349 research outputs found

### On the Birkhoff factorization problem for the Heisenberg magnet and nonlinear Schroedinger equations

A geometrical description of the Heisenberg magnet (HM) equation with
classical spins is given in terms of flows on the quotient space $G/H_+$ where
$G$ is an infinite dimensional Lie group and $H_+$ is a subgroup of $G$. It is
shown that the HM flows are induced by an action of $\mathbb{R}^2$ on $G/H_+$,
and that the HM equation can be integrated by solving a Birkhoff factorization
problem for $G$. For the HM flows which are Laurent polynomials in the spectral
variable we derive an algebraic transformation between solutions of the
nonlinear Schroedinger (NLS) and Heisenberg magnet equations. The Birkhoff
factorization for $G$ is treated in terms of the geometry of the Segal-Wilson
Grassmannian $Gr(H)$. The solution of the problem is given in terms of a pair
of Baker functions for special subspaces of $Gr(H)$. The Baker functions are
constructed explicitly for subspaces which yield multisoliton solutions of NLS
and HM equations.Comment: To appear in Journal of Mathematical Physic

### Integrable discrete nets in Grassmannians

We consider discrete nets in Grassmannians $\mathbb{G}^d_r$ which generalize
Q-nets (maps $\mathbb{Z}^N\to\mathbb{P}^d$ with planar elementary
quadrilaterals) and Darboux nets ($\mathbb{P}^d$-valued maps defined on the
edges of $\mathbb{Z}^N$ such that quadruples of points corresponding to
elementary squares are all collinear). We give a geometric proof of
integrability (multidimensional consistency) of these novel nets, and show that
they are analytically described by the noncommutative discrete Darboux system.Comment: 10 p

### Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top

We develop the theory of discrete time Lagrangian mechanics on Lie groups,
originated in the work of Veselov and Moser, and the theory of Lagrangian
reduction in the discrete time setting. The results thus obtained are applied
to the investigation of an integrable time discretization of a famous
integrable system of classical mechanics, -- the Lagrange top. We recall the
derivation of the Euler--Poinsot equations of motion both in the frame moving
with the body and in the rest frame (the latter ones being less widely known).
We find a discrete time Lagrange function turning into the known continuous
time Lagrangian in the continuous limit, and elaborate both descriptions of the
resulting discrete time system, namely in the body frame and in the rest frame.
This system naturally inherits Poisson properties of the continuous time
system, the integrals of motion being deformed. The discrete time Lax
representations are also found. Kirchhoff's kinetic analogy between elastic
curves and motions of the Lagrange top is also generalised to the discrete
context.Comment: LaTeX 2e, 44 pages, 1 figur

### Discrete constant mean curvature nets in space forms: Steiner's formula and Christoffel duality

We show that the discrete principal nets in quadrics of constant curvature
that have constant mixed area mean curvature can be characterized by the
existence of a K\"onigs dual in a concentric quadric.Comment: 12 pages, 10 figures, pdfLaTeX (plain pdfTeX source included as bak
file

### Minimal surfaces from circle patterns: Geometry from combinatorics

We suggest a new definition for discrete minimal surfaces in terms of sphere
packings with orthogonally intersecting circles. These discrete minimal
surfaces can be constructed from Schramm's circle patterns. We present a
variational principle which allows us to construct discrete analogues of some
classical minimal surfaces. The data used for the construction are purely
combinatorial--the combinatorics of the curvature line pattern. A
Weierstrass-type representation and an associated family are derived. We show
the convergence to continuous minimal surfaces.Comment: 30 pages, many figures, some in reduced resolution. v2: Extended
introduction. Minor changes in presentation. v3: revision according to the
referee's suggestions, improved & expanded exposition, references added,
minor mistakes correcte

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