1,784 research outputs found
Isospectral Flow and Liouville-Arnold Integration in Loop Algebras
A number of examples of Hamiltonian systems that are integrable by classical
means are cast within the framework of isospectral flows in loop algebras.
These include: the Neumann oscillator, the cubically nonlinear Schr\"odinger
systems and the sine-Gordon equation. Each system has an associated invariant
spectral curve and may be integrated via the Liouville-Arnold technique. The
linearizing map is the Abel map to the associated Jacobi variety, which is
deduced through separation of variables in hyperellipsoidal coordinates. More
generally, a family of moment maps is derived, identifying certain finite
dimensional symplectic manifolds with rational coadjoint orbits of loop
algebras. Integrable Hamiltonians are obtained by restriction of elements of
the ring of spectral invariants to the image of these moment maps. The
isospectral property follows from the Adler-Kostant-Symes theorem, and gives
rise to invariant spectral curves. {\it Spectral Darboux coordinates} are
introduced on rational coadjoint orbits, generalizing the hyperellipsoidal
coordinates to higher rank cases. Applying the Liouville-Arnold integration
technique, the Liouville generating function is expressed in completely
separated form as an abelian integral, implying the Abel map linearization in
the general case.Comment: 42 pages, 2 Figures, 1 Table. Lectures presented at the VIIIth
Scheveningen Conference, held at Wassenaar, the Netherlands, Aug. 16-21, 199
Darboux Coordinates and Liouville-Arnold Integration in Loop Algebras
Darboux coordinates are constructed on rational coadjoint orbits of the
positive frequency part \wt{\frak{g}}^+ of loop algebras. These are given by
the values of the spectral parameters at the divisors corresponding to
eigenvector line bundles over the associated spectral curves, defined within a
given matrix representation. A Liouville generating function is obtained in
completely separated form and shown, through the Liouville-Arnold integration
method, to lead to the Abel map linearization of all Hamiltonian flows induced
by the spectral invariants. Serre duality is used to define a natural
symplectic structure on the space of line bundles of suitable degree over a
permissible class of spectral curves, and this is shown to be equivalent to the
Kostant-Kirillov symplectic structure on rational coadjoint orbits. The general
construction is given for or , with
reductions to orbits of subalgebras determined as invariant fixed point sets
under involutive automorphisms. The case is shown to reproduce
the classical integration methods for finite dimensional systems defined on
quadrics, as well as the quasi-periodic solutions of the cubically nonlinear
Schr\"odinger equation. For , the method is applied to the
computation of quasi-periodic solutions of the two component coupled nonlinear
Schr\"odinger equation.Comment: 61 pg
Dual Isomonodromic Deformations and Moment Maps to Loop Algebras
The Hamiltonian structure of the monodromy preserving deformation equations
of Jimbo {\it et al } is explained in terms of parameter dependent pairs of
moment maps from a symplectic vector space to the dual spaces of two different
loop algebras. The nonautonomous Hamiltonian systems generating the
deformations are obtained by pulling back spectral invariants on Poisson
subspaces consisting of elements that are rational in the loop parameter and
identifying the deformation parameters with those determining the moment maps.
This construction is shown to lead to ``dual'' pairs of matrix differential
operators whose monodromy is preserved under the same family of deformations.
As illustrative examples, involving discrete and continuous reductions, a
higher rank generalization of the Hamiltonian equations governing the
correlation functions for an impenetrable Bose gas is obtained, as well as dual
pairs of isomonodromy representations for the equations of the Painleve
transcendents and .Comment: preprint CRM-1844 (1993), 28 pgs. (Corrected date and abstract.
Recommended from our members
Tank waste information network system II (TWINS2) year 2000 compliance assurance plan
The scope of this plan includes the Tank Waste Information Network System II (TWINS2) that contains the following major components: Tank Characterization Database (TCD), Tank Vapor Database (TVD), Data Source Access (DSA), automated Tank Characterization Report, Best-Basis Inventory Model (BBIM), and Tracker (corrective action tracking) function. The automated Tank Characterization Report application currently in development also will reside on-the TWINS system as will the BBIM. Critical inputs to TWINS occur from the following databases: Labcore and SACS. Output does not occur from TWINS to these two databases
Recommended from our members
Readiness to proceed: Characterization planning basis
This report summarizes characterization requirements, data availability, and data acquisition plans in support of the Phase 1 Waste Feed Readiness to Proceed Mid-Level Logic. It summarizes characterization requirements for the following program planning documents: Waste Feed Readiness Mid-Level Logic and Decomposition (in development); Master blue print (not available); Tank Waste Remediation System (TWRS) Operations and Utilization Plan and Privatization Contract; Enabling assumptions (not available); Privatization low-activity waste (LAW) Data Quality Objective (DQO); Privatization high-level waste (HLW) DQO (draft); Problem-specific DQOs (in development); Interface control documents (draft). Section 2.0 defines the primary objectives for this report, Section 3.0 discusses the scope and assumptions, and Section 4.0 identifies general characterization needs and analyte-specific characterization needs or potential needs included in program documents and charts. Section 4.0 also shows the analyses that have been conducted, and the archive samples that are available for additional analyses. Section 5.0 discusses current plans for obtaining additional samples and analyses to meet readiness-to-proceed requirements. Section 6.0 summarizes sampling needs based on preliminary requirements and discusses other potential characterization needs. Many requirements documents are preliminary. In many cases, problem-specific DQOs have not been drafted, and only general assumptions about the document contents could be obtained from the authors. As a result, the readiness-to-proceed characterization requirements provided in this document are evolving and may change
Classical and Quantum Integrable Systems in \wt{\gr{gl}}(2)^{+*} and Separation of Variables
Classical integrable Hamiltonian systems generated by elements of the Poisson
commuting ring of spectral invariants on rational coadjoint orbits of the loop
algebra \wt{\gr{gl}}^{+*}(2,{\bf R}) are integrated by separation of
variables in the Hamilton-Jacobi equation in hyperellipsoidal coordinates. The
canonically quantized systems are then shown to also be completely integrable
and separable within the same coordinates. Pairs of second class constraints
defining reduced phase spaces are implemented in the quantized systems by
choosing one constraint as an invariant, and interpreting the other as
determining a quotient (i.e., by treating one as a first class constraint and
the other as a gauge condition). Completely integrable, separable systems on
spheres and ellipsoids result, but those on ellipsoids require a further
modification of order \OO(\hbar^2) in the commuting invariants in order to
assure self-adjointness and to recover the Laplacian for the case of free
motion. For each case - in the ambient space , the sphere and the
ellipsoid - the Schr\"odinger equations are completely separated in
hyperellipsoidal coordinates, giving equations of generalized Lam\'e type.Comment: 28 page
Middle Convolution and Harnad Duality
We interpret the additive middle convolution operation in terms of the Harnad
duality, and as an application, generalize the operation to have a
multi-parameter and act on irregular singular systems.Comment: 50 pages; v2: Submitted version once revised according to referees'
comment
Renormalization scale uncertainty in tne DIS 2+1 jet cross-section
The deep inelastic scattering 2+1 jet cross- section is a useful observable
for precision tests of QCD, e.g. measuring the strong coupling constant
alpha(s). A consistent analysis requires a good understanding of the
theoretical uncertainties and one of the most fundamental ones in QCD is due to
the renormalization scheme and scale ambiguity. Different methods, which have
been proposed to resolve the scale ambiguity, are applied to the 2+1 jet
cross-section and the uncertainty is estimated. It is shown that the
uncertainty can be made smaller by choosing the jet definition in a suitable
way.Comment: 24 pages, uuencoded compressed tar file, DESY 94-082, TSL-ISV-94-009
Nuclear shadowing at low Q^2
We re-examine the role of vector meson dominance in nuclear shadowing at low
Q^2. We find that models which incorporate both vector meson and partonic
mechanisms are consistent with both the magnitude and the Q^2 slope of the
shadowing data.Comment: 7 pages, 2 figures; to appear in Phys. Rev.
Diffractive vector meson electroproduction at small Bjorken within GPD approach
We study light vector meson electroproduction at small within the
generalized parton distributions (GPDs) model. The modified perturbative
approach is used, where the quark transverse degrees of freedom in the vector
meson wave function and hard subprocess are considered. Our results on the
cross section and spin observables are in good agreement with experimentComment: 6 pages, 5 figures, presented at Symmetries and Spin meeting, Prague,
8- 14 July, 200
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