Solutions of the universal character hierarchy and BUC hierarchy by fermionic approach

In this paper, by means of the fermionic approach, we investigate the (m,n)-order τ-function of the universal character (UC) hierarchy which can be expressed as the determinants of the Schur functions. Meanwhile, we present the general τ-function for charged free fermions, from which soliton solutions of the UC hierarchy have been discussed. Furthermore, the algebraic structure and properties of the universal character hierarchy of B type (BUC hierarchy) have been developed. In terms of neutral free fermions and the generalized Q-function, the polynomial τ-functions and the extended formulas of the BUC and the 2-component BUC hierarchies have been analyzed. In addition, we derive the rational solutions of the BUC hierarchy. It is showed that soliton solutions of the BUC hierarchy can be expressed as multiplication of Pfaffians.