Symbolic Computational Algorithm for Hirota Bilinear Form to Higher-dimensional Nonlinear Partial Differential Equations in Nonlinear Sciences

Abstract

In the physical world, many real systems are governed by nonlinear partial differential equations from fluid dynamics and plasma phyasics to shallow-water waves and oceanographic systems. There is no uniform approach for solving nonlinear partial differential equations; consequently, we consider each equation as a separate problem. The most effective technique for building multi-soliton solutions of an integrable nonlinear PDE is the Hirota direct method from the several methods used to explore nonlinear PDEs to obtain solitons, lumps, rogue waves, breathers, and kink waves. To derive the multi-solitons of the non-linear PDEs, this method requires first transforming the equation into the bilinear form proposed by Hirota, and then applying the dependent variable transformation. Converting nonlinear evolution equations into Hirota bilinear form is the primary goal of the research study. This work investigated several well-known nonlinear PDEs, including the KdV Equation, Boussinesq Equation, GS Equation, KP equation, and other equations in order to comprehend and apply the concept. We design the algorithm using symbolic software Mathematica. These equations are widely recognized for multi-solitons and their integrability, which having applications in diverse fileds oceanography, fluid dynamics, plasma physics, mechanics, and other nonlinear sciences.