AI 中文总结
本文研究含气泡液体压力波的(1+2)维KS方程，通过Painlevé分析结合SMM及WTC算法证明其可积性，推导自Backlund变换并得到各类精确解，开展一致性检验并可视化解的几何特征。
AI 中文摘要
本研究工作中，我们考虑了一个非线性四阶(1+2)维Kudryashov-Sinelshchikov（KS）方程，该方程描述含气泡液体中的压力波传播。我们采用Painlevé分析结合奇异流形方法（SMM）分析KS方程的直接可积性，借助WTC算法证明该(1+2)维KS方程具有Painlevé可积性。随后通过截断Painlevé展开得到自Backlund变换（ABT），并基于所得自Backlund变换，选取合适的流形形式推导各类精确解，同时对这些解进行一致性检验，还给出了代表性解的二维和三维图，以理解解的几何视角。
英文摘要
In this research work, we consider a nonlinear fourth-order (1+2)-dimensional Kudryashov-Sinelshchikov (KS) equation which represents the wave propagation of pressures in liquids that contain gas bubbles. A direct Integrability of the KS equation is analysed using Painleve Analysis with Singular Manifold Method (SMM). With the help of WTC algorithm, we show that the (1+2) KS equation is Painleve integrable. Then by truncating the Painleve expansion we obtain the Auto-Backlund Transformation (ABT). By taking suitable forms of Manifold, various exact solutions based on the obtained Auto-Backlund transformation are derived. The consistancy check for these solutions are also performed. Representative solutions are presented in the from of 2D and 3D plots to understand the geometric perspective of the solutions.
