On investigation of kink-solitons and rogue waves to a new integrable (3+1)-dimensional KdV-type generalized equation in nonlinear sciences

This research study proposes a novel (3+1)-dimensional Painlevé integrable KdV-type equation that generalizes well-known equations in soliton theory and nonlinear sciences. It illustrates the Painlevé analysis to establish the complete integrability of the proposed equation. We employ the Cole-Hopf transformations to get the bilinear equation in an auxiliary function and further construct it into Hirota’s bilinear form. Utilizing the Hirota bilinear technique, we obtain the soliton solutions of kink types and their interactions up to the third order. It examines the rogue waves of the higher order using a direct symbolic approach up to the third order. For constructing the rogue waves, we transform the investigated equation from (3+1)-dimensional to a (1+1)-dimensional partial differential equation and form its Hirota bilinear form in transformed variables. It demonstrates the dynamics for the obtained kink-soliton and rogue wave solutions with appropriate parameter values using the symbolic system Mathematica. The interaction solutions of rogue waves show the dominating nature of more giant waves over smaller waves. We analyze the rogue dynamics in both the transformed and original variables. Solitons as solitary waves and rogue waves as extreme or monster waves are alluring concepts in various fields of nonlinear sciences, including oceanography, optical fibers, plasma physics, dynamical systems, and engineering.