4,685 research outputs found
A refined invariant subspace method and applications to evolution equations
The invariant subspace method is refined to present more unity and more
diversity of exact solutions to evolution equations. The key idea is to take
subspaces of solutions to linear ordinary differential equations as invariant
subspaces that evolution equations admit. A two-component nonlinear system of
dissipative equations was analyzed to shed light on the resulting theory, and
two concrete examples are given to find invariant subspaces associated with
2nd-order and 3rd-order linear ordinary differential equations and their
corresponding exact solutions with generalized separated variables.Comment: 16 page
Chaos in Symmetric Phase Oscillator Networks
Phase-coupled oscillators serve as paradigmatic models of networks of weakly
interacting oscillatory units in physics and biology. The order parameter which
quantifies synchronization was so far found to be chaotic only in systems with
inhomogeneities. Here we show that even symmetric systems of identical
oscillators may not only exhibit chaotic dynamics, but also chaotically
fluctuating order parameters. Our findings imply that neither inhomogeneities
nor amplitude variations are necessary to obtain chaos, i.e., nonlinear
interactions of phases give rise to the necessary instabilities.Comment: 4 pages; Accepted by Physical Review Letter
Symmetric coupling of four spin-1/2 systems
We address the non-binary coupling of identical angular momenta based upon
the representation theory for the symmetric group. A correspondence is pointed
out between the complete set of commuting operators and the
reference-frame-free subsystems. We provide a detailed analysis of the coupling
of three and four spin-1/2 systems and discuss a symmetric coupling of four
spin-1/2 systems.Comment: 20 pages, no figure
Hamiltonian Hopf bifurcation with symmetry
In this paper we study the appearance of branches of relative periodic orbits
in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry
groups that do not generically exist in the dissipative framework. The
theoretical study is illustrated with several examples.Comment: 35 pages, 3 figure
Open Systems Viewed Through Their Conservative Extensions
A typical linear open system is often defined as a component of a larger
conservative one. For instance, a dielectric medium, defined by its frequency
dependent electric permittivity and magnetic permeability is a part of a
conservative system which includes the matter with all its atomic complexity. A
finite slab of a lattice array of coupled oscillators modelling a solid is
another example. Assuming that such an open system is all one wants to observe,
we ask how big a part of the original conservative system (possibly very
complex) is relevant to the observations, or, in other words, how big a part of
it is coupled to the open system? We study here the structure of the system
coupling and its coupled and decoupled components, showing, in particular, that
it is only the system's unique minimal extension that is relevant to its
dynamics, and this extension often is tiny part of the original conservative
system. We also give a scenario explaining why certain degrees of freedom of a
solid do not contribute to its specific heat.Comment: 51 page
Variations of Hodge Structure Considered as an Exterior Differential System: Old and New Results
This paper is a survey of the subject of variations of Hodge structure (VHS)
considered as exterior differential systems (EDS). We review developments over
the last twenty-six years, with an emphasis on some key examples. In the
penultimate section we present some new results on the characteristic
cohomology of a homogeneous Pfaffian system. In the last section we discuss how
the integrability conditions of an EDS affect the expected dimension of an
integral submanifold. The paper ends with some speculation on EDS and Hodge
conjecture for Calabi-Yau manifolds
Pohlmeyer reduction revisited
A systematic group theoretical formulation of the Pohlmeyer reduction is
presented. It provides a map between the equations of motion of sigma models
with target-space a symmetric space M=F/G and a class of integrable
multi-component generalizations of the sine-Gordon equation. When M is of
definite signature their solutions describe classical bosonic string
configurations on the curved space-time R_t\times M. In contrast, if M is of
indefinite signature the solutions to those equations can describe bosonic
string configurations on R_t\times M, M\times S^1_\vartheta or simply M. The
conditions required to enable the Lagrangian formulation of the resulting
equations in terms of gauged WZW actions with a potential term are clarified,
and it is shown that the corresponding Lagrangian action is not unique in
general. The Pohlmeyer reductions of sigma models on CP^n and AdS_n are
discussed as particular examples of symmetric spaces of definite and indefinite
signature, respectively.Comment: 45 pages, LaTeX, more references added, accepted for publication in
JHE
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