Darboux Transformations And Additional Symmetries Of The Noncommuative BKP And CKP Hierarchies

In this paper,we define the noncommutative B type Kadomtsev-Petviashvili?ncBKP?and the noncommutative C type Kadomtsev-Petviashvili?ncCKP?hierarchies,then we construct Darboux transformations of the ncBKP hierarchy and the ncCKP hierarchy.From these Darboux transformations,the explicit differences between the BKP?or CKP?hierarchy and the ncBKP?or ncCKP?hierarchy can be seen clearly in this paper.What's more,a series of additional flows of the ncBKP and the ncCKP hierarchies will be defined and the additional symmetry flows constitute a infinite dimensional Lie algebra W_1+?.In addition,the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions.Further,the String equations of the ncBKP and the ncCKP hierarchies can be derived.At last,we give the recursion operator of the noncommutative constrained CKP hierarchy and the recursion operators of the noncommutative constrained KP hierarchy and the multi-component noncommutative constrained KP hierarchy,so that we can get any higher order flows in the following study.