A relationship between rational and multi-soliton solutions of the BKP hierarchy

Abstract

We consider a special class of solutions of the BKP hierarchy which we call $τ$-functions of hypergeometric type. These are series in Schur $Q$-functions over partitions, with coefficients parameterised by a function of one variable $ξ$, where the quantities $ξ(k)$, $k\in\mathbb{Z^+}$, are integrals of motion of the BKP hierarchy. We show that this solution is, at the same time, a infinite soliton solution of a dual BKP hierarchy, where the variables $ξ(k)$ are now related to BKP higher times. In particular, rational solutions of the BKP hierarchy are related to (finite) multi-soliton solution of the dual BKP hierarchy. The momenta of the solitons are given by the parts of partitions in the Schur $Q$-function expansion of the $τ$-function of hypergeometric type. We also show that the KdV and the NLS soliton $τ$-functions coinside the BKP $τ$-functions of hypergeometric type, evaluated at special point of BKP higher time; the variables $ξ$ (which are BKP integrals of motions) being related to KdV and NLS higher times.