Darboux transformations of noncommutative BKP and CKP hierarchies

In this paper, we construct Darboux transformations of the noncommutative BKP (ncBKP) hierarchy and the noncommutative CKP (ncCKP) hierarchy. From these Darboux transformations, the explicit differences between the BKP (CKP) hierarchy and the noncommutative BKP (ncCKP) hierarchy can be seen clearly in this paper. These studies might be useful in the M theory and string theory in future.

Gauge transformations of the multi-component BKP and CKP hierarchies

In this paper, we give the definition and multi-folds gauge transformations of the multi-component B-type Kakomtsev–Petviashvili (MBKP) hierarchy and multi-component CKP (MCKP) hierarchy. Besides, we derive new solutions after the one-fold gauge transformation of the MBKP hierarchy. The remarkable differences between the BKP (CKP) hierarchy and the MBKP(MCKP) hierarchy are given in this paper. What is more, we introduce some facts about the multi-component constrained BKP and multi-component constrained CKP hierarchies including some obstacles and some new multi-component equations.