Integrability and New Periodic, Kink–Antikink, and Complex Multiple Soliton Solutions to the (3 + 1)-Dimensional Extended Jimbo–Miwa Equation With Time-Dependent Variable Coefficients in Plasma Physics

This article presents the exact solutions for the time-dependent variable coefficients of the ( $3+1$ )-dimensional extended Jimbo–Miwa (JM) equation. The considered equation is demonstrated to be completely integrable via the Painlevé analysis method. The Laurent series, derived using the Painlevé analysis method, has been truncated to yield an auto-Bäcklund transformation (ABT), and this method is employed to construct analytical solutions. Three new categories of analytical solutions are effectively generated for the considered equation via the ABT. Also, multi-soliton solutions are obtained using the simplified Hirota method for the equation under consideration. All the determined results are illustrated in 3-D graphs through various functions and parameter settings. These graphs reflect the physical significance of the equation being studied.