In this paper, by using the Lie symmetry analysis, all of the geometric vector fields of the \((3+1)\)-Burgers system are obtained. We find the 1, 2, and 3-dimensional optimal system of the Burger system and then by applying the 3-dimensional optimal system reduce the order of the system. Also the nonclassical symmetries of the \((3+1)\)-Burgers system will be found by employing nonclassical methods. Finally, the ansatz solutions of BS equations with the aid of the tanh method has been presented. The achieved solutions are investigated through two- and three-dimensional plots for different values of parameters. The analytical simulations are presented to ensure the efficiency of the considered technique. The behavior of the obtained results for multiple cases of symmetries is captured in the present framework. The outcomes of the present investigation show that the considered scheme is efficient and powerful to solve nonlinear differential equations that arise in the sciences and technology.

本文利用李对称性分析，获得了（（3 + 1）\）- Burgers系统的所有几何矢量场。我们找到Burger系统的1、2和3维最佳系统，然后通过应用3维最佳系统来减少系统的阶数。也可以通过采用非经典方法找到\（（3 + 1）\）- Burgers系统的非经典对称性。最后，BS的ansatz解决方案提出了利用tanh方法的方程。通过针对不同参数值的二维图和三维图来研究获得的解决方案。提出了分析模拟以确保所考虑技术的效率。在本框架中捕获了多种对称情况下获得的结果的行为。本研究的结果表明，所考虑的方案对于解决科学和技术中出现的非线性微分方程是有效且强大的。