Darboux Transforms for the $hat B_{n}^{(1)}$-hierarchy

The $hat B_n^{(1)}$-hierarchy is constructed from the standard splitting of the affine Kac-Moody algebra $hat B_n^{(1)}$, the Drinfeld-Sokolov $hat B_n^{(1)}$-KdV hierarchy is obtained by pushing down the $hat B_n^{(1)}$-flows along certain gauge orbit to a cross section of the gauge action. In this paper, we (1) use loop group factorization to construct Darboux transforms (DTs) for the $hat B_n^{(1)}$-hierarchy, (2) give a Permutability formula and scaling transform for these DTs, (3) use DTs of the $hat B_{n}^{(1)}$-hierarchy to construct DTs for the $hat B_n^{(1)}$-KdV and the isotropic curve flows of B-type, (4) give algorithm to construct soliton solutions and write down explicit soliton solutions for the third $hat B_1^{(1}$-KdV, $hat B_2^{(1)}$-KdV flows and isotropic curve flows on $mathbb{R}^{2,1}$ and $mathbb{R}^{3,2}$ of B-type.