Generalization and analytic exploration of soliton solutions for nonlinear evolution equations via a novel symbolic approach in fluids and nonlinear sciences

In this work, we analyze the new generalized soliton solutions for the nonlinear partial differential equations with a novel symbolic bilinear technique. The proposed approach constructs the soliton solutions depending on the arbitrary parameters, which generalizes the soliton solutions with these additional parameters. Examining phase shifts and their dependence on the parameters influences how solitons collide, merge, or pass through each other, which is essential for the nonlinear analysis of solitons. Using the proposed technique, we examine the well-known (1＋1)-dimensional Korteweg–de Vries (KdV) and (2＋1)-dimensional Kadomtsev–Petviashvili (KP) equations with a comparative analysis of soliton solutions in the Hirota technique. We construct the generalized solitons solutions for both examined equations up to the third order, providing a better understanding of formed solitons with arbitrary parameter choices. The Cole-Hopf transformations are used to construct the bilinear form in the auxiliary function using Hirota’s $D$-operators for both investigated KdV and KP equations It discusses the phase shift depending on parameters and compares it to the phase shift in Hirota’s soliton solutions. We utilize Mathematica, a computer algebra system, to obtain the generalized solitons and analyze the dynamic behavior of the obtained solutions by finding the values for the parameters and the relationships among them. Solitons are localized waves that appear in different fields of nonlinear sciences, such as oceanography, plasmas, fluid mechanics, water engineering, optical fibers, and other sciences.