Semidiscrete Toda lattices

We study integrable cutoff constraints for a semidiscrete Toda lattice. We construct a Lax representation for a semidiscrete analogue of lattices corresponding to simple Lie algebras of the C series. We introduce nonlocal variables in terms of which the symmetries of the infinite semidiscrete lattice can be expressed, and we classify cutoff constraints of a certain form compatible with the symmetries of the infinite lattice.     

Toda lattices and positive-entropy integrable systems

This paper studies completely integrable hamiltonian systems on T^* where is a bundle over with an -split, free abelian monodromy group. For each periodic Toda lattice there is an integrable hamiltonian system on T^* with positive topological entropy. Bolsinov and Taimanovs example of an integrable geodesic flow with positive topological entropy fits into this general construction with the A⁽¹⁾₁ Toda lattice. Topological entropy is used to show that the flows associated to non-dual Toda lattices are typically topologically non-conjugate via an energy-preserving homeomorphism. The remaining cases are approached via the homology spectrum. An energy-preserving conjugacy implies the congruence of two rational quadratic forms over the unit group of a number field F. When F/ is normal a classification of flows is obtained. In degree 3, this results from a well-known result of Gelfond; in higher degrees, the result is conditional on the conjecture that a rationally independent set of logarithms of algebraic numbers is algebraically independent over . Mathematics Subject Classification (2000) 58F17, 53D25, 37D40     

Laplace Invariants of Two-Dimensional Open Toda Lattices

We show that Toda lattices with the Cartan matrices A _n , B _n , C _n , and D _n are Liouville-type systems. For these systems of equations, we obtain explicit formulas for the invariants and generalized Laplace invariants. We show how they can be used to construct conservation laws (x and y integrals) and higher symmetries.     

Complex high order Toda and Volterra lattices1

Given a solution of a high order Toda lattice we construct a one parameter family of new solutions. In our method, we use a set of Bäcklund transformations such that each new generalized Toda solution is related to a generalized Volterra solution.     

A Laplace Ladder of Discrete Laplace Equations

We present the notion of a Laplace ladder for a discrete analogue of the Laplace equation. We introduce the adjoint of the discrete Moutard equation and a discrete counterpart of the nonlinear representation for the Goursat equation.     

Discrete Laplace cycles of period four

We study discrete conjugate nets whose Laplace sequence is of period four. Corresponding points of opposite nets in this cyclic sequence have equal osculating planes in different net directions, that is, they correspond in an asymptotic transformation. We show that this implies that the connecting lines of corresponding points form a discrete W-congruence. We derive some properties of discrete Laplace cycles of period four and describe two explicit methods for their construction.     

A family of hyperbolic spin Calogero‐Moser systems and the spin Toda lattices

In this paper, we continue to develop a general scheme to study a broad class of integrable systems naturally associated with the coboundary dynamical Lie algebroids. In particular, we present a factorization method for solving the Hamiltonian flows. We also present two important classes of new examples, a family of hyperbolic spin Calogero‐Moser systems and the spin Toda lattices. To illustrate our factorization theory, we show how to solve these Hamiltonian systems explicitly. © 2004 Wiley Periodicals, Inc.     

Spectral Estimations for the Laplace Operator on the Discrete Heisenberg Group

We estimate the spectral measure of the Laplace operator for the discrete Heisenberg group with generators x and y in the vicinity of the unity. Bibliography: 7 titles.     

A Discrete Laplace–Beltrami Operator for Simplicial Surfaces

We define a discrete Laplace–Beltrami operator for simplicial surfaces (Definition 16). It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called “cotan formula”) except that it is based on the intrinsic Delaunay triangulation of the simplicial surface. This leads to new definitions of discrete harmonic functions, discrete mean curvature, and discrete minimal surfaces. The definition of the discrete Laplace–Beltrami operator depends on the existence and uniqueness of Delaunay tessellations in piecewise flat surfaces. While the existence is known, we prove the uniqueness. Using Rippa’s Theorem we show that, as claimed, Musin’s harmonic index provides an optimality criterion for Delaunay triangulations, and this can be used to prove that the edge flipping algorithm terminates also in the setting of piecewise flat surfaces. Research for this article was supported by the DFG Research Unit 565 “Polyhedral Surfaces” and the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.     

Discrete Fourier analysis with lattices on planar domains

A discrete Fourier analysis associated with translation lattices is developed recently by the authors. It permits two lattices, one determining the integral domain and the other determining the family of exponential functions. Possible choices of lattices are discussed in the case of lattices that tile ${{\mathbb R}}^2$ and several new results on cubature and interpolation by trigonometric, as well as algebraic, polynomials are obtained.     

Eigenvalues and Eigenfunctions of the Laplace Operator on an Equilateral Triangle for the Discrete Case

A discretized boundary value problem for the Laplace equation with the Dirichlet and Neumann boundary conditions on an equilateral triangle with a triangular mesh is transformed into a problem of the same type on a rectangle. Explicit formulae for all eigenvalues and all eigenfunctions are given.     

Discrete maximum principle for Galerkin approximations of the Laplace operator on arbitrary meshes

We derive a nonlinear stabilized Galerkin approximation of the Laplace operator for which we prove a discrete maximum principle on arbitrary meshes and for arbitrary space dimension without resorting to the well-known acute condition or generalizations thereof. We also prove the existence of a discrete solution and discuss the extension of the scheme to convection–diffusion–reaction equations. Finally, we present examples showing that the new scheme cures local minima produced by the standard Galerkin approach while maintaining first-order accuracy in the H¹-norm. To cite this article: E. Burman, A. Ern, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Change of the time for the periodic Toda lattices and natural systems on the plane with higher order integrals of motion

We discuss some special classes of canonical transformations of the time variable, which relate different integrable systems. Such dual systems have different integrals of motion, Lax equations, separated variables and bi-hamiltonian structures. As an example the two-dimensional periodic Toda lattices associated with the classical root systems and the dual natural systems on the palne are considered in detail.