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.