Abelian Functions for Cyclic Trigonal Curves of Genus Four

We discuss the theory of generalized Weierstrass $\sigma$ and $\wp$ functions defined on a trigonal curve of genus four, following earlier work on the genus three case. The specific example of the "purely trigonal" (or "cyclic trigonal") curve $y^3=x^5+\lambda_4 x^4 +\lambda_3 x^3+\lambda_2 x^2 +\lambda_1 x+\lambda_0$ is discussed in detail, including a list of some of the associated partial differential equations satisfied by the $\wp$ functions, and the derivation of an addition formulae.Comment: 23 page

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