In this study, the Sobolev-type equation is considered analytically to investigate the solitary wave solutions. The Sobolev-type equations are found in a broad range of fields, such as ecology, fluid dynamics, soil mechanics, and thermodynamics. There are two novel techniques used to explore the solitary wave structures namely as; generalized Riccati equation mapping and modified auxiliary equation (MAE) methods. The different types of abundant families of solutions in the form of dark soliton, bright soliton, solitary wave solutions, mixed singular soliton, mixed dark-bright soliton, periodic wave, and mixed periodic solutions. The linearized stability of the model has been investigated. Solitons behave differently in different circumstances, and their behaviour can be better understood by building unique physical problems with particular boundary conditions (BCs) and starting conditions (ICs) based on accurate soliton solutions. So, the choice of unique physical problems from various solutions is also carried out. The 3D, line graphs and corresponding contours are drawn with the help of the Mathematica software that explains the physical behavior of the state variable. This information can help the researchers in their understanding of the physical conditions.

在这项研究中，Sobolev 型方程被分析地考虑来研究孤立波解。索博列夫型方程广泛应用于生态学、流体动力学、土壤力学和热力学等领域。有两种新技术用于探索孤立波结构，即：广义 Riccati 方程映射和修正辅助方程 (MAE) 方法。暗孤子、亮孤子、孤立波解、混合奇异孤子、混合暗-亮孤子、周期波和混合周期解形式的不同类型的丰富族解。研究了模型的线性稳定性。孤子在不同情况下表现不同，通过基于准确的孤子解构建具有特定边界条件 (BC) 和起始条件 (IC) 的独特物理问题，可以更好地理解孤子的行为。因此，还进行了从各种解决方案中选择独特的物理问题。借助 Mathematica 软件绘制 3D 折线图和相应的轮廓，该软件解释了状态变量的物理行为。这些信息可以帮助研究人员了解身体状况。