Stochastic solutions of the geophysical KdV equation: Numerical simulations and white noise impact

Areej A. Almoneef; Abd-Allah Hyder; Mohamed A. Barakat; Abdelrheem M. Aly; Areej A. Almoneef; Abd-Allah Hyder; Mohamed A. Barakat; Abdelrheem M. Aly

doi:10.3934/math.2025269

AIMS Mathematics

2025, Volume 10, Issue 3: 5859-5879. doi: 10.3934/math.2025269

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Research article

Stochastic solutions of the geophysical KdV equation: Numerical simulations and white noise impact

1.
Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, College of Science, King Khalid University, Box 9004, 61413, Abha, Saudi Arabia
3.
Department of Basic Science, University College of Alwajh, University of Tabuk, Saudi Arabia

Received: 09 January 2025 Revised: 27 February 2025 Accepted: 04 March 2025 Published: 17 March 2025
MSC : 35Q53, 60H15, 37K40

This research explored stochastic soliton and periodic wave (SPW) solutions for the geophysical Korteweg-de Vries (KdV) equation with variable coefficients, incorporating the effects of Earth's rotation, fluid stratification, and topographical variations. The classical KdV equation, widely used to model nonlinear wave propagation, was extended to describe geophysical wave dynamics in atmospheric and oceanic systems. Exact solutions for both the deterministic and Wick-type stochastic (W-TS) forms of the geophysical KdV equation were obtained using white noise (WN) theory, the Hermite transform (HT), and the exp-function method. By employing the HT, the stochastic equation was transformed into a deterministic counterpart, facilitating the derivation of novel SPW solutions expressed as rational functions involving exponential terms. The inverse HT is then applied to retrieve stochastic SPW solutions under Gaussian WN conditions. Numerical analysis highlights the influence of Brownian motion (B-M) on the formation and behavior of SPWs in geophysical settings. Additionally, numerical simulations illustrate how random fluctuations affect wave stability and evolution, offering deeper insights into nonlinear wave interactions in oceanic and atmospheric environments.
- geophysical KdV equation,
- soliton and periodic wave solutions,
- white noise effects,
- exp-function method,
- numerical simulations
Citation: Areej A. Almoneef, Abd-Allah Hyder, Mohamed A. Barakat, Abdelrheem M. Aly. Stochastic solutions of the geophysical KdV equation: Numerical simulations and white noise impact[J]. AIMS Mathematics, 2025, 10(3): 5859-5879. doi: 10.3934/math.2025269

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Abstract

This research explored stochastic soliton and periodic wave (SPW) solutions for the geophysical Korteweg-de Vries (KdV) equation with variable coefficients, incorporating the effects of Earth's rotation, fluid stratification, and topographical variations. The classical KdV equation, widely used to model nonlinear wave propagation, was extended to describe geophysical wave dynamics in atmospheric and oceanic systems. Exact solutions for both the deterministic and Wick-type stochastic (W-TS) forms of the geophysical KdV equation were obtained using white noise (WN) theory, the Hermite transform (HT), and the exp-function method. By employing the HT, the stochastic equation was transformed into a deterministic counterpart, facilitating the derivation of novel SPW solutions expressed as rational functions involving exponential terms. The inverse HT is then applied to retrieve stochastic SPW solutions under Gaussian WN conditions. Numerical analysis highlights the influence of Brownian motion (B-M) on the formation and behavior of SPWs in geophysical settings. Additionally, numerical simulations illustrate how random fluctuations affect wave stability and evolution, offering deeper insights into nonlinear wave interactions in oceanic and atmospheric environments.

References

[1]	J. W. Miles, The Korteweg-de Vries equation: A historical essay, J. Fluid Mech., 106 (1981), 131–147.
[2]	K. R. Khusnutdinova, Y. A. Stepanyants, M. R. Tranter, Soliton solutions to the fifth-order Korteweg-de Vries equation and their applications to surface and internal water waves, Phys. Fluids, 30 (2018), 022104. https://doi.org/10.1063/1.5009965 doi: 10.1063/1.5009965
[3]	A. A. Hyder, A. H. Soliman, C. Cesarano, M. A. Barakat, Solving Schrödinger-Hirota equation in a stochastic environment and utilizing generalized derivatives of the conformable type, Mathematics, 9 (2021), 2760. https://doi.org/10.3390/math9212760 doi: 10.3390/math9212760
[4]	A. A. Hyder, M. A. Barakat, General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics, Phys. Scripta, 95 (2020), 045212. https://doi.org/10.1088/1402-4896/ab6526 doi: 10.1088/1402-4896/ab6526
[5]	Attaullah, M. Shakeel, M. K. Alaoui, A. M. Zidan, N. A. Shah, Closed form solutions for the generalized fifth-order KDV equation by using the modified exp-function method, J. Ocean Eng. Sci., In Press, 2022. https://doi.org/10.1016/j.joes.2022.06.037
[6]	K. R. Helfrich, W. K. Melville, Long nonlinear internal waves, Ann. Rev. Fluid Mech., 38 (2006), 395–425. https://doi.org/10.1146/annurev.fluid.38.050304.092129 doi: 10.1146/annurev.fluid.38.050304.092129
[7]	R. Grimshaw, E. Pelinovsky, T. Talipova, The modified Korteweg-de Vries equation in the theory of large-amplitude internal waves, Nonlinear Proce. Geoph., 4 (1997), 237–250. https://doi.org/10.5194/npg-4-237-1997 doi: 10.5194/npg-4-237-1997
[8]	S. T. R. Rizvi, A. R. Seadawy, F. Ashraf, M. Younis, H. Iqbal, D. Baleanu, Lump and interaction solutions of a geophysical Korteweg-de Vries equation, Results Phys., 19 (2020), 103661. https://doi.org/10.1016/j.rinp.2020.103661 doi: 10.1016/j.rinp.2020.103661
[9]	E. H. M. Zahran, A. Bekir, New unexpected behavior to the soliton arising from the geophysical Korteweg-de Vries equation, Mod. Phys. Lett. B, 36 (2022), 2150623. https://doi.org/10.1142/s0217984921506235 doi: 10.1142/s0217984921506235
[10]	P. L. F. Liu, X. M. Wang, A multi-layer model for nonlinear internal wave propagation in shallow water, J. Fluid Mech., 695 (2012), 341–365. https://doi.org/10.1017/jfm.2012.24 doi: 10.1017/jfm.2012.24
[11]	W. Hereman, Shallow water waves and solitary waves, Solitons, 2022,203–220. https://doi.org/10.1007/978-1-0716-2457-9_480
[12]	Q. Li, Nonlinear internal waves in the South China Sea, University of Rhode Island, 2010. https://doi.org/10.23860/diss-2353
[13]	J. P. Boyd, Nonlinear equatorial waves, Dynamics of the equatorial ocean, Heidelberg: Springer, 2017,329–404. https://doi.org/10.1007/978-3-662-55476-0_16
[14]	H. F. Ismael, M. A. S. Murad, H. Bulut, Various exact wave solutions for KdV equation with time-variable coefficients, J. Ocean Eng. Sci., 7 (2022), 409–418. https://doi.org/10.1016/j.joes.2021.09.014 doi: 10.1016/j.joes.2021.09.014
[15]	A. K. Gupta, S. S. Ray, On the solution of time-fractional KdV-Burgers equation using Petrov-Galerkin method for propagation of long wave in shallow water, Chaos, Soliton. Fract., 116 (2018), 376–380. https://doi.org/10.1016/j.chaos.2018.09.046 doi: 10.1016/j.chaos.2018.09.046
[16]	R. Grimshaw, Environmental stratified flows, Springer Science & Business Media, 2002. https://doi.org/10.1007/b100815
[17]	A. R. Osborne, Nonlinear ocean wave and the inverse scattering transform, Scattering, 2002,637–666. https://doi.org/10.1016/b978-012613760-6/50033-4
[18]	D. Kaya, Partial differential equations that lead to solitons, Solitons, 2022,193–201. https://doi.org/10.1007/978-1-0716-2457-9_380
[19]	A. Geyer, R. Quirchmayr, Shallow water equations for equatorial tsunami waves, Philos. T. Roy. Soc. A: Math. Phys. Eng. Sci., 376 (2018), 20170100. https://doi.org/10.1098/rsta.2017.0100 doi: 10.1098/rsta.2017.0100
[20]	H. Holden, B. Øksendal, J. Ubøe, T. Zhang, Stochastic partial differential equations, Springer, 2010. https://doi.org/10.1007/978-0-387-89488-1_5
[21]	H. Yépez-Martínez, J. F. Gómez-Aguilar, D. Baleanu, Beta-derivative and sub-equation method applied to the optical solitons in medium with parabolic law nonlinearity and higher order dispersion, Optik, 155 (2018), 357–365. https://doi.org/10.1016/j.ijleo.2017.10.104 doi: 10.1016/j.ijleo.2017.10.104
[22]	H. M. Baskonus, J. F. Gómez-Aguilar, New singular soliton solutions to the longitudinal wave equation in a magneto-electro-elastic circular rod with M-derivative, Mod. Phys. Lett. B, 33 (2019), 1950251. https://doi.org/10.1142/s0217984919502518 doi: 10.1142/s0217984919502518
[23]	B. Ghanbari, J. F. Gómez-Aguilar, Optical soliton solutions for the nonlinear Radhakrishnan-Kundu-Lakshmanan equation, Mod. Phys. Lett. B, 33 (2019), 1950402. https://doi.org/10.1142/s0217984919504025 doi: 10.1142/s0217984919504025
[24]	C. Qian, J. G. Rao, Y. B. Liu, J. S. He, Rogue waves in the three-dimensional Kadomtsev-Petviashvili equation, Chinese Phys. Lett., 33 (2016), 110201. https://doi.org/10.1088/0256-307x/33/11/110201 doi: 10.1088/0256-307x/33/11/110201
[25]	X. R. Hu, J. C. Chen, Y. Chen, Groups analysis and localized solutions of the (2+1)-dimensional Ito equation, Chinese Phys. Lett., 32 (2015), 070201. https://doi.org/10.1088/0256-307x/32/7/070201 doi: 10.1088/0256-307x/32/7/070201
[26]	X. Z. Liu, J. Yu, Z. M. Lou, X. M. Qian, A nonlocal Burgers equation in atmospheric dynamical system and its exact solutions, Chinese Phys. B, 28 (2019), 010201. https://doi.org/10.1088/1674-1056/28/1/010201 doi: 10.1088/1674-1056/28/1/010201
[27]	P. F. Zheng, M. Jia, A more general form of lump solution, lumpoff, and instanton/rogue wave solutions of a reduced (3+1)-dimensional nonlinear evolution equation, Chinese Phys. B, 27 (2018), 120201. https://doi.org/10.1088/1674-1056/27/12/120201 doi: 10.1088/1674-1056/27/12/120201
[28]	S. Q. Xu, X. G. Geng, N-soliton solutions for the nonlocal two-wave interaction system via the Riemann-Hilbert method, Chinese Phys. B, 27 (2018), 120202. https://doi.org/10.1088/1674-1056/27/12/120202 doi: 10.1088/1674-1056/27/12/120202
[29]	X. Y. Jiao, Truncated series solutions to the (2+1)-dimensional perturbed Boussinesq equation by using the approximate symmetry method, Chinese Phys. B, 27 (2018), 100202. https://doi.org/10.1088/1674-1056/27/10/100202 doi: 10.1088/1674-1056/27/10/100202
[30]	J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos, Soliton. Fract., 30 (2006), 700–708. https://doi.org/10.1016/j.chaos.2006.03.020 doi: 10.1016/j.chaos.2006.03.020
[31]	N. A. Kudryashov, N. B. Loguinova, Be careful with the exp-function method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 1881–1890. https://doi.org/10.1016/j.cnsns.2008.07.021. doi: 10.1016/j.cnsns.2008.07.021
[32]	N. A. Shah, Y. S. Hamed, K. M. Abualnaja, J. D. Chung, R. Shah, A. Khan, A Comparative analysis of fractional-order Kaup-Kupershmidt equation within different operators, Symmetry, 14 (2022), 986. https://doi.org/10.3390/sym14050986 doi: 10.3390/sym14050986
[33]	M. Shakeel, Attaullah, N. A. Shah, J. D. Chung, Application of modified exp-function method for strain wave equation for finding analytical solutions, Ain Shams Eng. J., 14 (2023), 101883. https://doi.org/10.1016/j.asej.2022.101883 doi: 10.1016/j.asej.2022.101883
[34]	S. Zhang, Exact solutions of Wick-type stochastic KdV equation, Can. J. Phys., 90 (2012), 181–186. https://doi.org/10.1139/p2012-002 doi: 10.1139/p2012-002
[35]	S. Zhang, H. W. Li, B. Xu, Inverse scattering integrability and fractional soliton solutions of a variable-coefficient fractional-order KdV-type equation, Fractal Fract., 8 (2024), 520. https://doi.org/10.3390/fractalfract8090520 doi: 10.3390/fractalfract8090520

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