Lax structure and tau function for large BKP hierarchy

In this paper, we mainly investigate Lax structure and tau function for the large BKP hierarchy, which is also known as Toda hierarchy of B type. Firstly, the large BKP hierarchy can be derived from fermionic BKP hierarchy by using a special bosonization, which is presented in the form of bilinear equation. Then from bilinear equation, the corresponding Lax equation is given, where in particular the relation of flow generator with Lax operator is obtained. Also starting from Lax equation, the corresponding bilinear equation and existence of tau function are discussed. After that, large BKP hierarchy is viewed as sub-hierarchy of modified Toda (mToda) hierarchy, also called two-component first modified KP hierarchy. Finally by using two basic Miura transformations from mToda to Toda, we understand two typical relations between large BKP tau function \(\tau _n(\textbf{t})\) ${\tau }_n(\mathbf{t})$ and Toda tau function \(\tau _n^\textrm{Toda}(\textbf{t},-\textbf{t})\) ${\tau }_n^{\text{Toda}}(\mathbf{t},-\mathbf{t})$ , that is, \(\tau _n^{\textrm{Toda}}(\textbf{t},-{\textbf{t}})=\tau _n(\textbf{t})\tau _{n-1}(\textbf{t})\) ${\tau }_n^{\text{Toda}}(\mathbf{t},-\mathbf{t})={\tau }_n(\mathbf{t}){\tau }_{n-1}(\mathbf{t})$ and \(\tau _n^{\textrm{Toda}}(\textbf{t},-{\textbf{t}})=\tau _n^2(\textbf{t})\) ${\tau }_n^{\text{Toda}}(\mathbf{t},-\mathbf{t})={\tau }_n^2(\mathbf{t})$ . Further, we find \(\big (\tau _n(\textbf{t})\tau _{n-1}(\textbf{t}),\tau _n^2(\textbf{t})\big )\) $({\tau }_n(\mathbf{t}){\tau }_{n-1}(\mathbf{t}),{\tau }_n^2(\mathbf{t}))$ satisfies bilinear equation of mToda hierarchy.