In this paper it is shown that, if you have two planes in the Sato Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by Bäcklund-Darboux (shortly BD-)transformations. The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from so-called elementary BD-transformations and is given here in a closed form. The corresponding action on the tau-function, associated to a plane in the Grassmannian, is also determined §1. The KP-HierarchyThe KP-hierarchy consists of a tower of nonlinear evolution equations in infinitely many variables {t n |n ≥ 1}. It is named after the simplest nontrivial equation in this tower, the Khadomtsev-Petviashvili equation:The compact form in which these equations are usually presented, is the socalled Lax form. To give some insight in this form and to formulate precisely
In this paper we present a geometric realization of infinite dimensional analogues of the finite dimensional representations of the general linear group. This requires a detailed analysis of the structure of the flag varieties involved and the line bundles over them. In general the action of the restricted linear group can not be lifted to the line bundles and thus leads to central extensions of this group. It is determined exactly when these extensions are non-trivial. These representations are of importance in quantum field theory and in the framework of integrable systems. As an application, it is shown how the flag varieties occur in the latter context.
Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the complex plane. Inspired by the analogy between the rational (complex) and the elliptic (i.e., doubly periodic meromorphic) functions, a theory on the class of so-called Elliptic Newton flows is developed.With respect to an appropriate topology on the set of all elliptic functions f of fixed order r( 2) we prove: For almost all functions f , the corresponding Newton flows are structurally stable i.e., topologically invariant under small perturbations of the zeros and poles for f [genericity]. They can be described in terms of nondegeneracy-properties of f similar to the rational case [characterization].
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