Cohomological structure in soliton equations and jacobian varieties

Abstract

The structure of the orbits of the dynamical system defined by the total KP hierarchy is studied. It is shown that every orbit is locally isomorphic to a certain cohomology group associated with a commutative algebra. The KP dynamical system restricted to each orbit determines a dynamical system of linear motions on it with respect to the linear structure of the cohomology group. Remarkably, it is proved that an orbit is finite dimensional if and only if it is essentially a Jacobian variety of an algebraic curve. Using this fact, the problem of characterization of Jacobians among Abelian varieties (Scho?tky Problem) is solved. It is also shown that our cohomology group describes complete families of iso-spectral deformations of linear ordinary differential operators. © 1984 Lehigh University. All rights reserved.