On the Soliton Solutions for the Stochastic Konno–Oono System in Magnetic Field with the Presence of Noise
Abstract
:1. Introduction
- is taken for the Stratonovich sense;
- is taken for the It sense.
2. Wiener Process
- is a continuous functionif .
- = 0.
- For , is independent.
- has a Gaussian distribution .
3. Soliton Solutions for K–O System with Stratonovich Sense
- Family-I: When , and , then . So, we obtain the soliton solutions of Equation (5) as follows:
- Family-II: When , and , then . So, we obtained the Soliton solutions of Equation (5) as follow,
- Family-III: When , and , then or . So, we obtain the soliton solutions of Equation (5) as follows:
- Family-IV: When , and , then or . So, we obtain the soliton solutions of Equation (5) as follows:
- Family-V: When , and , then or . So, we obtain the trigonometric solutions of Equation (5) as follows:
- Family-VI: When , and , then or . So, we obtain the trigonometric solutions of Equation (5) as follows:
- Family-VII: When , and , then . So, we obtain the trigonometric solutions of Equation (5) as follows:
- Family-VIII: When , and , then . So, we obtain the trigonometric solutions of Equation (5) as follows:
- Family-IX: When , and , then . So, we obtain the rational solutions of Equation (5) as follows:
4. Soliton Solutions for K–O System with It Sense
- Case 1.
- When we take , we obtain
- Case 2.
- When we take , we obtain
- Case 3.
- When we take , but , then we obtain
- Family-I: When , and , then . So, we obtain the soliton solutions of Equation (45) as follows:
- Family-II: When , and , then . So, we obtain the soliton solutions of Equation (45) as follows:
5. Graphical Discussion
Algorithm 1 Pseudocode for WhiteNoise |
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6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Shaikh, T.S.; Baber, M.Z.; Ahmed, N.; Shahid, N.; Akgül, A.; De la Sen, M. On the Soliton Solutions for the Stochastic Konno–Oono System in Magnetic Field with the Presence of Noise. Mathematics 2023, 11, 1472. https://doi.org/10.3390/math11061472
Shaikh TS, Baber MZ, Ahmed N, Shahid N, Akgül A, De la Sen M. On the Soliton Solutions for the Stochastic Konno–Oono System in Magnetic Field with the Presence of Noise. Mathematics. 2023; 11(6):1472. https://doi.org/10.3390/math11061472
Chicago/Turabian StyleShaikh, Tahira Sumbal, Muhammad Zafarullah Baber, Nauman Ahmed, Naveed Shahid, Ali Akgül, and Manuel De la Sen. 2023. "On the Soliton Solutions for the Stochastic Konno–Oono System in Magnetic Field with the Presence of Noise" Mathematics 11, no. 6: 1472. https://doi.org/10.3390/math11061472
APA StyleShaikh, T. S., Baber, M. Z., Ahmed, N., Shahid, N., Akgül, A., & De la Sen, M. (2023). On the Soliton Solutions for the Stochastic Konno–Oono System in Magnetic Field with the Presence of Noise. Mathematics, 11(6), 1472. https://doi.org/10.3390/math11061472