In this paper, the Allen–Cahn equation with time noise is under consideration. The extended fan-sub technique is used to find the exact solutions. The solutions are successfully extracted in the form of hyperbolic, trigonometric and mixed forms of solitons. Importantly, the physical unique value problems of the solutions are discussed using the different values of parameters. The 2D, 3D, and their corresponding contour behaviors of these solutions are depicted by choosing the different values of parameters. The stability is controlled through the Borel coefficient in the noise term.

中文翻译：

---

在孤子理论方法框架下求解随机 Allen-Cahn 方程

在本文中，考虑了具有时间噪声的 Allen-Cahn 方程。扩展的扇子技术用于找到精确的解决方案。解决方案以双曲线、三角和混合形式的孤子形式成功提取。重要的是，使用不同的参数值讨论了解决方案的物理唯一值问题。通过选择不同的参数值来描述这些解的 2D、3D 及其相应的轮廓行为。稳定性通过噪声项中的 Borel 系数来控制。