Dynamical properties of discrete breathers in curved chains with first
and second neighbor interactions
M. Iban
˜
es,
1
J. M. Sancho,
1
and G. P. Tsironis
1,2,
*
1
Department d’Estructura i Constituents de la Mate `ria, Universitat de Barcelona, Diagonal 647, E– 08028 Barcelona, Spain
2
Center for Nonlinear Studies, MS-B258, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
Received 16 November 2000; revised manuscript received 2 October 2001; published 15 March 2002
We present the study of discrete breather dynamics in curved polymerlike chains consisting of masses
connected via nonlinear springs. The polymer chains are one dimensional but not rectilinear and their motion
takes place on a plane. After constructing breathers following numerically accurate procedures, we launch them
in the chains and investigate properties of their propagation dynamics. We find that breather motion is strongly
affected by the presence of curved regions of polymers, while the breathers themselves show a very strong
resilience and remarkable stability in the presence of geometrical changes. For chains with strong angular
rigidity we find that breathers either pass through bent regions or get reflected while retaining their frequency.
Their motion is practically lossless and seems to be determined through local energy conservation. For less
rigid chains modeled via second neighbor interactions, we find similarly that chain geometry typically does not
destroy the localized breather states but, contrary to the angularly rigid chains, it induces some small but
constant energy loss. Furthermore, we find that a curved segment acts as an active gate reflecting or refracting
the incident breather and transforming its velocity to a value that depends on the discrete breathers frequency.
We analyze the physical reasoning behind these seemingly general breather properties.
DOI: 10.1103/PhysRevE.65.041902 PACS numbers: 87.15.-v, 63.20.Pw, 36.20.-r
I. INTRODUCTION
The basic question to be addressed in this paper relates to
the dynamics of space localized lattice oscillation modes re-
ferred to as intrinsic localized modes ILM’s or discrete
breathers DB’s in elastic polymeric chains with rigidity
1–20. Unlike solitons, DB’s appear to be generic modes of
nonlinear lattices provided the latter are equipped with these
two basic ingredients: nonlinearity in the interactions and
lattice discreteness. During the last several years there has
been an abundance of theoretical work dealing with various
aspects of DB properties, including generation 1–8, rigor-
ous existence 6, dynamics and mobility 9, thermodynam-
ics and statistical properties 11,13, quantum features
10,16, etc. As a result, many of the basic DB properties
have been revealed and are now relatively well understood.
On the experimental front, a recent avalanche of results in a
large variety of systems demonstrating DB presence, or at
least strong indications for it, has set the whole area on very
solid and promising new grounds 17–20. There can now be
more precise studies of specific condensed matter, chemical
and biological systems, as well as discussion on the design of
breather based materials. The work to be presented in this
paper points also in this direction, as it attempts to deviate
from simple one-dimensional lattice models by introducing
one additional new element, that of single chain elasticity.
We will thus be concerned here with polymeric chains of
masses coupled with springs that can move, in principle, in
the whole ( x , y ) plane and are characterized by local and
global elastic properties. Our basic question is how energy
localization in the form of DB’s interplays with single chain
polymer elasticity.
Our basic model will be an arbitrarily shaped chain of
equal masses coupled typically via nonlinear springs involv-
ing only two-body polynomial-type interactions. Unlike re-
cent work on the same general topic 21, we will not con-
sider long-range interactions presently so that we keep the
complexity of the model and the number of parameters used
to a minimum. We will instead use first and second neighbor
interactions that have the same form but enter with different
strengths and equilibrium distances. After the first stage of
our analysis we will initially ‘‘turn off’’ the second neighbor
interaction and study the simplest possible nonlinear bead
and spring model, while subsequently the second neighbor
interaction will be included and comparisons will be made
between the two cases. Since our main interest is in under-
standing the physics of breathers in biomolecules such as
proteins 22 rather than general homopolymers, we will
have to somehow restrict our study to rigid and quasirigid
polymer geometries. This can only be done artificially
through constraints when only first neighbor interactions are
taken into account due to the high level of degeneracy of the
chain. The source of the latter is geometric since there are
multiple equilibrium states that are distributed in various
configurations on the plane while preserving the nearest
neighbor equilibrium distances. However, when the second
neighbor interaction is turned on, the chain degeneracy re-
duces substantially and there is no need for additional exter-
nal constraints.
As mentioned earlier, the questions to be addressed here
will focus on the interplay of energy localization in the form
of DB’s and biopolymer spatial structure. Since this presents
a rather broad topic we will narrow the questions in this
study down to the following three: a can a stable breather
*Permanent address: Department of Physics, University of Crete
and Foundation for Research and Technology–Hellas, P. O. Box
2208, 71003 Heraklion, Crete, Greece.
PHYSICAL REVIEW E, VOLUME 65, 041902
1063-651X/2002/654/04190213/$20.00 ©2002 The American Physical Society 65 041902-1