423 research outputs found
Thresholds for breather solutions on the Discrete Nonlinear Schr\"odinger Equation with saturable and power nonlinearity
We consider the question of existence of periodic solutions (called breather
solutions or discrete solitons) for the Discrete Nonlinear Schr\"odinger
Equation with saturable and power nonlinearity. Theoretical and numerical
results are proved concerning the existence and nonexistence of periodic
solutions by a variational approach and a fixed point argument. In the
variational approach we are restricted to DNLS lattices with Dirichlet boundary
conditions. It is proved that there exists parameters (frequency or
nonlinearity parameters) for which the corresponding minimizers satisfy
explicit upper and lower bounds on the power. The numerical studies performed
indicate that these bounds behave as thresholds for the existence of periodic
solutions. The fixed point method considers the case of infinite lattices.
Through this method, the existence of a threshold is proved in the case of
saturable nonlinearity and an explicit theoretical estimate which is
independent on the dimension is given. The numerical studies, testing the
efficiency of the bounds derived by both methods, demonstrate that these
thresholds are quite sharp estimates of a threshold value on the power needed
for the the existence of a breather solution. This it justified by the
consideration of limiting cases with respect to the size of the nonlinearity
parameters and nonlinearity exponents.Comment: 26 pages, 10 figure
Abelian functions associated with a cyclic tetragonal curve of genus six
We develop the theory of Abelian functions defined using a tetragonal curve of genus six, discussing in detail the cyclic curve y^4 = x^5 + λ[4]x^4 + λ[3]x^3 + λ[2]x^2 + λ[1]x + λ[0]. We construct Abelian functions using the multivariate sigma-function associated with the curve, generalizing the theory of theWeierstrass℘-function.
We demonstrate that such functions can give a solution to the KP-equation, outlining how a general class of solutions could be generated using a wider class of curves. We also present the associated partial differential equations
satisfied by the functions, the solution of the Jacobi inversion problem, a power series expansion for σ(u) and a new addition formula
Sigma, tau and Abelian functions of algebraic curves
We compare and contrast three different methods for the construction of the
differential relations satisfied by the fundamental Abelian functions
associated with an algebraic curve. We realize these Abelian functions as
logarithmic derivatives of the associated sigma function. In two of the
methods, the use of the tau function, expressed in terms of the sigma function,
is central to the construction of differential relations between the Abelian
functions.Comment: 25 page
Periods of second kind differentials of (n,s)-curves
For elliptic curves, expressions for the periods of elliptic integrals of the
second kind in terms of theta-constants, have been known since the middle of
the 19th century. In this paper we consider the problem of generalizing these
results to curves of higher genera, in particular to a special class of
algebraic curves, the so-called -curves. It is shown that the
representations required can be obtained by the comparison of two equivalent
expressions for the projective connection, one due to Fay-Wirtinger and the
other from Klein-Weierstrass. As a principle example, we consider the case of
the genus two hyperelliptic curve, and a number of new Thomae and
Rosenhain-type formulae are obtained. We anticipate that our analysis for the
genus two curve can be extended to higher genera hyperelliptic curves, as well
as to other classes of non-hyperelliptic curves.Comment: 21 page
Breathers and kinks in a simulated crystal experiment
We develop a simple 1D model for the scattering of an incoming particle
hitting the surface of mica crystal, the transmission of energy through the
crystal by a localized mode, and the ejection of atom(s) at the incident or
distant face. This is the first attempt to model the experiment described in
Russell and Eilbeck in 2007 (EPL, v. 78, 10004). Although very basic, the model
shows many interesting features, for example a complicated energy dependent
transition between breather modes and a kink mode, and multiple ejections at
both incoming and distant surfaces. In addition, the effect of a heavier
surface layer is modelled, which can lead to internal reflections of breathers
or kinks at the crystal surface.Comment: 15 pages, 12 figures, based on a talk given at the conference
"Localized Excitations in Nonlinear Complex Systems (LENCOS)", Sevilla
(Spain) July 14-17, 200
Aharonov-Bohm effect for an exciton in a finite width nano-ring
We study the Aharonov-Bohm effect for an exciton on a nano-ring using a 2D attractive fermionic Hubbard model. We extend previous results obtained for a 1D ring in which only azimuthal motion is considered, to a more general case of 2D annular lattices. In general, we show that the
existence of the localization effect, increased by the nonlinearity, makes the phenomenon in the 2D system similar to the 1D case. However, the introduction of radial motion introduces extra frequencies, different from the original isolated frequency corresponding to the excitonic Aharonov-
Bohm oscillations. If the circumference of the system becomes large enough, the Aharonov-Bohm effect is suppressed
Influence of moving breathers on vacancies migration
A vacancy defect is described by a Frenkel--Kontorova model with a
discommensuration. This vacancy can migrate when interacts with a moving
breather. We establish that the width of the interaction potential must be
larger than a threshold value in order that the vacancy can move forward. This
value is related to the existence of a breather centred at the particles
adjacent to the vacancy.Comment: 11 pages, 10 figure
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