In present study, the Boussinesq equation is obtained by means of the Sardar Sub-Equation Technique (SSET) to create unique soliton solutions containing parameters. Using this technique, different solutions are obtained, such as the singular soliton, the dark-bright soliton, the bright soliton and the periodic soliton. The graphs of these solutions are plotted for a batter understanding of the model. The results show that the technique is very effective in solving nonlinear partial differential equations (PDEs) arising in mathematical physics.

A solitary wave is the dispersion-less solution of nonlinear evolutionary equations that travels at a constant speed without dissipating its energy. The purpose of this article is to provide insight into the discovery and history of solitons. The different types of the solitons are discussed in brief that is helpful for the researchers. For the discussion of the nature of solitons, the solution behavior of the Korteweg de Vries equation (KdV), the sine-Gordon (SG), the Camassa-Holm (CH) equation, and the nonlinear Schrodinger (NLS) equation are considered. This article deals with the various applications of solitons in different fields such as biophysics, nonlinear optics, Bose-Einstein condensation, plasma physics, Josephson junction, etc. focusing on the properties of solitons based on their classification.

Partial differential equations are a vital component of the study of mathematical models in science and engineering. There are various tools and techniques developed by the researchers to solve the differential equations. The radial basis functions have proven to be an efficient basis function for approximating the solutions to ordinary and partial differential equations. There are different types of radial basis function methods that have been developed by the researchers to solve various well known differential equation. It has been developed for approximation of the solution with various approaches that lead to the development of hybrid methods. Radial basis function methods are widely used in numerical analysis and statistics because of their ability to deal with meshless domain. In this work, the different radial basis function approaches were investigated along with the focus on the strategies being addressed to find the shape parameter value. The mathematical formulations of the various radial basis function methods are discussed along with the available shape parameters to get the optimal value of the numerical solutions. The present work will lay a foundation to understand the development of the radial basis functions that could lead to a play an important role in development of method thereafter.