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

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

The relation between a 2D Lotka-Volterra equation and a 2D Toda lattice

It is shown that the 2-discrete dimensional Lotka-Volterra lattice, the two dmensional Toda lattice equation and the recent 2-discrete dimensional Toda lattice equation of Santini et al can be obtained from a 2-discrete 2-continuous dimensional Lotka-Volterra lattice

A direct approach to the ultradiscrete KdV equation with negative

A generalisation of the ultra-discrete KdV equation is investigated using a direct approach. We show that evolution through one time step serves to reveal the entire solitonic content of the system

On Darboux transformations for the derivative nonlinear Schr\"odinger equation

We consider Darboux transformations for the derivative nonlinear Schr\"odinger equation. A new theorem for Darboux transformations of operators with no derivative term are presented and proved. The solution is expressed in quasideterminant forms. Additionally, the parabolic and soliton solutions of the derivative nonlinear Schr\"odinger equation are given as explicit examples.Comment: 14 page

A bilinear approach to a Pfaffian self-dual Yang-Mills equation

By using the bilinear technique of soliton theory, a pfaffian version of the SU(2) self-dual Yang-Mills equation and its solution is constructed