G.F. Helminck

An analytic description of the vector constrained KP hierarchy

Abstract:In this paper we give a geometric description in terms of the Grassmann manifold of Segal and Wilson, of the reduction of the KP hierarchy known as the vector

The Strict AKNS Hierarchy: Its Structure and Solutions

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Newton flows for elliptic functions I Structural stability : characterization & genericity

-constrained KP hierarchy. We also show in a geometric way that these hierarchies are equivalent to Krichevers general rational reductions of the KP hierarchy.

Newton flows for elliptic functions: a pilot study†

We discuss an integrable hierarchy of compatible Lax equations that is obtained by a wider deformation of a commutative algebra in the loop space of than that in the AKNS case and whose Lax equations are based on a different decomposition of this loop space. We show the compatibility of these Lax equations and that they are equivalent to a set of zero curvature relations. We present a linearization of the system and conclude by giving a wide construction of solutions of this hierarchy.

Newton flows for elliptic functions II

Abstract 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 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].

Newton flows for elliptic functions III & IV: Pseudo Newton graphs: bifurcation and creation of flows

Elliptic Newton flows are generated by a continuous, desingularized Newton method for doubly periodic meromorphic functions on the complex plane. In the special case, where the functions underlying these elliptic Newton flows are of second-order, we introduce various, closely related, concepts of structural stability. In particular, within the class of all (second order) elliptic Newton flows, structural stability turns out to be a generic property. Moreover, the phase portraits of all such structurally stable flows are equal, up to conjugacy. As an illustration, we treat the elliptic Newton flows for the Jacobi functions sn, cn and dn. †Dedicated to H.Th. Jongen on the occasion of his 60th birthday.

An integrable hierarchy including the AKNS hierarchy and its strict version

In our previous paper we associated to each non-constant elliptic function f on a torus T a dynamical system, the elliptic Newton flow corresponding to f. We characterized the functions for which these flows are structurally stable and showed a genericity result. In the present paper we focus on the classification and representation of these structurally stable flows. The phase portrait of a structurally stable elliptic Newton flow generates a connected, cellularly embedded graph

A formal infinite dimensional Cauchy problem and its relation to integrable hierarchies

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