8 research outputs found
Periodic orbits, localization in normal mode space, and the Fermi-Pasta-Ulam problem
The Fermi-Pasta-Ulam problem was one of the first computational experiments.
It has stirred the physics community since, and resisted a simple solution for
half a century. The combination of straightforward simulations, efficient
computational schemes for finding periodic orbits, and analytical estimates
allows us to achieve significant progress. Recent results on -breathers,
which are time-periodic solutions that are localized in the space of normal
modes of a lattice and maximize the energy at a certain mode number, are
discussed, together with their relation to the Fermi-Pasta-Ulam problem. The
localization properties of a -breather are characterized by intensive
parameters, that is, energy densities and wave numbers. By using scaling
arguments, -breather solutions are constructed in systems of arbitrarily
large size. Frequency resonances in certain regions of wave number space lead
to the complete delocalization of -breathers. The relation of these features
to the Fermi-Pasta-Ulam problem are discussed.Comment: 19 pages, 9 figures, to appear in Am. J. Phy
q-breathers in Discrete Nonlinear Schroedinger lattices
-breathers are exact time-periodic solutions of extended nonlinear systems
continued from the normal modes of the corresponding linearized system. They
are localized in the space of normal modes. The existence of these solutions in
a weakly anharmonic atomic chain explained essential features of the
Fermi-Pasta-Ulam (FPU) paradox. We study -breathers in one- two- and
three-dimensional discrete nonlinear Sch\"{o}dinger (DNLS) lattices --
theoretical playgrounds for light propagation in nonlinear optical waveguide
networks, and the dynamics of cold atoms in optical lattices. We prove the
existence of these solutions for weak nonlinearity. We find that the
localization of -breathers is controlled by a single parameter which depends
on the norm density, nonlinearity strength and seed wave vector. At a critical
value of that parameter -breathers delocalize via resonances, signaling a
breakdown of the normal mode picture and a transition into strong mode-mode
interaction regime. In particular this breakdown takes place at one of the
edges of the normal mode spectrum, and in a singular way also in the center of
that spectrum. A stability analysis of -breathers supplements these
findings. For three-dimensional lattices, we find -breather vortices, which
violate time reversal symmetry and generate a vortex ring flow of energy in
normal mode space.Comment: 19 pages, 9 figure
q-breathers in finite two- and three-dimensional nonlinear acoustic lattices
Nonlinear interaction between normal modes dramatically affects energy
equipartition, heat conduction and other fundamental processes in extended
systems. In their celebrated experiment Fermi, Pasta and Ulam (FPU, 1955)
observed that in simple one-dimensional nonlinear atomic chains the energy must
not always be equally shared among the modes. Recently, it was shown that exact
and stable time-periodic orbits, coined -breathers (QBs), localize the mode
energy in normal mode space in an exponential way, and account for many aspects
of the FPU problem. Here we take the problem into more physically important
cases of two- and three-dimensional acoustic lattices to find existence and
principally different features of QBs. By use of perturbation theory and
numerical calculations we obtain that the localization and stability of QBs is
enhanced with increasing system size in higher lattice dimensions opposite to
their one-dimensional analogues.Comment: 4 pages, 5 figure
Stability of non-time-reversible phonobreathers
Non-time reversible phonobreathers are non-linear waves that can transport
energy in coupled oscillator chains by means of a phase-torsion mechanism. In
this paper, the stability properties of these structures have been considered.
It has been performed an analytical study for low-coupling solutions based upon
the so called {\em multibreather stability theorem} previously developed by
some of the authors [Physica D {\bf 180} 235]. A numerical analysis confirms
the analytical predictions and gives a detailed picture of the existence and
stability properties for arbitrary frequency and coupling.Comment: J. Phys. A.:Math. and Theor. In Press (2010