## 1 Introduction  
Finding the solution to non-linear problems still faces many difficulties in the field of mathematical physics. Non-linear partial differential equations (NPDEs) [1] play a significant role in physical and mathematical models [2]. They define their ranges from gravitation [3] to fluid dynamics [4], describing many different physical systems. They are mostly related to the fact that these types of equations face the problem of finding their integrability. There are almost no pervasive techniques that can be used for all problems, and usually, every separate model works as an individual problem. Partial differential equations (PDEs) give solutions in an ultimate state from the past few years while, to exemplify the solutions of NPDEs [5], we can write their solutions with some special clarification. To exemplify the most important applications of NPDEs from the historical point of view, we can highlight one of our focuses as the Westervelt equation [6], which is a non-linear mathematical model, widely used for wave propagation, that can be specified by the possible physical measurements leading to over-posed data. Other fundamental models, which can also be mentioned here, are the Euler and Navier-Stokesequations in fluid dynamics [7], non-linear Schrodinger [8], Klein-Gordon equation [9], the Boltzmann equation in gas dynamics [10], and many more. To write the solutions of NPDEs explicitly [11], we can reduce the given equation to the equation of one dimension, for which, the process of conversion is applied on NPDE to convert it into an ordinary differential equation (ODE). In the past few years, many analytical and numerical techniques have been projected to obtain solutions for NPDEs, for example, Bernoulli functional methodology [12], the F-expansion technique [13], the auxiliary equation technique [14], the simplest extended equation technique [15], the (G\({}^{\prime}\)/G)-expansion technique [16], the sub-ODE technique [1], the generalized Kudryashov technique(GKM) [17], and many more. The collective theme of all these techniques is to convert the PDEs to ODEs using wave transformations [18]. In this field, the study of solitons [19] is playing an important role in constructing various families of analytic traveling wave solutions [20], which defines the dynamics of solitons leaving a remarkable position in non-linear optics. According to some theoretical research, various modes of plasma (i.e., periodic, rational, solitons, shock-like, explosive) [21] show wave propagation in different natures of non-linear waves. The reason behind choosing the generalized form of the Kudryashov method is that it approaches the most consistent solutions of the NPDEs. It is also a very useful and efficient approach to finding the solutions to non-linear evolution equations. The modified Kudryashov method is a very strong solution scheme that shows many ways towards the exact solution to the NPDE problem in mathematical physics and biology. Due to the efficient work of this method in the field of mathematics, it has received significant attention towards it. Similarly, a higher-order non-linear Schrodinger equation (NDNLSE) can be solved with the help of this powerful method. It is a successful application that can be performed in several works just like in [22,23]. Ultrasound imaging [24,25] is being used in a well-established way to produce pictures of tissues inside the body of human beings. They are modeled in non-linear wave equations with high intensity. In many medical and industrial applications, high intensity-focused ultrasound (HIFU) is one of the crucial procedures, which uses high-energy sound waves directly at an area of abnormal tissues of the body to tighten and lift the skin. It also treats tremors, uterine fibroids, and tumors in certain conditions. It includes ultrasound or welding, thermotherapy, sonochemistry, and lithotripsy. For the purpose of medical imaging, a spatially varying coefficient can be used, which is called acoustic nonlinearity parameter tomography [26].