| Abstract: |
| Some stochastic Boussinesq equations under Brown motion are investigated. The integrability of the stochastic Boussinesq equations are established by demonstrating the existence of a Lax pair. By using the Darboux transformation, we construct a variety of exact solutions, including multi-soliton solutions, periodic solutions and rational solutions, under integrated Brownian motion. Most importantly, the dynamic behaviors of the stochastic soliton solutions are analyzed through two key probabilistic characteristics: the statistical mean and variance. Specifically, the long-time asymptotic expressions for these characteristics are rigorously derived, revealing a high degree of consistency with the statistical mean and variance obtained from sampled solutions. |
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