A novel analysis of Cole–Hopf transformations in different dimensions, solitons, and rogue waves for a (2 + 1)-dimensional shallow water wave equation of ion-acoustic waves in plasmas

This work investigates a (2 + 1)-dimensional shallow water wave equation of ion-acoustic waves in plasma physics. It comprehensively analyzes Cole–Hopf transformations concerning dimensions x, y, and t and obtains the dispersion for a phase variable of this equation. We show that the soliton solutions are independent of the different logarithmic transformations for the investigated equation. We also explore the linear equations in the auxiliary function f present in Cole–Hopf transformations. We study this equation's first- and second-order rogue waves using a generalized N-rogue wave expression from the N-soliton Hirota technique. We generate the rogue waves by applying a symbolic technique with β and γ as center parameters. We create rogue wave solutions for first- and second-order using direct computation for appropriate choices of several constants in the equation and center parameters. We obtain a trilinear equation by transforming variables ξ and y via logarithmic transformation for u in the function F. We harness the computational power of the symbolic tool Mathematica to demonstrate the graphics of the soliton and center-controlled rogue wave solutions with suitable choices of parameters. The outcomes of this study transcend the confines of plasma physics, shedding light on the interaction dynamics of ion-acoustic solitons in three-dimensional space. The equation's implications resonate across diverse scientific domains, encompassing classical shallow water theory, fluid dynamics, optical fibers, nonlinear dynamics, and many other nonlinear fields.