关键词: Bose-Fermi, mixture, prolongation structure, Lax pair, Darboux transformation

Abstract: The widespread popularization and application of laser technology have provided a powerful tool for a deeper understanding of the material world and given birth to several emerging research fields. This study mainly focuses on the following three key aspects. First, the classical ensemble method is adopted to conduct a comprehensive and in-depth analysis of two-dimensional (2D) matter-wave pulses in Bose-Fermi mixed gases (including linear and nonlinear pulses). Second, under the strict constraints of unitary systems, a coupled $\mathrm{KdV}$ equation is successfully derived, and the prolongation structure theory is skillfully used to carry out detailed calculations and analyses on this equation. Thus, the prolongation algebra of this equation is accurately determined, and the corresponding $\mathrm{Lax}$ pair is rigorously derived. Finally, based on the carefully obtained $\mathrm{Lax}$ pair from the prolongation structure theory, the soliton solutions of this equation are further analyzed in depth, and intuitive images of each soliton solution are carefully drawn. This lays a solid foundation for subsequent detailed research on these soliton characteristics and provides great convenience.

Key words: Bose-Fermi, mixture, prolongation structure, Lax pair, Darboux transformation