Non-Abelian Bell Polynomials and Their some Applications for Integrable Systems - Zhang, Yufeng (Discontinuity, Nonlinearity, and Complexity, 2026)
The noncommutative Bell polynomials and their dual Bell polynomials are presented, respectively, which are extensively applied to mathematics and physics. We make use of them to exhibit a method for generating integrable hierarchies of evolution equations. As applications, we obtain the Burgers hierarchy and a convection-diffusion equation which can be applied to fluid mechanics, specially, be used to represent mass transformations in fluid systems under some constrained conditions. As reduced cases, the Burgers equation which has extensive applications in physics is followed to produce. Furthermore, we obtain a set of nonlinear evolution equations with four potential functions which reduces to a new nonlinear equation similar to the Calogero-Degasperis-Fokas equation. Finally, we discrete the convection-diffusion equation and obtain its three kinds of finite-difference schemes, that is,the weighted implicit difference scheme and the Lax difference scheme. Their some properties including truncation errors, compatibilities and stabilities based on the Von Neumann condition are discussed in detail.