1,094 research outputs found
Periodic travelling waves in convex Klein-Gordon chains
We study Klein-Gordon chains with attractive nearest neighbour forces and
convex on-site potential, and show that there exists a two-parameter family of
periodic travelling waves (wave trains) with unimodal and even profile
functions. Our existence proof is based on a saddle-point problem with
constraints and exploits the invariance properties of an improvement operator.
Finally, we discuss the numerical computation of wave trains.Comment: 12 pages, 3 figure
Discrete Breathers
Nonlinear classical Hamiltonian lattices exhibit generic solutions in the
form of discrete breathers. These solutions are time-periodic and (typically
exponentially) localized in space. The lattices exhibit discrete translational
symmetry. Discrete breathers are not confined to certain lattice dimensions.
Necessary ingredients for their occurence are the existence of upper bounds on
the phonon spectrum (of small fluctuations around the groundstate) of the
system as well as the nonlinearity in the differential equations. We will
present existence proofs, formulate necessary existence conditions, and discuss
structural stability of discrete breathers. The following results will be also
discussed: the creation of breathers through tangent bifurcation of band edge
plane waves; dynamical stability; details of the spatial decay; numerical
methods of obtaining breathers; interaction of breathers with phonons and
electrons; movability; influence of the lattice dimension on discrete breather
properties; quantum lattices - quantum breathers. Finally we will formulate a
new conceptual aproach capable of predicting whether discrete breather exist
for a given system or not, without actually solving for the breather. We
discuss potential applications in lattice dynamics of solids (especially
molecular crystals), selective bond excitations in large molecules, dynamical
properties of coupled arrays of Josephson junctions, and localization of
electromagnetic waves in photonic crystals with nonlinear response.Comment: 62 pages, LaTeX, 14 ps figures. Physics Reports, to be published; see
also at http://www.mpipks-dresden.mpg.de/~flach/html/preprints.htm
Numerical analysis of solitons profiles in a composite model for DNA to rsion dynamics
We present the results of our numerical analysis of a "composite" model of
DNA which generalizes a well-known elementary torsional model of Yakushevich by
allowing bases to move independently from the backbone. The model shares with
the Yakushevich model many features and results but it represents an
improvement from both the conceptual and the phenomenological point of view. It
provides a more realistic description of DNA and possibly a justification for
the use of models which consider the DNA chain as uniform. It shows that the
existence of solitons is a generic feature of the underlying nonlinear dynamics
and is to a large extent independent of the detailed modelling of DNA. As
opposite to the Yakushevich model, where it is needed to use an unphysical
value for the torsion in order to induce the correct velocity of sound, the
model we consider supports solitonic solutions, qualitatively and
quantitatively very similar to the Yakushevich solitons, in a fully realistic
range of all the physical parameters characterizing the DNA.Comment: 16 pages, 9 figure
Periodic and compacton travelling wave solutions of discrete nonlinear Klein-Gordon lattices
We prove the existence of periodic travelling wave solutions for general
discrete nonlinear Klein-Gordon systems, considering both cases of hard and
soft on-site potentials. In the case of hard on-site potentials we implement a
fixed point theory approach, combining Schauder's fixed point theorem and the
contraction mapping principle. This approach enables us to identify a ring in
the energy space for non-trivial solutions to exist, energy (norm) thresholds
for their existence and upper bounds on their velocity. In the case of soft
on-site potentials, the proof of existence of periodic travelling wave
solutions is facilitated by a variational approach based on the Mountain Pass
Theorem. The proof of the existence of travelling wave solutions satisfying
Dirichlet boundary conditions establishes rigorously the presence of compactons
in discrete nonlinear Klein-Gordon chains. Thresholds on the averaged kinetic
energy for these solutions to exist are also derived.Comment: 21 pages, 1 figur
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