857 research outputs found

### Discrete soliton collisions in a waveguide array with saturable nonlinearity

We study the symmetric collisions of two mobile breathers/solitons in a model
for coupled wave guides with a saturable nonlinearity. The saturability allows
the existence of breathers with high power. Three main regimes are observed:
breather fusion, breather reflection and breather creation. The last regime
seems to be exclusive of systems with a saturable nonlinearity, and has been
previously observed in continuous models. In some cases a ``symmetry breaking''
can be observed, which we show to be an numerical artifact.Comment: 5 pages, 7 figure

### The Discrete Nonlinear Schr\"odinger equation - 20 Years on

We review work on the Discrete Nonlinear Schr\"odinger (DNLS) equation over
the last two decades.Comment: 24 pages, 1 figure, Proceedings of the conference on "Localization
and Energy Transfer in Nonlinear Systems", June 17-21, 2002, San Lorenzo de
El Escorial, Madrid, Spain; to be published by World Scientifi

### Evolution of the Sequence Ontology terms and relationships

The Sequence Ontology is undergoing reform to meet the standards of the OBO Foundry. Here we report some of the incremental changes and improvements made to SO. We also propose new relationships to better define the mereological, spatial and temporal aspects of biological sequence

### Quantum Lattice Solitons

The number state method is used to study soliton bands for three anharmonic
quantum lattices: i) The discrete nonlinear Schr\"{o}dinger equation, ii) The
Ablowitz-Ladik system, and iii) A fermionic polaron model. Each of these
systems is assumed to have $f$-fold translational symmetry in one spatial
dimension, where $f$ is the number of freedoms (lattice points). At the second
quantum level $(n=2)$ we calculate exact eigenfunctions and energies of pure
quantum states, from which we determine binding energy $(E_{\rm b})$, effective
mass $(m^{*})$ and maximum group velocity $(V_{\rm m})$ of the soliton bands as
functions of the anharmonicity in the limit $f \to \infty$. For arbitrary
values of $n$ we have asymptotic expressions for $E_{\rm b}$, $m^{*}$, and
$V_{\rm m}$ as functions of the anharmonicity in the limits of large and small
anharmonicity. Using these expressions we discuss and describe wave packets of
pure eigenstates that correspond to classical solitons.Comment: 21 pages, 1 figur

### On Hyperelliptic Abelian Functions of Genus 3

The affine ring A of the affine Jacobian variety of a hyperelliptic curve of
genus 3 is studied as a D-module. The conjecture on the minimal D-free
resolution previously proposed is proved in this case. As a by-product a linear
basis of A is explicitly constructed in terms of derivatives of Klein's
hyperelliptic pe functions.Comment: 40 page

### Abelian functions associated with genus three algebraic curves

We develop the theory of Abelian functions associated with algebraic curves.
The growth in computer power and an advancement of efficient symbolic
computation techniques has allowed for recent progress in this area. In this
paper we focus on the genus three cases, comparing the two canonical classes of
hyperelliptic and trigonal curves. We present new addition formulae, derive
bases for the spaces of Abelian functions and discuss the differential
equations such functions satisfy.Comment: 34 page

### 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

### 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

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