Research Article

Year 2024,
Volume: 21 Issue: 1, 6 - 17, 01.05.2024
### Abstract

### Keywords

### References

This article deals with the dynamical behavior of weakly nonlinear gravity waves propagating including the viscosity and surfactant on the ocean free surface. Through the bifurcation method, we could predict the nature of solutions of the nonlinear Schrödinger equation (NLSE) and reduce it to the nonlinear ordinary differential equation, easily solvable. Then, bright soliton, dark soliton, and Jacobi elliptic functions solutions of this NLSE under the viscosity and surfactant effect have been derived. These obtained results are central in hydrodynamics and can predict physical phenomena such as gravity wave propagation in deep water. Moreover, they allow to enhance the decision-making process and the acquisition of radar and lidar data on the ocean surface.

nonlinear Schrödinger equation viscosity Nonlinear gravity waves Surfactant and Bifurcation Method

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- A. R. Seadawy, “Exact solutions of a two-dimensional nonlinear Schrödinger equation,” Applied Mathematics Letters, vol. 25, no. 4, pp. 687-691, 2012.
- D. H. PEREGRINE, “Water waves, nonlinear Schrodinger equations and their solutions,” J. Austral. Math. Soc. Ser. B, vol. 25, pp. 16-43, 1983

Year 2024,
Volume: 21 Issue: 1, 6 - 17, 01.05.2024
### Abstract

### References

- D. Augustin, M. E. Honore, N. C. Marie, and C. M. Biouele, “Application of Benjamin-Feir Equations to Tornadoes’ Rogue Waves Modulational Instability in oceans,” Inter. J. Phys. Sci., vol. 7, no. 46, pp. 6053-6061, 2012.
- T. R. Akylas, “Three-dimensional long water-wave phenomena,” Annu. Rev. Fluid Mech., vol. 26, pp. 191–210, 1994.
- C. Mei, The applied dynamics of ocean surface waves, World Scientific Pub Co Inc., 1989.
- D. Augustin, and M. B. Cesar, “Relationship between Sea Surface Single Carrier Waves and Decreasing Pressures of Atmosphere Lower Boundary” Open Journal of Marine Science, vol. 5, no. 1, pp. 45-54, 2015.
- D. Augustin, N. C. Marie, and M. B. Cesar, “Correlation between Atmosphere’s Low-Pressure Systems and Ocean Surface Gravity Waves Formation: Geneses and Predictability,” American Journal of Geophysics, Geochemistry and Geosystems, vol. 6, no. 3, pp. 96-101, 2020.
- T. Gao, P. A. Milewski, D.T. Papageorgiou, J-M. Vanden-Broeck, “Dynamics of fully nonlinear capillary–gravity solitary waves under normal electric fields,” Journal of Engineering Mathematics, vol. 108, pp. 107–122, 2018.
- M. J. Ablowitz, and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.
- A. A. Nwatchok Stéphane, D. Augustin, and M. B. César, “Extended (G’/G) method applied to the modified nonlinear Schrödinger equation in the case of ocean rogue waves,” Open J. Mar. Sci., vol. 4, no. 4, pp. 246–256, 2014.
- J-M. Vanden-Broeck, Gravity–Capillary Free-Surface Flows, Cambridge University Press, Cambridge, 2010.
- A. Bekir, M. S. M. Shehata, and E. H. M. Zahran, “Comparison between the exact solutions of three distinct shallow water equations using the painlevé approach and its numerical solutions,” Russian J. Nonlinear Dynamics, vol. 16, no. 3, pp. 463–477, 2020.
- D. Augustin, T. N. Nkomom, and M. B. Cesar, “Application of Stationary Phase Method to Wind Stress and Breaking Impacts on Ocean Relatively High Waves,” Open Journal of Marine Science, vol. 4, no. 1, pp. 18-24, 2014.
- C. C. Mei, M. Stiassnie, and D. K.- P Yue, Theory and Applications of Ocean Surface Waves, Advanced Series on Ocean Engineering, vol. 23, Copyright © 2005 by World Scientific Publishing Co. Pte. Ltd, 2005.
- A. R. Osborne, Nonlinear Ocean Waves and the Inverse Scattering Transform, International Geophysics Series, Elsevier, vol. 97, 2010.
- D. Augustin, “Etude des instabilités modulationnelles des ondes de gravité générées par les interactions atmosphèreNappe d’eau étendue et profonde: le cas des vagues scélérates ou meurtrières,” PhD Thesis, University of Yaounde 1, pp. 166, 2018.
- T. J. Bridges, and F. Dias, “Spatially quasi periodic capillary-gravity waves,” Contemp. Math., vol. 200, pp. 31–4, 1996.
- F. Dias, C. Kharif, “Nonlinear gravity and capillary-gravity waves,” Annu. Rev. Fluid Mech., vol. 31, pp. 301–46, 1999.
- M. S. Longuet-Higgins, “Capillary–gravity waves of solitary type and envelope solitons on deep water,” Journal of Fluid Mechanics, vol. 252, pp. 703–711, 1993.
- Z. Wang and P. A. Milewski, “Dynamics of gravity–capillary solitary waves in deep water,” Journal of Fluid Mechanics, vol. 708, pp. 480–501, 2012.
- D. Augustin, and M. B. César, “Gravity waves’ modulational instability under the effect of drag coefficient in the ocean,” Physica Scripta, vol. 98, no. 12, pp. 125014, 2023.
- Z. Wang, “Stability and dynamics of two-dimensional fully nonlinear gravity–capillary solitary waves in deep water,” Journal of Fluid Mechanics, vol. 809, pp. 530–552, 2016.
- S. W JOO, A. F. Messiter, and W.W. Schultz, “Evolution of weakly nonlinear water waves in the presence of viscosity and surfactant,” J. Fluid Mech., vol. 229, pp. 135-158, 1991.
- G. S. Lapham, D. R. Dowling, and W. W. Schultz, “Linear and nonlinear gravity-capillary water waves with a soluble surfactant,” Experiments in Fluids, vol. 30, pp. 448-457, 2001.
- D. Augustin, and M. B. César, “Nonlinear Evolution of Gravity Waves on the Surface Deep-water Under the Action of Viscosity and Surfactant,” American Journal of Geophysics, Geochemistry and Geosystems, vol. 6, no. 2, pp. 50-57, 2020.
- T. F. Waffo, T. J. B. Gonpe, A. Chamgoue, M. N. C. Tsague, F. Kenmogne, and L. Nana, “Analytic Solutions of 2D Cubic Quintic Complex Ginzburg-Landau Equation,” Journal of Applied Mathematics, vol. 2023, no. 2549560, 2023.
- D. Bienvenue, A. Houwe, and H. Rezazadeh, “New explicit and exact traveling waves solutions to the modified complex Ginzburg Landau equation,” Opt. Quant Electron, vol. 54, no.237, 2020.
- A. R. Seadawy, “Exact solutions of a two-dimensional nonlinear Schrödinger equation,” Applied Mathematics Letters, vol. 25, no. 4, pp. 687-691, 2012.
- D. H. PEREGRINE, “Water waves, nonlinear Schrodinger equations and their solutions,” J. Austral. Math. Soc. Ser. B, vol. 25, pp. 16-43, 1983

There are 27 citations in total.

Primary Language | English |
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Subjects | Geophysical and Environmental Fluid Flows |

Journal Section | Articles |

Authors | |

Publication Date | May 1, 2024 |

Submission Date | January 11, 2024 |

Acceptance Date | April 15, 2024 |

Published in Issue | Year 2024 Volume: 21 Issue: 1 |