In this manuscript, the well-known stochastic Burgers’ equation in under investigation numerically and analytically. The stochastic Burgers’ equation plays an important role in the fields of applied mathematics such as fluid dynamics, gas dynamics, traffic flow, and nonlinear acoustics. This study is presented the existence, approximate, and exact stochastic solitary wave results. The existence of results is shown by the help of Schauder fixed point theorem. For the approximate results the proposed stochastic finite difference scheme is constructed. The analysis of the proposed scheme is analyzed by presented the consistency and stability of scheme. The consistency is checked under the mean square sense while the stability condition is gained by the help of Von-Neumann criteria. Meanwhile, the stochastic exact solutions are constructed by using the generalized exponential rational function method. These exact stochastic solutions are obtained in the form of hyperbolic, trigonometric and exponential functions. Mainly, the comparison of both numerical and exact solutions are analyzed via simulations. The unique physical problems are constructed from the newly constructed soliton solutions to compare the numerical results with exact solutions under the presence of randomness. The 3D and line plots are dispatched that are shown the similar behavior by choosing the different values of parameters. These results are the main innovation of this study under the noise effects.

在这份手稿中，我们对著名的随机伯格斯方程进行了数值和分析研究。随机伯格斯方程在流体动力学、气体动力学、交通流和非线性声学等应用数学领域发挥着重要作用。这项研究提出了随机孤立波的存在性、近似性和精确性结果。借助Schauder不动点定理证明了结果的存在性。为了获得近似结果，构建了所提出的随机有限差分格式。对所提出的方案进行了分析，提出了方案的一致性和稳定性。一致性检验是在均方意义下进行的，稳定性条件是通过冯诺依曼准则获得的。同时，利用广义指数有理函数方法构造随机精确解。这些精确的随机解以双曲函数、三角函数和指数函数的形式获得。主要通过模拟分析数值解和精确解的比较。从新构造的孤子解构造独特的物理问题，以将数值结果与随机性存在下的精确解进行比较。调度 3D 和线图，通过选择不同的参数值显示类似的行为。这些结果是本研究在噪声影响下的主要创新点。