Darboux Transforms for the ${\widehat{B}}_n^{(1)}$-hierarchy

The ${\widehat{B}}_n^{(1)}$-hierarchy is constructed from the standard splitting of the affine Kac-Moody algebra ${\widehat{B}}_n^{(1)}$, the Drinfeld-Sokolov $\hat B_n^{(1)}$-KdVhierarchyisobtainedbypushingdownthe$\hat B_n^{(1)}$-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 ${\widehat{B}}_n^{(1)}$-hierarchy, (2) give a Permutability formula and scaling transform for these DTs, (3) use DTs of the ${\widehat{B}}_n^{(1)}$-hierarchy to construct DTs for the ${\widehat{B}}_n^{(1)}$-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 ${\widehat{B}}_1^{(1}$-KdV, ${\widehat{B}}_2^{(1)}$-KdV flows and isotropic curve flows on ${\mathbb{R}}^{2,1}$ and ${\mathbb{R}}^{3,2}$ of B-type.