Exploration traveling solitary solutions to the modified Korteweg-de Vries equation with advection noise

Sofian T. Obeidat; Doaa Rizk; Wael W. Mohammed; Sofian T. Obeidat; Doaa Rizk; Wael W. Mohammed

doi:10.3934/math.20251158

AIMS Mathematics

2025, Volume 10, Issue 11: 26334-26350. doi: 10.3934/math.20251158

Previous Article Next Article

Research article Special Issues

Exploration traveling solitary solutions to the modified Korteweg-de Vries equation with advection noise

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, College of Science, Qassim University, Saudi Arabia
3.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 10 September 2025 Revised: 20 October 2025 Accepted: 10 November 2025 Published: 14 November 2025
MSC : 35A20, 83C15, 60H10, 60H15, 35Q51

In this study, we investigate the stochastic modified Korteweg-de Vries (SmKdV) equation, which is driven in the Itô sense by advection noise. We show that by solving certain deterministic counterparts of the modified Korteweg-de Vries with an extra diffusion term (for short DmKdV), and then combining the results with a solution of stochastic ordinary differential equations, the exact solution of the SmKdV equation may be discovered. We derive soliton solutions for the DmKdV problem using two distinct methods: the extended tanh function approach and the $ \exp (-\psi (\eta)) $ -expansion method. Additionally, we study how the advection noise affects the solutions of the SmKdV equation by presenting several 3D graphs using a MATLAB program.
- stochastic solutions,
- simulation,
- mathematical model,
- exact solutions,
- analytical techniques,
- noise intensity
Citation: Sofian T. Obeidat, Doaa Rizk, Wael W. Mohammed. Exploration traveling solitary solutions to the modified Korteweg-de Vries equation with advection noise[J]. AIMS Mathematics, 2025, 10(11): 26334-26350. doi: 10.3934/math.20251158

Related Papers:

Abstract

In this study, we investigate the stochastic modified Korteweg-de Vries (SmKdV) equation, which is driven in the Itô sense by advection noise. We show that by solving certain deterministic counterparts of the modified Korteweg-de Vries with an extra diffusion term (for short DmKdV), and then combining the results with a solution of stochastic ordinary differential equations, the exact solution of the SmKdV equation may be discovered. We derive soliton solutions for the DmKdV problem using two distinct methods: the extended tanh function approach and the $ \exp (-\psi (\eta)) $ -expansion method. Additionally, we study how the advection noise affects the solutions of the SmKdV equation by presenting several 3D graphs using a MATLAB program.

References

[1]	A. M. Wazwaz, G. Q. Xu, Variety of optical solitons for perturbed Fokas–Lenells equation through modified exponential rational function method and other distinct schemes, Optik, 287 (2023), 171011. https://doi.org/10.1016/j.ijleo.2023.171011 doi: 10.1016/j.ijleo.2023.171011
[2]	Y. Y. Wang, Y. P. Zhang, C. Q. Dai, Re-study on localized structures based on variable separation solutions from the modified tanh-function method, Nonlinear Dyn., 83 (2016), 1331–1339. https://doi.org/10.1007/s11071-015-2406-5 doi: 10.1007/s11071-015-2406-5
[3]	Y. Kai, Z. Yin, Linear structure and soliton molecules of Sharma–Tasso–Olver–Burgers equation, Phys. Lett. A, 452 (2022), 128430. https://doi.org/10.1016/j.physleta.2022.128430 doi: 10.1016/j.physleta.2022.128430
[4]	M. S. Ullah, M. Z. Ali, H. O. Roshid, Bifurcation analysis and new wave forms to the fractional KFG equation, Partial Differ. Equ. Appl. Math., 10 (2024), 100716. https://doi.org/10.1016/j.padiff.2024.100716 doi: 10.1016/j.padiff.2024.100716
[5]	M. S. Ullah, Interaction solution to the (3 + 1)-D negative-order KdV first structure, Partial Differ. Equ. Appl. Math., 8 (2023), 100566. https://doi.org/10.1016/j.padiff.2023.100566 doi: 10.1016/j.padiff.2023.100566
[6]	T. Usman, M. S. Ullah, E. Safitri, M. Z. Ali, Soliton outcomes and stability study of the Schrödinger equation With cubic nonlinearity, J. Comput. Nonlinear Dynam., 20 (2025), 1–11. https://doi.org/10.1115/1.4069111 doi: 10.1115/1.4069111
[7]	M. S. Rahaman, M. S. Ullah, M. N. Islam, Novel dynamics of the nonlinear fractional soliton neuron model with sensitivity analysis, AIP Adv., 15 (2025), 085116. https://doi.org/10.1063/5.0282276 doi: 10.1063/5.0282276
[8]	M. S. Ullah, M. Z. Ali, H. O. Roshid, Stability analysis, $ \varphi ^{6}$ model expansion method, and diverse chaos-detecting tools for the DSKP model, Sci. Rep., 15 (2025), 13658. https://doi.org/10.1038/s41598-025-98275-7 doi: 10.1038/s41598-025-98275-7
[9]	M. Abdalla, M. M. Roshid, M. S. Ullah, I. Hossain, Dynamical analysis, and the effect of fractional parameters on optical soliton solution for Yajima–Oikawa model in short-wave and long-wave, Chaos Soliton. Fract., 199 (2025), 116697. https://doi.org/10.1016/j.chaos.2025.116697 doi: 10.1016/j.chaos.2025.116697
[10]	I. Alraddadi, F. Alsharif, S. Malik, H. Ahmad, T. Radwan, K. K. Ahmed, Innovative soliton solutions for a (2 + 1)-dimensional generalized KdV equation using two effective approaches, AIMS Math., 9 (2024), 34966–34980. https://doi.org/10.3934/math.20241664 doi: 10.3934/math.20241664
[11]	I. Samir, K. K. Ahmed, H. M. Ahmed, H. Emadifar, W. B. Rabie, Extraction of newly soliton wave structure of generalized stochastic NLSE with standard Brownian motion, quintuple power law of nonlinearity and nonlinear chromatic dispersion, Phys. Open, 21 (2024), 100232. https://doi.org/10.1016/j.physo.2024.100232 doi: 10.1016/j.physo.2024.100232
[12]	M. A. Helal, Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics, Chaos Soliton. Fract., 13 (2002), 1917–1929. https://doi.org/10.1016/S0960-0779(01)00189-8 doi: 10.1016/S0960-0779(01)00189-8
[13]	A. H. Khater, O. H. El-Kalaawy, D. K. Callebaut, Bäcklund transformations and exact solutions for Alfvén solitons in a relativistic electron–positron plasma, Phys. Scr., 58 (1998), 545. https://doi.org/10.1088/0031-8949/58/6/001 doi: 10.1088/0031-8949/58/6/001
[14]	Z. P. Li, Y. C. Liu, Analysis of stability and density waves of traffic flow model in an ITS environment, Eur. Phys. J. B, 53 (2006), 367–374. https://doi.org/10.1140/epjb/e2006-00382-7 doi: 10.1140/epjb/e2006-00382-7
[15]	H. Leblond, D. Mihalache, Few-optical-cycle solitons: modified Kortewegde Vries sine-Gordon equation versus other non-slowly-varying-envelopeapproximation models, Phys. Rev. A, 79 (2009), 063835. https://doi.org/10.1103/PhysRevA.79.063835 doi: 10.1103/PhysRevA.79.063835
[16]	R. R. Yuan, Y. Shi, S. L. Zhao, W. Z. Wang, The mKdV equation under the Gaussian white noise and Wiener process: Darboux transformation and stochastic soliton solutions, Chaos Soliton. Fract., 181 (2024), 114709. https://doi.org/10.1016/j.chaos.2024.114709 doi: 10.1016/j.chaos.2024.114709
[17]	C. M. Wei, Y. H. Ren, Z. Q. Xia, Symmetry reductions and soliton-like solutions for stochastic MKdV equation, Chaos Soliton. Fract., 26 (2005), 1507–1513. https://doi.org/10.1016/j.chaos.2005.04.018 doi: 10.1016/j.chaos.2005.04.018
[18]	A. M. Wazwaz, The tanh method for generalized forms of nonlinear heat conduction and Burgers–Fisher equations, Appl. Math. Comput., 169 (2005), 321–338. https://doi.org/10.1016/j.amc.2004.09.054 doi: 10.1016/j.amc.2004.09.054
[19]	Y. Yang, Exact solutions of the mKdV equation, IOP Conf. Ser.: Earth Environ. Sci., 769 (2021), 042040. https://doi.org/10.1088/1755-1315/769/4/042040 doi: 10.1088/1755-1315/769/4/042040
[20]	K. R. Raslan, The application of He's Exp-function method for MKdV and Burgers' equations with variable coefficients, Int. J. Nonlinear Sci., 7 (2009), 174–181.
[21]	N. Taghizadeh, Comparison of solutions of mKdV equation by using the first integral method and ($G^{\prime }/G$)-expansion method, Math. AEterna, 2 (2012), 309–320.
[22]	A. M. Wazwaz, The tanh method for generalized forms of nonlinear heat conduction and Burgers–Fisher equations, Appl. Math. Comput., 169 (2005), 321–338. https://doi.org/10.1016/j.amc.2004.09.054 doi: 10.1016/j.amc.2004.09.054
[23]	A. A. Elmandouha, A. G. Ibrahim, Bifurcation and travelling wave solutions for a (2+1)-dimensional KdV equation, J. Taibah Univ. Sci., 14 (2020), 139–147. https://doi.org/10.1080/16583655.2019.1709271 doi: 10.1080/16583655.2019.1709271
[24]	A. Zafara, A. R. Seadawy, The conformable space-time fractional mKdV equations and their exact solutions, J. King Saud Univ. Sci., 31 (2019), 1478–1484. https://doi.org/10.1016/j.jksus.2019.09.003 doi: 10.1016/j.jksus.2019.09.003
[25]	H. Ur Rehman, M. Inc, M. I. Asjad, A. Habib, Q. Munir, New soliton solutions for the space-time fractional modified third order Korteweg–de Vries equation, J. Ocean Eng. Sci., In Press. https://doi.org/10.1016/j.joes.2022.05.032
[26]	W. W. Mohammed, F. M. Al-Askar, C. Cesarano, The analytical solutions of the stochastic mKdV equation via the mapping method, Mathematics, 10 (2022), 4212. https://doi.org/10.3390/math10224212 doi: 10.3390/math10224212
[27]	W. W. Mohammed, F. M. Al-Askar, New stochastic solitary solutions for the modified Korteweg-de Vries equation with stochastic term/random variable coefficients, AIMS Math., 9 (2024), 20467–20481. https://doi.org/10.3934/math.2024995 doi: 10.3934/math.2024995
[28]	Q. Liu, Various types of exact solutions for stochastic mKdV equation via a modified mapping method, Europhys. Lett., 74 (2006), 377. https://doi.org/10.1209/epl/i2005-10556-5 doi: 10.1209/epl/i2005-10556-5
[29]	C. Q. Dai, J. F. Zhang, Application of He's exp-ftmction method to the stochastic mKdV equation, Int. J. Nonlinear Sci. Numer. Simul., 10 (2009), 675–680. https://doi.org/10.1515/IJNSNS.2009.10.5.675 doi: 10.1515/IJNSNS.2009.10.5.675
[30]	O. Calin, An informal introduction to stochastic calculus with applications, Singapore: World Scientific, 2015. https://doi.org/10.1142/9620
[31]	K. Adjibi, A. Martinez, M. Mascorro, C. Montes, T. Oraby, R. Sandoval, et al., Exact solutions of stochastic Burgers–Korteweg de Vries type equation with variable coefficients, Partial Differ. Equ. Appl. Math., 11 (2024), 100753. https://doi.org/10.1016/j.padiff.2024.100753 doi: 10.1016/j.padiff.2024.100753
[32]	B. $\phi $ksendal, Stochastic differential equations: an introduction with applications, Universitext, Springer-Verlag, Berlin, 1998.
[33]	K. Khan, M. A. Akbar, Application of $ exp(-\varphi (\eta))$-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation, World Appl. Sci. J., 24 (2013), 1373–1377.
[34]	M. G. Hafez, M. Y. Ali, M. K. H. Chowdury, M. A. Kauser, Application of the $exp(-\varphi (\eta))$ expansion method for solving nonlinear TRLW and Gardner equations, Int. J. Appl. Math. Comput. Sci., 27 (2016), 44–56.
[35]	E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, 277 (2000), 212–218. https://doi.org/10.1016/s0375-9601(00)00725-8 doi: 10.1016/s0375-9601(00)00725-8

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)