Solving constrained BKP hierarchy by boson–fermion correspondence

Abstract

Solutions of $(k,m)$–constrained BKP hierarchy $L^k={\left(L^k\right)}_{\ge 0}+{\sum }_{i=1}^m(q_{1,i}{\partial }^{-1}{\left(q_{2,i}\right)}_x-q_{2,i}{\partial }^{-1}{\left(q_{1,i}\right)}_x)$ with $k\in Z_{\ge 0}^{\mathrm{odd}}$ are investigated by boson–fermion correspondence. Firstly, $(k,m)$–constrained BKP hierarchy is converted into the fermionic form ${\Omega }_k({\tau }_0\otimes {\tau }_0)=\frac{1}{2}{\sum }_{i=1}^m({\tau }_{2,i}\otimes {\tau }_{1,i}-{\tau }_{1,i}\otimes {\tau }_{2,i})$. Then we determine the conditions for the fermionic field β, which keeps the fermionic form of constrained BKP hierarchy after the transformations ${\tau }_{\alpha }\to \beta {\tau }_{\alpha }$ for $\alpha =0$, $\{1,i\}$ and $\{2,i\}$. Finally based upon this, the corresponding solutions are given by boson–fermion correspondence.