Kink, kink-lump, kink-breather and rough wave soliton solutions of the (3 + 1)-dimensional space-time fractional Jimbo-Miwa equation

In this study, we construct soliton solutions to the space-time fractional (3 + 1)-dimensional Jimbo-Miwa (JM) equation with a modified Riemann-Liouville derivative. Analysing this equation holds significant importance due to its utility in clarifying nonlinear wave behaviours in a variety of physical settings, such as plasma science, nuclear explosions, the solar system, tsunami waves and lightning pressure blasts. Using the Hirota bilinear method in conjunction with the homoclinic test technique and a complex transformation, based on the modified Riemann–Liouville fractional derivative, exact soliton solutions for the given equation are successfully established. A simplified algorithm is proposed to convert any nonlinear fractional partial differential equation into Hirota's bilinear form through complex transformation and the Cole-Hopf transformation. First, we apply the fractional complex transformation to the aforementioned equation to convert it into a solvable classical ordinary differential equation. After successfully implementing the proposed algorithm, multi-soliton, multi-kink, lump, breather and rough wave solutions are obtained for the fractional JM equation. To further explore the dynamical characteristics of the model, complex hybrid solutions, such as kink-breather and kink-lump wave solutions, are investigated. Furthermore, the impact of fractional parameters, along with other variables, on various wave structures is analysed using numerical graphs. The study illustrates different dynamic behaviours of solitons and their collisions as the fractional parameters change.