  We show that the trigonometric solitons of the KP hierarchy enjoy a
differential-difference bispectral property, which becomes transparent when
translated on two suitable spaces of pairs of matrices satisfying certain rank
one conditions. The result can be seen as a non-self-dual illustration of
Wilson's fundamental idea [Invent. Math. 133 (1998), 1-41] for understanding
the (self-dual) bispectral property of the rational solutions of the KP
hierarchy. It also gives a bispectral interpretation of a (dynamical) duality
between the hyperbolic Calogero-Moser system and the rational
Ruijsenaars-Schneider system, which was first observed by Ruijsenaars [Comm.
Math. Phys. 115 (1988), 127-165].
