456 research outputs found
A minimal-variable symplectic method for isospectral flows
Isospectral flows are abundant in mathematical physics; the rigid body, the
the Toda lattice, the Brockett flow, the Heisenberg spin chain, and point
vortex dynamics, to mention but a few. Their connection on the one hand with
integrable systems and, on the other, with Lie--Poisson systems motivates the
research for optimal numerical schemes to solve them. Several works about
numerical methods to integrate isospectral flows have produced a large
varieties of solutions to this problem. However, many of these algorithms are
not intrinsically defined in the space where the equations take place and/or
rely on computationally heavy transformations. In the literature, only few
examples of numerical methods avoiding these issues are known, for instance,
the \textit{spherical midpoint method} on \SO(3). In this paper we introduce
a new minimal-variable, second order, numerical integrator for isospectral
flows intrinsically defined on quadratic Lie algebras and symmetric matrices.
The algorithm is isospectral for general isospectral flows and Lie--Poisson
preserving when the isospectral flow is Hamiltonian. The simplicity of the
scheme, together with its structure-preserving properties, makes it a
competitive alternative to those already present in literature.Comment: 17 pages, 9 figure
Isospectral flows on a class of finite-dimensional Jacobi matrices
We present a new matrix-valued isospectral ordinary differential equation
that asymptotically block-diagonalizes zero-diagonal Jacobi
matrices employed as its initial condition. This o.d.e.\ features a right-hand
side with a nested commutator of matrices, and structurally resembles the
double-bracket o.d.e.\ studied by R.W.\ Brockett in 1991. We prove that its
solutions converge asymptotically, that the limit is block-diagonal, and above
all, that the limit matrix is defined uniquely as follows: For even, a
block-diagonal matrix containing blocks, such that the
super-diagonal entries are sorted by strictly increasing absolute value.
Furthermore, the off-diagonal entries in these blocks have the same
sign as the respective entries in the matrix employed as initial condition. For
odd, there is one additional block containing a zero that is
the top left entry of the limit matrix. The results presented here extend some
early work of Kac and van Moerbeke.Comment: 19 pages, 3 figures, conjecture from previous version is added as
assertion (iv) of the main theorem including a proof; other major change
Post-Lie Algebras and Isospectral Flows
In this paper we explore the Lie enveloping algebra of a post-Lie algebra
derived from a classical -matrix. An explicit exponential solution of the
corresponding Lie bracket flow is presented. It is based on the solution of a
post-Lie Magnus-type differential equation
Post-Lie Algebras, Factorization Theorems and Isospectral-Flows
In these notes we review and further explore the Lie enveloping algebra of a
post-Lie algebra. From a Hopf algebra point of view, one of the central
results, which will be recalled in detail, is the existence of a second Hopf
algebra structure. By comparing group-like elements in suitable completions of
these two Hopf algebras, we derive a particular map which we dub post-Lie
Magnus expansion. These results are then considered in the case of
Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined
in terms of solutions of modified classical Yang-Baxter equation. In this
context, we prove a factorization theorem for group-like elements. An explicit
exponential solution of the corresponding Lie bracket flow is presented, which
is based on the aforementioned post-Lie Magnus expansion.Comment: 49 pages, no-figures, review articl
An integrable shallow water equation with peaked solitons
We derive a new completely integrable dispersive shallow water equation that
is biHamiltonian and thus possesses an infinite number of conservation laws in
involution. The equation is obtained by using an asymptotic expansion directly
in the Hamiltonian for Euler's equations in the shallow water regime. The
soliton solution for this equation has a limiting form that has a discontinuity
in the first derivative at its peak.Comment: LaTeX file. Figure available from authors upon reques
Symplectic methods for isospectral flows and 2D ideal hydrodynamics
The numerical solution of non-canonical Hamiltonian systems is an active and still growing field of research. At the present time, the biggest challenges concern the realization of structure preserving algorithms for differential equations on infinite dimensional manifolds. Several classical PDEs can indeed be set in this framework, and in particular the 2D hydrodynamical Euler equations. In this thesis, I have developed a new class of numerical schemes for Hamiltonian and non-Hamiltonian isospectral flows, in order to solve the 2D hydrodynamical Euler equations. The use of a conservative scheme has revealed new insights in the 2D ideal hydrodynamics, showing clear connections between geometric mechanics, statistical mechanics and integrability theory. The results are presented in four papers.In the first paper, we derive a general framework for the isospectral flows, providing a new class of numerical methods of arbitrary order, based on the Lie--Poisson reduction of Hamiltonian systems. Avoiding the use of any constraint, we obtain geometric integrators for a large class of Hamiltonian and non-Hamiltonian isospectral flows. One of the advantages of these methods is that, together with the isospectrality, they exhibit near conservation of the Hamiltonian and, indeed, they are Lie--Poisson integrators.In the second paper, using the results of paper I and III, we present a numerical method based on the geometric quantization of the Poisson algebra of the smooth functions on a sphere, which gives an approximate solution of the Euler equations with a number of discrete first integrals which is consistent with the level of discretization. The conservative properties of these schemes have allowed a more precise analysis of the statistical state of a fluid on a sphere. On the one hand, we show the link of the statistical state with some conserved quantities, on the other hand, we suggest a mechanism of formation of coherent structures related to the integrability theory of point-vortices.In the third paper, I present and analyse a minimal variable isospectral Lie--Poisson integrator for quadratic matrix Lie algebras. This result comes from a more careful analysis of the isospectral midpoint method derived in paper I. I also present a detailed description of quadratic Lie algebras, showing under which conditions the related Lie--Poisson systems are also isospectral flows.In the fourth paper, we give a survey on the integrability theory of the point-vortex dynamics. In particular, we show that all the results found in literature can be derived in the framework of symplectic reduction theory. Furthermore, our work aims to connect the 2D Euler equations with the point-vortex dynamics, as suggested in paper II
Isomonodromic deformations and supersymmetric gauge theories
Seiberg-Witten solutions of four-dimensional supersymmetric gauge theories
possess rich but involved integrable structures. The goal of this paper is to
show that an isomonodromy problem provides a unified framework for
understanding those various features of integrability. The Seiberg-Witten
solution itself can be interpreted as a WKB limit of this isomonodromy problem.
The origin of underlying Whitham dynamics (adiabatic deformation of an
isospectral problem), too, can be similarly explained by a more refined
asymptotic method (multiscale analysis). The case of SU()
supersymmetric Yang-Mills theory without matter is considered in detail for
illustration. The isomonodromy problem in this case is closely related to the
third Painlev\'e equation and its multicomponent analogues. An implicit
relation to t\tbar fusion of topological sigma models is thereby expected.Comment: Several typos are corrected, and a few sentenses are altered. 19 pp +
a list of corrections (page 20), LaTe
Isospectral deformations of the Dirac operator
We give more details about an integrable system in which the Dirac operator
D=d+d^* on a finite simple graph G or Riemannian manifold M is deformed using a
Hamiltonian system D'=[B,h(D)] with B=d-d^* + i b. The deformed operator D(t) =
d(t) + b(t) + d(t)^* defines a new exterior derivative d(t) and a new Dirac
operator C(t) = d(t) + d(t)^* and Laplacian M(t) = d(t) d(t)^* + d(t)* d(t) and
so a new distance on G or a new metric on M.Comment: 32 pages, 8 figure
Integrators on homogeneous spaces: Isotropy choice and connections
We consider numerical integrators of ODEs on homogeneous spaces (spheres,
affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a
built-in symmetry. A numerical integrator respects this symmetry if it is
equivariant. One obtains homogeneous space integrators by combining a Lie group
integrator with an isotropy choice. We show that equivariant isotropy choices
combined with equivariant Lie group integrators produce equivariant homogeneous
space integrators. Moreover, we show that the RKMK, Crouch--Grossman or
commutator-free methods are equivariant. To show this, we give a novel
description of Lie group integrators in terms of stage trees and motion maps,
which unifies the known Lie group integrators. We then proceed to study the
equivariant isotropy maps of order zero, which we call connections, and show
that they can be identified with reductive structures and invariant principal
connections. We give concrete formulas for connections in standard homogeneous
spaces of interest, such as Stiefel, Grassmannian, isospectral, and polar
decomposition manifolds. Finally, we show that the space of matrices of fixed
rank possesses no connection
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