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271
Jacobi operators and completely integrable nonlinear lattices
 MATHEMATICAL SURVEYS AND MONOGRAPHS
, 2000
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Liegroup methods
 ACTA NUMERICA
, 2000
"... Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having ..."
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Cited by 149 (24 self)
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Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Liegroup structure, highlighting theory, algorithmic issues and a number of applications.
NONLINEAR EQUATIONS OF KORTEWEGDE VRIES TYPE, FINITEZONE LINEAR OPERATORS, AND ABELIAN VARIETIES
 RUSS MATH SURV
, 1976
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Infinite wedge and random partitions
 Selecta Mathematica (new series
"... The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example, ..."
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Cited by 96 (7 self)
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The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example,
The problem of integrable discretization: Hamiltonian approach
 Progress in Mathematics, Volume 219. Birkhäuser
"... this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the e ..."
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Cited by 71 (2 self)
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this paper. Hence, the nature of this discretization as a member of the Toda hierarchy was not understood properly. A complete account was given in Suris (1995). The work Rutishauser (1954) contains not only the denition of the qd algorithm from the viewpoint of the numerical analist, but also the equations of motion of the Toda lattice (under the name of a \continuous analogue of the qd algorithm")! The relation of the qd algorithm to integrable systems might have important implications for the numerical analysis, cf. Deift et al. (1991), Nagai and Satsuma (1995).
Geometric numerical integration illustrated by the StörmerVerlet method
, 2003
"... The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved longtime behaviour. This article illustrates concepts and results of geometric nume ..."
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Cited by 65 (6 self)
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The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved longtime behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Störmer–Verlet method. It thus presents a crosssection of the recent monograph by the authors, enriched by some additional material. After an introduction to the Newton–Störmer–Verlet–leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent longtime behaviour of the method: longtime energy conservation, linear error growth and preservation of invariant tori in nearintegrable systems, a discrete virial theorem, and preservation of adiabatic invariants.
Discrete breathers
 Advances in theory and applications,” Phys. Rep. 467, 1
, 2008
"... We give definitions for different types of moving spatially localized objects in discrete nonlinear lattices. We derive general analytical relations connecting frequency, velocity and localization length of moving discrete breathers and kinks in nonlinear onedimensional lattices. Then we propose nu ..."
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Cited by 59 (3 self)
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We give definitions for different types of moving spatially localized objects in discrete nonlinear lattices. We derive general analytical relations connecting frequency, velocity and localization length of moving discrete breathers and kinks in nonlinear onedimensional lattices. Then we propose numerical algorithms to find these solutions. I.
Numerical solution of isospectral flows
 Math. of Comp
, 1997
"... Abstract. In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation L ′ =[B(L),L], L(0) = L0, where L0 is a d × d symmetric matrix, B(L) is a skewsymmetric matrix function of L and [B, L] is the Lie bracket ..."
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Cited by 54 (23 self)
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Abstract. In this paper we are concerned with the problem of solving numerically isospectral flows. These flows are characterized by the differential equation L ′ =[B(L),L], L(0) = L0, where L0 is a d × d symmetric matrix, B(L) is a skewsymmetric matrix function of L and [B, L] is the Lie bracket operator. We show that standard Runge–Kutta schemes fail in recovering the main qualitative feature of these flows, that is isospectrality, since they cannot recover arbitrary cubic conservation laws. This failure motivates us to introduce an alternative approach and establish a framework for generation of isospectral methods of arbitrarily high order. 1. Background and notation 1.1. Introduction. The interest in solving isospectral flows is motivated by their relevance in a wide range of applications, from molecular dynamics to micromagnetics to linear algebra. The general form of an isospectral flow is the differential
Perturbations of orthogonal polynomials with periodic recursion coefficients
, 2007
"... We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well ada ..."
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Cited by 46 (16 self)
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We extend the results of Denisov–Rakhmanov, Szegő–Shohat– Nevai, and Killip–Simon from asymptotically constant orthogonal polynomials on the real line (OPRL) and unit circle (OPUC) to asymptotically periodic OPRL and OPUC. The key tool is a characterization of the isospectral torus that is well adapted to the study of perturbations.
Computation of Conserved Densities for Systems of Nonlinear DifferentialDifference Equations
, 1997
"... A new method for the computation of conserved densities of nonlinear dierentialdi erence equations is applied to Toda lattices and discretizations of the Kortewegde Vries and nonlinear Schrodinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, ..."
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Cited by 44 (25 self)
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A new method for the computation of conserved densities of nonlinear dierentialdi erence equations is applied to Toda lattices and discretizations of the Kortewegde Vries and nonlinear Schrodinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, can be used as an indicator of integrability. Keywords: Conserved densities; Integrability; Semidiscrete equations; Lattice 1 Introduction Nonlinear dierentialdierence equations (DDEs) describe many interesting phenomena such as vibrations of particles in lattices, charge uctuations in networks, Langmuir waves in plasmas, interactions between competing populations. Mathematically, DDEs also occur as spatially discrete analogues of partial dierential equations (PDEs). As such, lattices play a key role in numerical solvers for PDEs [1]. In [24], we introduced an algorithm to nd the analytical form of polynomial conserved densities for systems of nonlinear evolution equations. We use...