64 research outputs found

### 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

### 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

### Localized moving breathers in a 2-D hexagonal lattice

We show for the first time that highly localized in-plane breathers can
propagate in specific directions with minimal lateral spreading in a model 2-D
hexagonal non-linear lattice. The lattice is subject to an on-site potential in
addition to longitudinal nonlinear inter-particle interactions. This study
investigates the prediction that stable breather-like solitons could be formed
as a result of energetic scattering events in a given layered crystal and would
propagate in atomic-chain directions in certain atomic planes. This prediction
arose from a long-term study of previously unexplained dark lines in natural
crystals of muscovite mica.Comment: 6 pages, 2 Figs. Submitted to PR

### Fast energy transfer mediated by multi-quanta bound states in a nonlinear quantum lattice

By using a Generalized Hubbard model for bosons, the energy transfer in a
nonlinear quantum lattice is studied, with special emphasis on the interplay
between local and nonlocal nonlinearity. For a strong local nonlinearity, it is
shown that the creation of v quanta on one site excites a soliton band formed
by bound states involving v quanta trapped on the same site. The energy is
first localized on the excited site over a significant timescale and then
slowly delocalizes along the lattice. As when increasing the nonlocal
nonlinearity, a faster dynamics occurs and the energy propagates more rapidly
along the lattice. Nevertheless, the larger is the number of quanta, the slower
is the dynamics. However, it is shown that when the nonlocal nonlinearity
reaches a critical value, the lattice suddenly supports a very fast energy
propagation whose dynamics is almost independent on the number of quanta. The
energy is transfered by specific bound states formed by the superimposition of
states involving v-p quanta trapped on one site and p quanta trapped on the
nearest neighbour sites, with p=0,..,v-1. These bound states behave as
independent quanta and they exhibit a dynamics which is insensitive to the
nonlinearity and controlled by the single quantum hopping constant.Comment: 28 pages, 8 figure

### Properties of the series solution for Painlevé I

We present some observations on the asymptotic behaviour of the coefficients in the Laurent series expansion of solutions of the first Painlevé equation. For the general solution, explicit recursive formulae for the Taylor expansion of the tau-function around a zero are given, which are natural extensions of analogous formulae for the elliptic sigma function, as given by Weierstrass. Numerical and exact results on the symmetric solution which is singular at the origin are also presented

### Linear $r$-Matrix Algebra for Systems Separable\\ in Parabolic Coordinates

We consider a hierarchy of many particle systems on the line with polynomial
potentials separable in parabolic coordinates. Using the Lax representation,
written in terms of $2\times 2$ matrices for the whole hierarchy, we construct
the associated linear $r$-matrix algebra with the $r$-matrix dependent on the
dynamical variables. A dynamical Yang-Baxter equation is discussed.Comment: 10 pages, LaTeX. Submitted to Phys.Lett.

### Spectral Curves of Operators with Elliptic Coefficients

A computer-algebra aided method is carried out, for determining geometric objects associated to differential operators that satisfy the elliptic ansatz. This results in examples of Lamé curves with double reduction and in the explicit reduction of the theta function of a Halphen curve

### Generalised Elliptic Functions

We consider multiply periodic functions, sometimes called Abelian functions,
defined with respect to the period matrices associated with classes of
algebraic curves. We realise them as generalisations of the Weierstras
P-function using two different approaches. These functions arise naturally as
solutions to some of the important equations of mathematical physics and their
differential equations, addition formulae, and applications have all been
recent topics of study.
The first approach discussed sees the functions defined as logarithmic
derivatives of the sigma-function, a modified Riemann theta-function. We can
make use of known properties of the sigma function to derive power series
expansions and in turn the properties mentioned above. This approach has been
extended to a wide range of non hyperelliptic and higher genus curves and an
overview of recent results is given.
The second approach defines the functions algebraically, after first
modifying the curve into its equivariant form. This approach allows the use of
representation theory to derive a range of results at lower computational cost.
We discuss the development of this theory for hyperelliptic curves and how it
may be extended in the future.Comment: 16 page

### Transport behaviour of a Bose Einstein condensate in a bichromatic optical lattice

The Bloch and dipole oscillations of a Bose Einstein condensate (BEC) in an
optical superlattice is investigated. We show that the effective mass increases
in an optical superlattice, which leads to localization of the BEC, in
accordance with recent experimental observations [16]. In addition, we find
that the secondary optical lattice is a useful additional tool to manipulate
the dynamics of the atoms.Comment: Modified manuscrip

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