Advanced Algebraic Approaches to Soliton Solutions in Shallow Water Wave Equations

In this paper, soliton solutions of the (1+1)-dimensional Boussinesq equation are studied with an improved scheme of the direct algebraic method. The Boussinesq equation as concerns the propagation of long waves in shallow water in addition to other physical systems, exhibit both dispersal and nonlinearity. With this new approach we obtain analytical soliton solutions and make a study of their characteristics. The obtained solutions are said in terms of positive forms and this is an improvement of the perception of the dynamics of the equation. The performance of the method is illustrated by calculation, and further use of the method in the related nonlinear systems is also discussed. This work adds to an ongoing research on the analysis of the solution of nonlinear partial differential equations and provides understanding on the solitons in the dispersive media. The obtained solutions as well have been depicted graphically for better understanding. © 2025 NSP Natural Sciences Publishing Cor.