Bispectral KP Solutions and Linearization of Calogero-Moser Particle Systems

Rational and soliton solutions of the KP hierarchy in the subgrassmannianGr1 are studied within the context of finite dimensional dual grassmannians. In the rational case, properties of the tau function, τ , which are equivalent to bispectrality of the associated wave function, ψ, are identified. In particular, it is shown that there exists a bound on the degree of all time variables in τ if and only if ψ is a rank one bispectral wave function. The action of the bispectral involution, β, in the generic rational case is determined explicitly in terms of dual grassmannian parameters. Using the correspondence between rational solutions and particle systems, it is demonstrated that β is a linearizing map of the Calogero-Moser particle system and is essentially the map σ introduced by Airault, McKean and Moser in 1977 [2].