Analytical Study of the (3+1)-Dimensional Hirota-Jimbo-Miwa Equation: Exact Solutions and Localized Excitations in Optical Field

This paper explores the (3+1)-dimensional Hirota-Jimbo-Miwa equation, which is significant in fluid dynamics, nonlinear optics, and plasma physics. The complete integrability, bilinear form, Bäcklund transformation, and various exact solutions of this equation have not been thoroughly investigated before. Using the Hirota bilinear method and variable separation approach, we successfully construct multiple types of localized excitations, including multi-kink solitons, lump solutions, breather-kink hybrids, and folded solitons. We also develop two distinct bilinear Bäcklund transformations with free parameters, which provide a powerful tool for generating a wide range of solutions. Our findings reveal the rich solution structures of the Hirota-Jimbo-Miwa equation and their parametric regulation, offering new analytical models for complex wave phenomena in optical fields and fluid dynamics. The novelty of this work lies in the systematic study of the Hirota-Jimbo-Miwa equation and the proposal of new mathematical methods for constructing exact solutions of complex high-dimensional nonlinear systems.