1,306 research outputs found
Symmetry and resonance in periodic FPU chains
The symmetry and resonance properties of the Fermi Pasta Ulam chain with
periodic boundary conditions are exploited to construct a near-identity
transformation bringing this Hamiltonian system into a particularly simple
form. This `Birkhoff-Gustavson normal form' retains the symmetries of the
original system and we show that in most cases this allows us to view the
periodic FPU Hamiltonian as a perturbation of a nondegenerate Liouville
integrable Hamiltonian. According to the KAM theorem this proves the existence
of many invariant tori on which motion is quasiperiodic. Experiments confirm
this qualitative behaviour. We note that one can not expect it in lower-order
resonant Hamiltonian systems. So the FPU chain is an exception and its special
features are caused by a combination of special resonances and symmetries.Comment: 21 page
Open problems, questions, and challenges in finite-dimensional integrable systems
The paper surveys open problems and questions related to different aspects
of integrable systems with finitely many degrees of freedom. Many of the open
problems were suggested by the participants of the conference “Finite-dimensional
Integrable Systems, FDIS 2017” held at CRM, Barcelona in July 2017.Postprint (updated version
Gravitational descendants in symplectic field theory
It was pointed out by Y. Eliashberg in his ICM 2006 plenary talk that the
rich algebraic formalism of symplectic field theory leads to a natural
appearance of quantum and classical integrable systems, at least in the case
when the contact manifold is the prequantization space of a symplectic
manifold. In this paper we generalize the definition of gravitational
descendants in SFT from circle bundles in the Morse-Bott case to general
contact manifolds. After we have shown that for the basic examples of
holomorphic curves in SFT, that is, branched covers of cylinders over closed
Reeb orbits, the gravitational descendants have a geometric interpretation in
terms of branching conditions, we compute the corresponding sequences of
Poisson-commuting functions when the contact manifold is the unit cotangent
bundle of a Riemannian manifold.Comment: 44 pages, no figure
Asymptotics of action variables near semi-toric singularities
The presence of focus-focus singularities in semi-toric integrables
Hamiltonian systems is one of the reasons why there cannot exist global
Action-Angle coordinates on such systems. At focus-focus critical points, the
Liouville-Arnold-Mineur theorem does not apply. In particular, the affine
structure of the image of the moment map around has non-trivial monodromy. In
this article, we establish that the singular behaviour and the multi-valuedness
of the Action integrals is given by a complex logarithm. This extends a
previous result by Vu Ngoc to any dimension. We also calculate the monodromy
matrix for these systems.Comment: 24 pages, 2 figures, 1 table; modifications in the introduction,
accepted for publicatio
Duality in Integrable Systems and Gauge Theories
We discuss various dualities, relating integrable systems and show that these
dualities are explained in the framework of Hamiltonian and Poisson reductions.
The dualities we study shed some light on the known integrable systems as well
as allow to construct new ones, double elliptic among them. We also discuss
applications to the (supersymmetric) gauge theories in various dimensions.Comment: harvmac 45 pp.; v4. minor corrections, to appear in JHE
Poisson vertex algebras in the theory of Hamiltonian equations
We lay down the foundations of the theory of Poisson vertex algebras aimed at
its applications to integrability of Hamiltonian partial differential
equations. Such an equation is called integrable if it can be included in an
infinite hierarchy of compatible Hamiltonian equations, which admit an infinite
sequence of linearly independent integrals of motion in involution. The
construction of a hierarchy and its integrals of motion is achieved by making
use of the so called Lenard scheme. We find simple conditions which guarantee
that the scheme produces an infinite sequence of closed 1-forms \omega_j, j in
Z_+, of the variational complex \Omega. If these forms are exact, i.e. \omega_j
are variational derivatives of some local functionals \int h_j, then the latter
are integrals of motion in involution of the hierarchy formed by the
corresponding Hamiltonian vector fields. We show that the complex \Omega is
exact, provided that the algebra of functions V is "normal"; in particular, for
arbitrary V, any closed form in \Omega becomes exact if we add to V a finite
number of antiderivatives. We demonstrate on the examples of KdV, HD and CNW
hierarchies how the Lenard scheme works. We also discover a new integrable
hierarchy, which we call the CNW hierarchy of HD type. Developing the ideas of
Dorfman, we extend the Lenard scheme to arbitrary Dirac structures, and
demonstrate its applicability on the examples of the NLS, pKdV and KN
hierarchies.Comment: 95 page
Semiclassical Quantisation of Finite-Gap Strings
We perform a first principle semiclassical quantisation of the general
finite-gap solution to the equations of a string moving on R x S^3. The
derivation is only formal as we do not regularise divergent sums over stability
angles. Moreover, with regards to the AdS/CFT correspondence the result is
incomplete as the fluctuations orthogonal to this subspace in AdS_5 x S^5 are
not taken into account. Nevertheless, the calculation serves the purpose of
understanding how the moduli of the algebraic curve gets quantised
semiclassically, purely from the point of view of finite-gap integration and
with no input from the gauge theory side. Our result is expressed in a very
compact and simple formula which encodes the infinite sum over stability angles
in a succinct way and reproduces exactly what one expects from knowledge of the
dual gauge theory. Namely, at tree level the filling fractions of the algebraic
curve get quantised in large integer multiples of hbar = 1/lambda^{1/2}. At
1-loop order the filling fractions receive Maslov index corrections of hbar/2
and all the singular points of the spectral curve become filled with small
half-integer multiples of hbar. For the subsector in question this is in
agreement with the previously obtained results for the semiclassical energy
spectrum of the string using the method proposed in hep-th/0703191.
Along the way we derive the complete hierarchy of commuting flows for the
string in the R x S^3 subsector. Moreover, we also derive a very general and
simple formula for the stability angles around a generic finite-gap solution.
We also stress the issue of quantum operator orderings since this problem
already crops up at 1-loop in the form of the subprincipal symbol.Comment: 53 pages, 22 figures; some significant typos corrected, references
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