arXiv subjects

G. F. Helminck

Publications and source records attributed to G. F. Helminck.

6 recordsLinked to original sources

arXiv2018

Newton flows for elliptic functions III Classification of $3^{\text{rd}}$ order Newton graphs

A Newton graph of order $r( \geqslant 2)$ is a cellularly embedded toroidal graph on $r$ vertices, $2r$ edges and $r$ faces that fulfils certain combinatorial properties (Euler, Hall). The significance of these graphs relies on their role in the study of structurally stable elliptic Newton flows - say $\bar{\bar{\mathcal{N}}} (f)$ - of order $r$, i.e. desingularized continuous versions of Newton's iteration method for finding zeros for an elliptic function $f$ (of order $r$). In previous work we established a representation of these flows in terms of Newton graphs. The present paper results into the classification of all $3^{\text{rd}}$ order Newton graphs, implying a list of all nine possible $3^{\text{rd}}$ order flows $\bar{\bar{\mathcal{N}}} (f)$ (up to conjugacy and duality).

math.DS↗

arXiv2017

Newton flows for elliptic functions I Structural stability: Characterization & Genericity

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 (\geqslant 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].

math.DS↗

arXiv2017

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

An elliptic Newton flow is a dynamical system that can be interpreted as a continuous version of Newton's iteration method for finding the zeros of an elliptic function f. Previous work focusses on structurally stable flows (i.e., the phase portraits are topologically invariant under perturbations of the poles and zeros for f), including a classification / representation result for such flows in terms of Newton graphs (i.e., cellularly embedded toroidal graphs fulfilling certain combinatorial properties). The present paper deals with non-structurally stable elliptic Newton flows determined by pseudo Newton graphs (i.e., cellularly embedded toroidal graphs, either generated by a Newton graph, or the so called nuclear Newton graph, exhibiting only one vertex and two edges). Our study results into a deeper insight in the creation of structurally stable Newton flows and the bifurcation of non-structurally stable Newton flows.

math.DS↗

arXiv2016

Newton flows for elliptic functions II Structural stability: Classification & Representation

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 $\mathcal{G}(f)$ on a torus $T$ with $r$ vertices, 2$r$ edges and $r$ faces that fulfil certain combinatorial properties ( Euler, Hall) on some of its subgraphs. The graph $\mathcal{G}(f)$ determines the conjugacy class of the flow. [classification] A connected, cellularly embedded toroidal graph $\mathcal{G}$ with the above Euler and Hall properties, is called a Newton graph. Any Newton graph $\mathcal{G}$ can be realized as the graph $\mathcal{G}(f)$ of the structurally stable Newton flow for some function $f$. This leads to: up till conjugacy between flows and (topological) equivalency between graphs, there is a one to one correspondence between the structurally stable Newton flows and Newton graphs, both with respect to the same order $r$ of the underlying functions $f$.[representation] Finally, we clarify the analogy between rational and elliptic Newton flows, and show that the detection of elliptic Newton flows is possible in polynomial time.

math.DS↗

arXiv1998

Geometric Bäcklund--Darboux transformations for the KP hierarchy

We shown that, if you have two planes in the Segal-Wilson Grassmannian that have an intersection of finite codimension, then the corresponding solutions of the KP hierarchy are linked by Bäcklund-Darboux transformations (BDT). The pseudodifferential operator that performs this transformation is shown to be built up in a geometric way from elementary BDT's and is given here in a closed form. The geometric description of elementary DBT's requires that one has a geometric interpretation of the dual wavefunctions involved. This is done here with the help of a suitable algebraic characterization of the wavefunction. The BDT's also induce transformations of the tau-function associated to a plane in the Grassmannian. For the Gelfand-Dickey hierarchies we derive a geometric characterization of the BDT'ss that preserves these subsystems of the KP hierarchy. This generalizes the classical Darboux-transformations. we also determine an explicit expression for the squared eigenfunction potentials. Next a connection is laid between the KP hierarchy and the 1-Toda lattice hierarchy. It is shown that infinite flags in the Grassmannian yield solutions of the latter hierarchy. these flags can be constructed by means of BDT's, starting from some plane. Other applications of these BDT's are a geometric way to characterize Wronskian solutions of the $m$-vector $k$-constrained KP hierarchy and the construction of a vast collection of orthogonal polynomials, playing a role in matrix models.

solv-int↗

arXiv1997

An analytic description of the vector constrained KP hierarchy

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 $k$-constrained KP hierarchy. We also show in a geometric way that these hierarchies are equivalent to Krichever's general rational reductions of the KP hierarchy.

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