On the rational solutions and the solitons of the KP hierarchy

Yuji Kodama, Ohio State University

 It is well known that the Schur polynomials give rational solutions of the KP hierarchy, and that each Schur polynomial can be parametrized by a unique Young diagram. We also know that the KP solitons (exponential solutions) can be parametrized by certain decomposition of the Grassmannians. In the talk, I will explain the connection between the rational solutions and the KP solitons in terms of the Young diagrams. More explicitly,  I will show how one gets a rational solution describing the most degenerate zero locus of the tau-function from a KP soliton. I will also discuss a connection between quasi-periodic solutions (theta or sigma functions) and the KP solitons. The rational solutions then give theta divisors of certain algebraic curves.

This work is in progress under a collaboration with A. Mikhailov and J.-P. Wang.