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