Results 1  10
of
71
The Pentagram Map: A Discrete Integrable System
, 2008
"... The pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons. (A twisted polygon is a map from Z into the projective plane that is periodic modulo a projective transformation.) We find a Poisson structure on the space of twisted polygons and ..."
Abstract

Cited by 51 (20 self)
 Add to MetaCart
The pentagram map is a projectively natural iteration defined on polygons, and also on objects we call twisted polygons. (A twisted polygon is a map from Z into the projective plane that is periodic modulo a projective transformation.) We find a Poisson structure on the space of twisted polygons and show that the pentagram map relative to this Poisson structure is completely integrable in the sense of ArnoldLiouville. For certain families of twisted polygons, such as those we call universally convex, we translate the integrability into a statement about the quasiperiodic motion for the dynamics of the pentagram map. We also explain how the pentagram map, in the continuous limit, corresponds to the classical Boussinesq equation. The Poisson structure we attach to the pentagram map is a discrete version of the first Poisson structure associated with the Boussinesq equation.
AN OVERVIEW OF VARIATIONAL INTEGRATORS
, 2003
"... The purpose of this paper is to survey some recent advances in variational integrators for both finite dimensional mechanical systems as well as continuum mechanics. These advances include the general development of discrete mechanics, applications to dissipative systems, collisions, spacetime int ..."
Abstract

Cited by 17 (3 self)
 Add to MetaCart
The purpose of this paper is to survey some recent advances in variational integrators for both finite dimensional mechanical systems as well as continuum mechanics. These advances include the general development of discrete mechanics, applications to dissipative systems, collisions, spacetime integration algorithms, AVI’s (Asynchronous Variational Integrators), as well as reduction for discrete mechanical systems. To keep the article within the set limits, we will only
Painlevé Tests, Singularity Structure, and Integrability
"... Summary. After a brief introduction to the Painlevé property for ordinary differential equations, we present a concise review of the various methods of singularity analysis which are commonly referred to as Painlevé tests. The tests are applied to several different examples, and the connection betwe ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
(Show Context)
Summary. After a brief introduction to the Painlevé property for ordinary differential equations, we present a concise review of the various methods of singularity analysis which are commonly referred to as Painlevé tests. The tests are applied to several different examples, and the connection between singularity structure and integrability for ordinary and partial differential equation is discussed. 1
DESARGUES MAPS AND THE HIROTA–MIWA EQUATION
"... Abstract. We study the Desargues maps φ: Z N → P M, which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multidimensional consistency of the map is equivalent to the Desargues theorem and its higherdimensional generalizations. The nonlin ..."
Abstract

Cited by 15 (9 self)
 Add to MetaCart
(Show Context)
Abstract. We study the Desargues maps φ: Z N → P M, which generate lattices whose points are collinear with all their nearest (in positive directions) neighbours. The multidimensional consistency of the map is equivalent to the Desargues theorem and its higherdimensional generalizations. The nonlinear counterpart of the map is the noncommutative (in general) Hirota–Miwa system. In the commutative case of the complex field we apply the nonlocal ¯ ∂dressing method to construct Desargues maps and the corresponding solutions of the equation. In particular, we identify the Fredholm determinant of the integral equation inverting the nonlocal ¯ ∂dressing problem with the τfunction. Finally, we establish equivalence between the Desargues maps and quadrilateral lattices provided we take into consideration also their Laplace transforms.
Integrable hierarchies and the mirror model of local CP 1. Physica D: Nonlinear Phenomena
, 2011
"... ar ..."
(Show Context)
Local conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik hierarchy
, 2007
"... We derive a systematic and recursive approach to local conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik (AL) hierarchy. Our methods rely on a recursive approach to the AL hierarchy using Laurent polynomials and on asymptotic expansions of the Green’s function of the AL Lax oper ..."
Abstract

Cited by 10 (5 self)
 Add to MetaCart
(Show Context)
We derive a systematic and recursive approach to local conservation laws and the Hamiltonian formalism for the Ablowitz–Ladik (AL) hierarchy. Our methods rely on a recursive approach to the AL hierarchy using Laurent polynomials and on asymptotic expansions of the Green’s function of the AL Lax operator, a fivediagonal finite difference operator.
DISCRETE HAMILTON–JACOBI THEORY
"... Abstract. We develop a discrete analogue of the Hamilton–Jacobi theory in the framework of the discrete Hamiltonian mechanics. We first reinterpret the discrete Hamilton–Jacobi equation derived by Elnatanov and Schiff in the language of discrete mechanics. The resulting discrete Hamilton– Jacobi equ ..."
Abstract

Cited by 9 (6 self)
 Add to MetaCart
(Show Context)
Abstract. We develop a discrete analogue of the Hamilton–Jacobi theory in the framework of the discrete Hamiltonian mechanics. We first reinterpret the discrete Hamilton–Jacobi equation derived by Elnatanov and Schiff in the language of discrete mechanics. The resulting discrete Hamilton– Jacobi equation is discrete only in time, and is shown to recover the Hamilton–Jacobi equation in the continuoustime limit. The correspondence between discrete and continuous Hamiltonian mechanics naturally gives rise to a discrete analogue of Jacobi’s solution to the Hamilton–Jacobi equation. We also prove a discrete analogue of the geometric Hamilton–Jacobi theorem of Abraham and Marsden. These results are readily applied to discrete optimal control setting, and some wellknown results in discrete optimal control theory, such as the Bellman equation (discretetime Hamilton–Jacobi–Bellman equation) of dynamic programming, follow immediately. We also apply the theory to discrete linear Hamiltonian systems, and show that the discrete Riccati equation follows as a special case of the discrete Hamilton–Jacobi equation.
Suris: On the Hamiltonian structure of HirotaKimura discretization of the Euler top
, 707
"... Abstract. This paper deals with a remarkable integrable discretization of the so(3) Euler top introduced by Hirota and Kimura. Such a discretization leads to an explicit map, whose integrability has been understood by finding two independent integrals of motion and a solution in terms of elliptic fu ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
Abstract. This paper deals with a remarkable integrable discretization of the so(3) Euler top introduced by Hirota and Kimura. Such a discretization leads to an explicit map, whose integrability has been understood by finding two independent integrals of motion and a solution in terms of elliptic functions. Our goal is the construction of its Hamiltonian formulation. After giving a simplified and streamlined presentation of their results, we provide a biHamiltonian structure for this discretization, thus proving its integrability in the standard LiouvilleArnold sense. 1.