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

TL;DR

This paper develops Darboux transforms for the $ obreak B_n^{(1)}$-hierarchy using loop group factorization, enabling explicit soliton solutions and extending to related flows and KdV hierarchies.

Contribution

It introduces a method to construct Darboux transforms for the $ obreak B_n^{(1)}$-hierarchy and applies these to generate soliton solutions and related flows.

Findings

Constructed Darboux transforms for the hierarchy

Derived permutability and scaling formulas

Explicit soliton solutions for specific flows

Abstract

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.