This article deals with the study of ultrasound propagation, which propagates the mechanical vibration of the molecules or of the particles of a material. It measures the speed of sound in air. For this reason, the third-order non-linear model of the Westervelt equation was chosen to be studied, as the solutions to such problems have much importance for physical purposes. In this article, we discuss the exact solitary wave solutions of the third-order non-linear model of the Westervelt equation for an acoustic pressure p representing the equation of ultrasound with high intensity, as used in acoustic tomography. Moreover, the non-linear coefficient $B/A$ (being a part of space-dependent coefficient K), has also been investigated in this literature. This problem is solved using the Generalized Kudryashov method along with a comparison of the Modified Kudryashov method. All of the solutions have been discussed with both surface and contour plots, which shows the behavior of the solution. The images are prepared in a well-established way, showing the production of tissues inside the human body.

In this article, stochastic behavior of reaction diffusion brusselator model is under consideration. There are many physical phenomena which are related to chemical concentrations. One chemical concentration coincide with the other chemical concentration and their inter-diffusion is a major question to be addressed and to be understood. So, that is why Brusselator model is very proto-type and standard model that lays the foundation of any kind of that matter chemical concentrations of different substances. It also has the application in physical species as well. That is why we are considering such model. The existence of solution is guaranteed with fix-point operator, self mapping and pre-compact conditions. Nonstandard finite difference scheme and Crank-Nicolson schemes are used to show the graphical behavior of the model. The consistency and stability of the schemes are discussed and both schemes are unconditionally stable. The 3D and 2D graphs represents the concentration of the models.

The Biswas–Mollivic equation is a special type of nonlinear Schrödinger equation, which explains the spatio-temporal behaviour of excitable media. In this paper, we investigate the optical soliton solutions of the Biswas–Mollivic equation with cubic–quintic–septic–nonic nonlinearities using the generalized Riccati equation mapping method. This method is efficient and provides new perspectives. It also provides novel insights into the dynamics of excitable media. Our findings add to a better understanding of the complex spatio-temporal patterns that develop in excitable media and have potential applications in the design of new technologies for controlling and manipulating pattern formation. To depict optical soliton solutions graphically, we use the MATLAB software.

In this article, a numerical model for a Brusselator advection–reaction–diffusion (BARD) system by using an elegant numerical scheme is developed. The consistency and stability of the proposed scheme is demonstrated. Positivity preserving property of the proposed scheme is also verified. The designed scheme is compared with the two well-known existing classical schemes to validate the certain physical properties of the continuous system. A test problem is also furnished for simulations to support our claim. Prior to computations, the existence and uniqueness of solutions for more generic problems is investigated. In the underlying system, the nonlinearities depend not only on the desired solution but also on the advection term that reflects the pivotal importance of the study.