4,562 research outputs found
Periodic solutions of second order Hamiltonian systems bifurcating from infinity
The goal of this article is to study closed connected sets of periodic
solutions, of autonomous second order Hamiltonian systems, emanating from
infinity. The main idea is to apply the degree for SO(2)-equivariant gradient
operators defined by the second author. Using the results due to Rabier we show
that we cannot apply the Leray-Schauder degree to prove the main results of
this article. It is worth pointing out that since we study connected sets of
solutions, we also cannot use the Conley index technique and the Morse theory.Comment: 24 page
Subharmonic solutions for nonautonomous sublinear first order Hamiltonian systems
In this paper, the existence of subharmonic solutions for a class of
non-autonomous first-order Hamiltonian systems is investigated. We also study
the minimality of periods for such solutions. Our results which extend and
improve many previous results will be illustrated by specific examples. Our
main tools are the minimax methods in critical point theory and the least
action principle. {\bf Key words.} Hamiltonian systems. Critical point theory.
Least action principle. Subharmonic solutions.Comment: 17 page
Quasiperiodic Dynamics in Bose-Einstein Condensates in Periodic Lattices and Superlattices
We employ KAM theory to rigorously investigate quasiperiodic dynamics in
cigar-shaped Bose-Einstein condensates (BEC) in periodic lattices and
superlattices. Toward this end, we apply a coherent structure ansatz to the
Gross-Pitaevskii equation to obtain a parametrically forced Duffing equation
describing the spatial dynamics of the condensate. For shallow-well,
intermediate-well, and deep-well potentials, we find KAM tori and Aubry-Mather
sets to prove that one obtains mostly quasiperiodic dynamics for condensate
wave functions of sufficiently large amplitude, where the minimal amplitude
depends on the experimentally adjustable BEC parameters. We show that this
threshold scales with the square root of the inverse of the two-body scattering
length, whereas the rotation number of tori above this threshold is
proportional to the amplitude. As a consequence, one obtains the same dynamical
picture for lattices of all depths, as an increase in depth essentially only
affects scaling in phase space. Our approach is applicable to periodic
superlattices with an arbitrary number of rationally dependent wave numbers.Comment: 29 pages, 6 figures (several with multiple parts; higher-quality
versions of some of them available at
http://www.its.caltech.edu/~mason/papers), to appear very soon in Journal of
Nonlinear Scienc
A variational approach to the study of the existence of invariant Lagrangian graphs
This paper surveys some results by the author and collaborators on the
existence of invariant Lagrangian graphs for Tonelli Hamiltonian systems. The
presentation is based on an invited talk by the author at XIX Congresso Unione
Matematica Italiana (Bologna, 12-17 Sept. 2011).Comment: 28 page
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