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The mKdV equation under the Gaussian white noise and Wiener process: Darboux transformation and stochastic soliton solutions

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  • Yuan, Rui-rui
  • Shi, Ying
  • Zhao, Song-lin
  • Wang, Wen-zhuo

Abstract

In this paper, we propose a novel integrable system named the stochastic mKdV equation, along with its corresponding Lax pair. We aim to extend the methodology of deterministic integrable systems to construct and solve stochastic integrable systems. The Darboux transformation effectively obtains analytic solutions for the integrable stochastic mKdV equation. Using the Darboux transformation, soliton solutions incorporating stochastic terms are obtained as Wronskian determinants. Furthermore, we conduct an in-depth analysis of the dynamics exhibited by the stochastic one-soliton and the two-soliton solutions.

Suggested Citation

  • Yuan, Rui-rui & Shi, Ying & Zhao, Song-lin & Wang, Wen-zhuo, 2024. "The mKdV equation under the Gaussian white noise and Wiener process: Darboux transformation and stochastic soliton solutions," Chaos, Solitons & Fractals, Elsevier, vol. 181(C).
    • Handle: RePEc:eee:chsofr:v:181:y:2024:i:c:s0960077924002613
    DOI: 10.1016/j.chaos.2024.114709
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    References listed on IDEAS

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    1. Raut, Santanu & Ma, Wen-Xiu & Barman, Ranjan & Roy, Subrata, 2023. "A non-autonomous Gardner equation and its integrability: Solitons, positons and breathers," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    2. Liu, Qing, 2007. "Some exact solutions for stochastic mKdV equation," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1224-1230.
    3. Herman, Russell L. & Rose, Andrew, 2009. "Numerical realizations of solutions of the stochastic KdV equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(1), pages 164-172.
    4. Chen, Yong & Wang, Qi & Li, Biao, 2005. "The stochastic soliton-like solutions of stochastic KdV equations," Chaos, Solitons & Fractals, Elsevier, vol. 23(4), pages 1465-1473.
    5. Biswas, Swapan & Ghosh, Uttam & Raut, Santanu, 2023. "Construction of fractional granular model and bright, dark, lump, breather types soliton solutions using Hirota bilinear method," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
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