5 research outputs found
Exact static solutions for discrete models free of the Peierls-Nabarro barrier: Discretized first integral approach
We propose a generalization of the discrete Klein-Gordon models free of the
Peierls-Nabarro barrier derived in Nonlinearity {\bf 12}, 1373 (1999) and Phys.
Rev. E {\bf 72}, 035602(R) (2005), such that they support not only kinks but a
one-parameter set of exact static solutions. These solutions can be obtained
iteratively from a two-point nonlinear map whose role is played by the
discretized first integral of the static Klein-Gordon field, as suggested in J.
Phys. A {\bf 38}, 7617 (2005). We then discuss some discrete models
free of the Peierls-Nabarro barrier and identify for them the full space of
available static solutions, including those derived recently in Phys. Rev. E
{\bf 72} 036605 (2005) but not limited to them. These findings are also
relevant to standing wave solutions of discrete nonlinear Schr{\"o}dinger
models. We also study stability of the obtained solutions. As an interesting
aside, we derive the list of solutions to the continuum equation that
fill the entire two-dimensional space of parameters obtained as the continuum
limit of the corresponding space of the discrete models.Comment: Accepted for publication in PRE; the M/S has been revised in line
with the referee repor
Discrete Klein-Gordon models with static kinks free of the Peierls-Nabarro potential
For the nonlinear Klein-Gordon type models, we describe a general method of
discretization in which the static kink can be placed anywhere with respect to
the lattice. These discrete models are therefore free of the {\it static}
Peierls-Nabarro potential. Previously reported models of this type are shown to
belong to a wider class of models derived by means of the proposed method. A
relevant physical consequence of our findings is the existence of a wide class
of discrete Klein-Gordon models where slow kinks {\it practically} do not
experience the action of the Peierls-Nabarro potential. Such kinks are not
trapped by the lattice and they can be accelerated by even weak external
fields.Comment: 6 pages, 2 figure
Kinks in dipole chains
It is shown that the topological discrete sine-Gordon system introduced by
Speight and Ward models the dynamics of an infinite uniform chain of electric
dipoles constrained to rotate in a plane containing the chain. Such a chain
admits a novel type of static kink solution which may occupy any position
relative to the spatial lattice and experiences no Peierls-Nabarro barrier.
Consequently the dynamics of a single kink is highly continuum like, despite
the strongly discrete nature of the model. Static multikinks and kink-antikink
pairs are constructed, and it is shown that all such static solutions are
unstable. Exact propagating kinks are sought numerically using the
pseudo-spectral method, but it is found that none exist, except, perhaps, at
very low speed.Comment: Published version. 21 pages, 5 figures. Section 3 completely
re-written. Conclusions unchange
Travelling kinks in discrete phi^4 models
In recent years, three exceptional discretizations of the phi^4 theory have
been discovered [J.M. Speight and R.S. Ward, Nonlinearity 7, 475 (1994); C.M.
Bender and A. Tovbis, J. Math. Phys. 38, 3700 (1997); P.G. Kevrekidis, Physica
D 183, 68 (2003)] which support translationally invariant kinks, i.e. families
of stationary kinks centred at arbitrary points between the lattice sites. It
has been suggested that the translationally invariant stationary kinks may
persist as 'sliding kinks', i.e. discrete kinks travelling at nonzero
velocities without experiencing any radiation damping. The purpose of this
study is to check whether this is indeed the case. By computing the Stokes
constants in beyond-all-order asymptotic expansions, we prove that the three
exceptional discretizations do not support sliding kinks for most values of the
velocity - just like the standard, one-site, discretization. There are,
however, isolated values of velocity for which radiationless kink propagation
becomes possible. There is one such value for the discretization of Speight and
Ward and three 'sliding velocities' for the model of Kevrekedis.Comment: To be published in Nonlinearity. 22 pages, 5 figures. Extensive
clarifications to the text have been mad