, S.; Ponomarenko, S.A. Self-similar pulses in coherent linear amplifiers. Optics Express
**2011**, 19, 9750–9758. [Google Scholar] [CrossRef] [PubMed] - Kruglov, V.; Peacock, A.; Harvey, J. Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients. Phys. Rev. Lett.
**2003**, 90, 113902. [Google Scholar] [CrossRef] [PubMed] - Kruglov, V.; Peacock, A.; Harvey, J. Exact solutions of the generalized nonlinear Schrödinger equation with distributed coefficients. Phys. Rev. E
**2005**, 71, 056619. [Google Scholar] [CrossRef] [PubMed] - Fermann, M.; Kruglov, V.; Thomsen, B.; Dudley, J.; Harvey, J. Self-similar propagation and amplification of parabolic pulses in optical fibers. Phys. Rev. Lett.
**2000**, 84, 6010. [Google Scholar] [CrossRef] [PubMed] - Hamedi, H.R. Optical bistability and multistability via magnetic field intensities in a solid. Appl. Opt.
**2014**, 53, 5391–5397. [Google Scholar] [CrossRef] [PubMed] - Munk, W.; Snodgrass, F. Measurements of southern swell at Guadalupe Island. Deep Sea Res.
**1957**, 4, 272–286. [Google Scholar] [CrossRef] - Kruglov, V.; Peacock, A.; Dudley, J.; Harvey, J. Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers. Opt. Lett.
**2000**, 25, 1753–1755. [Google Scholar] [CrossRef] [PubMed] - Osborne, A.R.; Onorato, M.; Serio, M. The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains. Phys. Lett. A
**2000**, 275, 386–393. [Google Scholar] [CrossRef] - 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] - Serkin, V.; Hasegawa, A.; Belyaeva, T. Nonautonomous solitons in external potentials. Phys. Rev. Lett.
**2007**, 98, 074102. [Google Scholar] [CrossRef] [PubMed] - Dai, C.Q.; Wang, D.S.; Wang, L.L.; Zhang, J.F.; Liu, W. Quasi-two-dimensional Bose–Einstein condensates with spatially modulated cubic–quintic nonlinearities. Ann. Phys.
**2011**, 326, 2356–2368. [Google Scholar] [CrossRef] - Peregrine, D. Water waves, nonlinear Schrödinger equations and their solutions. ANZIAM J.
**1983**, 25, 16–43. [Google Scholar] [CrossRef] - Zakharov, V.; Shabat, A. Interaction between solitons in a stable medium. Sov. Phys. JETP
**1973**, 37, 823–828. [Google Scholar] - Matveev, V.B.; Matveev, V. Darb. Trans. Solitons; Springer-Verlag: Berlin, Germany, 1991. [Google Scholar]
- Akhmediev, N.; Korneev, V. Modulation instability and periodic solutions of the nonlinear Schrödinger equation. Theor. Math. Phys.
**1986**, 69, 1089–1093. [Google Scholar] [CrossRef] - Dysthe, K.B.; Trulsen, K. Note on breather type solutions of the NLS as models for freak-waves. Phys. Scr.
**1999**, 1999, 48. [Google Scholar] [CrossRef] - Voronovich, V.V.; Shrira, V.I.; Thomas, G. Can bottom friction suppress ‘freak wave’formation? J. Fluid Mech.
**2008**, 604, 263–296. [Google Scholar] [CrossRef] - Benjamin, T.B.; Feir, J. The disintegration of wave trains on deep water Part 1. Theory. J. Fluid Mech.
**1967**, 27, 417–430. [Google Scholar] [CrossRef] - Bespalov, V.; Talanov, V. Filamentary structure of light beams in nonlinear liquids. ZhETF Pisma Redaktsiiu
**1966**, 3, 471. [Google Scholar] - Kim, D.S.; Markowsky, G.; Lee, S.G. Mobile Sage-Math for linear algebra and its application. Electron. J. Math. Technol.
**2010**, 4, 285–298. [Google Scholar] - SageMath Mathematics Software, Version 6.5. 2015. Available online: http://www.sagemath.org/ (accessed on 5 June 2017).
- Kharif, C.; Pelinovsky, E. Physical mechanisms of the rogue wave phenomenon. Eur. J. Mech. B Fluids
**2003**, 22, 603–634. [Google Scholar] [CrossRef] [Green Version] - Matsuno, Y. Bilinear Transformation Method; Elsevier: New York, NY, USA, 1984. [Google Scholar]
- Hirota, R. A new form of Bäcklund transformations and its relation to the inverse scattering problem. Prog. Theor. Phys.
**1974**, 52, 1498–1512. [Google Scholar] [CrossRef] - Matveev, V.B. Positons: Slowly decreasing analogues of solitons. Theor. Math. Phys.
**2002**, 131, 483–497. [Google Scholar] [CrossRef] - Osborne, A. Soliton physics and the periodic inverse scattering transform. Phys. D Nonlinear Phenom.
**1995**, 86, 81–89. [Google Scholar] [CrossRef] - Osborne, A. Solitons in the periodic Korteweg–de Vries equation, the FTHETA-function representation, and the analysis of nonlinear, stochastic wave trains. Phys. Rev. E
**1995**, 52, 1105. [Google Scholar] [CrossRef] - Dysthe, K.B. Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. R. Soc. Lond. A
**1979**, 369, 105–114. [Google Scholar] [CrossRef] - Longuet-Higgins, M. The instability of gravity waves of infinite amplitude in deep water. II. Subharmonics. Proc. R. Soc. Lond. A
**1978**, 360, 489–506. [Google Scholar] [CrossRef] - Trulsen, K.; Dysthe, K. Freak waves—A three-dimensional wave simulation. In Proceedings of the 21st Symposium on Naval Hydrodynamics, Trondheim, Norway, 24–28 June 1996; National Academy Press: Washington, DC, USA, 1997; Volume 550, p. 558. [Google Scholar]
- Trulsen, K.; Dysthe, K.B. A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion
**1996**, 24, 281–289. [Google Scholar] [CrossRef] - Trulsen, K.; Kliakhandler, I.; Dysthe, K.B.; Velarde, M.G. On weakly nonlinear modulation of waves on deep water. Phys. Fluids
**2000**, 12, 2432–2437. [Google Scholar] [CrossRef] - Majda, A.; McLaughlin, D.; Tabak, E. A one-dimensional model for dispersive wave turbulence. J. Nonlinear Sci.
**1997**, 7, 9–44. [Google Scholar] [CrossRef] - Pushkarev, A.; Zakharov, V. Quasibreathers in the MMT model. Phys. D Nonlinear Phenom.
**2013**, 248, 55–61. [Google Scholar] [CrossRef] [Green Version] - Zakharov, V.; Dias, F.; Pushkarev, A. One-dimensional wave turbulence. Phys. Rep.
**2004**, 398, 1–65. [Google Scholar] [CrossRef] - Zakharov, V.; Guyenne, P.; Pushkarev, A.; Dias, F. Wave turbulence in one-dimensional models. Phys. D Nonlinear Phenom.
**2001**, 152, 573–619. [Google Scholar] [CrossRef] [Green Version] - Komen, G.J.; Cavaleri, L.; Donelan, M. Dynamics and Modelling of Ocean Waves; Cambridge University Press: Cambridge, UK, 1996. [Google Scholar]
- Lavrenov, I. The wave energy concentration at the Agulhas current off South Africa. Nat. hazards
**1998**, 17, 117–127. [Google Scholar] [CrossRef] - Zakharov, V.; Kuznetsov, E. Optical solitons and quasisolitons. J. Exp. Theor. Phys.
**1998**, 86, 1035–1046. [Google Scholar] [CrossRef] - Tao, Y.; He, J. Multisolitons, breathers, and rogue waves for the Hirota equation generated by the Darboux transformation. Phys. Rev. E
**2012**, 85, 026601. [Google Scholar] [CrossRef] [PubMed] - Ablowitz, M.J.; Kaup, D.J.; Newell, A.C.; Segur, H. Method for solving the sine-Gordon equation. Phys. Rev. Lett.
**1973**, 30, 1262. [Google Scholar] [CrossRef] - Ankiewicz, A.; Soto-Crespo, J.; Akhmediev, N. Rogue waves and rational solutions of the Hirota equation. Phys. Rev. E
**2010**, 81, 046602. [Google Scholar] [CrossRef] [PubMed] - He, J.; Zhang, H.; Wang, L.; Porsezian, K.; Fokas, A. Generating mechanism for higher-order rogue waves. Phys. Rev. E
**2013**, 87, 052914. [Google Scholar] [CrossRef] [PubMed] - Ablowitz, M.J.; Musslimani, Z.H. Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett.
**2013**, 110, 064105. [Google Scholar] [CrossRef] [PubMed] - Ablowitz, M.J.; Musslimani, Z.H. Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity
**2016**, 29, 915. [Google Scholar] [CrossRef] - Ablowitz, M.J.; Musslimani, Z.H. Integrable nonlocal nonlinear equations. Stud. Appl. Math.
**2017**, 139, 7–59. [Google Scholar] [CrossRef] - Ablowitz, M.J.; Musslimani, Z.H. Integrable discrete P T symmetric model. Phys. Rev. E
**2014**, 90, 032912. [Google Scholar] [CrossRef] [PubMed] - Musslimani, Z.; Makris, K.G.; El-Ganainy, R.; Christodoulides, D.N. Optical Solitons in P T Periodic Potentials. Phys. Rev. Lett.
**2008**, 100, 030402. [Google Scholar] [CrossRef] [PubMed] - Ablowitz, M.J.; Chakravarty, S.; Takhtajan, L.A. A self-dual Yang-Mills hierarchy and its reductions to integrable systems in 1+1 and 2+1 dimensions. Commun. Math. Phys.
**1993**, 158, 289–314. [Google Scholar] [CrossRef] - Ablowitz, M.J.; Luo, X.D.; Musslimani, Z.H. Inverse scattering transform for the nonlocal nonlinear Schrödinger equation with nonzero boundary conditions. J. Math. Phys.
**2018**, 59, 011501. [Google Scholar] [CrossRef] [Green Version] - Yang, B.; Yang, J. General rogue waves in the nonlocal PT-symmetric nonlinear Schrodinger equation. arXiv, 2017; arXiv:1711.05930. [Google Scholar]
- Yu, F. Dynamics of nonautonomous discrete rogue wave solutions for an Ablowitz–Musslimani equation with PT-symmetric potential. Chaos Interdiscip. J. Nonlinear Sci.
**2017**, 27, 023108. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The laser readings of the most extreme rogue wave registered (

**top**), which hit the the North Alwyn platform east of Shetland [5], reaching 18.04 m and a ratio of 3.19× the surrounding waves. (

**Bottom**) The New Years wave registered in 1995 on the Draupner platform in the North Sea.

**Figure 4.**The root functions for the standard NLSE and the inhomogenous variable coefficient NLSE. (

**Top**) The seed impulse used in the solutions to the standard NLSE, ${e}^{ix}$ [19]. (

**Bottom**) A generic form of the seed impulse used in the solutions of the inhomogenous variable coefficient NLSE [17] $f\left(x\right)={e}^{i(1-{x}^{2}/2)+x}$. Real part (blue) and imaginary part (red).

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Manzetti, S.
Mathematical Modeling of Rogue Waves: A Survey of Recent and Emerging Mathematical Methods and Solutions. *Axioms* **2018**, *7*, 42.
https://doi.org/10.3390/axioms7020042

**AMA Style**

Manzetti S.
Mathematical Modeling of Rogue Waves: A Survey of Recent and Emerging Mathematical Methods and Solutions. *Axioms*. 2018; 7(2):42.
https://doi.org/10.3390/axioms7020042

**Chicago/Turabian Style**

Manzetti, Sergio.
2018. "Mathematical Modeling of Rogue Waves: A Survey of Recent and Emerging Mathematical Methods and Solutions" *Axioms* 7, no. 2: 42.
https://doi.org/10.3390/axioms7020042