316 research outputs found
The Bianchi-Darboux transform of L-isothermic surfaces
We study an analogue of the classical Bianchi-Darboux transformation for
L-isothermic surfaces in Laguerre geometry, the Bianchi-Darboux transformation.
We show how to construct the Bianchi-Darboux transforms of an L-isothermic
surface by solving an integrable linear differential system. We then establish
a permutability theorem for iterated Bianchi-Darboux transforms.Comment: 13 pages, amstex, to be published in IJ
A discrete version of the Darboux transform for isothermic surfaces
We study Christoffel and Darboux transforms of discrete isothermic nets in
4-dimensional Euclidean space: definitions and basic properties are derived.
Analogies with the smooth case are discussed and a definition for discrete
Ribaucour congruences is given. Surfaces of constant mean curvature are special
among all isothermic surfaces: they can be characterized by the fact that their
parallel constant mean curvature surfaces are Christoffel and Darboux
transforms at the same time. This characterization is used to define discrete
nets of constant mean curvature. Basic properties of discrete nets of constant
mean curvature are derived.Comment: 30 pages, LaTeX, a version with high quality figures is available at
http://www-sfb288.math.tu-berlin.de/preprints.htm
Darboux transforms and spectral curves of constant mean curvature surfaces revisited
We study the geometric properties of Darboux transforms of constant mean
curvature (CMC) surfaces and use these transforms to obtain an
algebro-geometric representation of constant mean curvature tori. We find that
the space of all Darboux transforms of a CMC torus has a natural subset which
is an algebraic curve (called the spectral curve) and that all Darboux
transforms represented by points on the spectral curve are themselves CMC tori.
The spectral curve obtained using Darboux transforms is not bi-rational to, but
has the same normalisation as, the spectral curve obtained using a more
traditional integrable systems approach.Comment: 7 figure
Darboux transforms on Band Matrices, Weights and associated Polynomials
Classically, it is well known that a single weight on a real interval leads
to orthogonal polynomials. In "Generalized orthogonal polynomials, discrete KP
and Riemann-Hilbert problems", Comm. Math. Phys.
207, pp. 589-620 (1999), we have shown that -periodic sequences of weights
lead to "moments", polynomials defined by determinants of matrices involving
these moments and -step relations between them, thus leading to
-band matrices . Given a Darboux transformations on , which effect
does it have on the -periodic sequence of weights and on the associated
polynomials ? These questions will receive a precise answer in this paper. The
methods are based on introducing time parameters in the weights, making the
band matrix evolve according to the so-called discrete KP hierarchy.
Darboux transformations on that translate into vertex operators acting on
the -function.Comment: 43 page
Non-commutative NLS-type hierarchies: dressing & solutions
We consider the generalized matrix non-linear Schrodinger (NLS) hierarchy. By
employing the universal Darboux-dressing scheme we derive solutions for the
hierarchy of integrable PDEs via solutions of the matrix
Gelfand-Levitan-Marchenko equation, and we also identify recursion relations
that yield the Lax pairs for the whole matrix NLS-type hierarchy. These results
are obtained considering either matrix-integral or general order
matrix-differential operators as Darboux-dressing transformations. In this
framework special links with the Airy and Burgers equations are also discussed.
The matrix version of the Darboux transform is also examined leading to the
non-commutative version of the Riccati equation. The non-commutative Riccati
equation is solved and hence suitable conserved quantities are derived. In this
context we also discuss the infinite dimensional case of the NLS matrix model
as it provides a suitable candidate for a quantum version of the usual NLS
model. Similarly, the non-commutitave Riccati equation for the general dressing
transform is derived and it is naturally equivalent to the one emerging from
the solution of the auxiliary linear problem.Comment: 29 pages, LaTex. Minor modification
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