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 $\tau$ -functions of hypergeometric type. These are series in Schur $Q$ -functions over partitions, with coefficients parameterised by a function of one variable $\xi$ , where the quantities $\xi(k)$ , $k\in\mathbb{Z^+}$ , 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 $\xi(k)$ 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 $\tau$ -function of hypergeometric type. We also show that the KdV and the NLS soliton $\tau$ -functions coinside the BKP $\tau$ -functions of hypergeometric type, evaluated at special point of BKP higher time; the variables $\xi$ (which are BKP integrals of motions) being related to KdV and NLS higher times.