89 research outputs found
Integrable Quartic Potentials and Coupled KdV Equations
We show a surprising connection between known integrable Hamiltonian systems
with quartic potential and the stationary flows of some coupled KdV systems
related to fourth order Lax operators. In particular, we present a connection
between the Hirota-Satsuma coupled KdV system and (a generalisation of) the
integrable case quartic potential. A generalisation of the case
is similarly related to a different (but gauge related) fourth order Lax
operator. We exploit this connection to derive a Lax representation for each of
these integrable systems. In this context a canonical transformation is derived
through a gauge transformation.Comment: LaTex, 11 page
Quasiperiodic Solutions of the Fibre Optics Coupled Nonlinear Schr{\"o}dinger Equations
We consider travelling periodical and quasiperiodical waves in single mode
fibres, with weak birefringence and under the action of cross-phase modulation.
The problem is reduced to the ``1:2:1" integrable case of the two-particle
quartic potential. A general approach for finding elliptic solutions is given.
New solutions which are associated with two-gap Treibich-Verdier potentials are
found. General quasiperiodic solutions are given in terms of two dimensional
theta functions with explicit expressions for frequencies in terms of theta
constants. The reduction of quasiperiodic solutions to elliptic functions is
discussed.Comment: 24 page
Particle motion in Horava-Lifshitz black hole space-times
We study the particle motion in the space-time of a Kehagias-Sfetsos (KS)
black hole. This is a static spherically symmetric solution of a
Horava-Lifshitz gravity model that reduces to General Relativity in the IR
limit and deviates slightly from detailed balance. Taking the viewpoint that
the model is essentially a (3+1)-dimensional modification of General Relativity
we use the geodesic equation to determine the motion of massive and massless
particles. We solve the geodesic equation exactly by using numerical
techniques. We find that neither massless nor massive particles with
non-vanishing angular momentum can reach the singularity at r=0. Next to bound
and escape orbits that are also present in the Schwarzschild space-time we find
that new types of orbits exist: manyworld bound orbits as well as two-world
escape orbits. We also discuss observables such as the perihelion shift and the
light deflection.Comment: 16 pages including 13 figures; minor changes, to match version
accepted for publication in Phys. Rev.
Double Pendulum and θ -Divisor
The equations of motion of integrable systems involving hyperelliptic Riemann surfaces of genus 2 and one relevant degree of freedom are integrated in the framework of the Jacobi inversion problem, using a reduction to the θ -divisor on the Jacobi variety, i.e., to the set of zeros of the θ -function. Explicit solutions are given in terms of Kleinian σ -functions and their derivatives. The procedure is applied to the planar double pendulum without gravity, but it is worked out for any Abelian integral of first or second kind
The Lax pairs for the Holt system
By using non-canonical transformation between the Holt system and the
Henon-Heiles system the Lax pairs for all the integrable cases of the Holt
system are constructed from the known Lax representations for the Henon-Heiles
system.Comment: 7 pages, LaTeX2e, a4.st
On the calculation of finite-gap solutions of the KdV equation
A simple and general approach for calculating the elliptic finite-gap
solutions of the Korteweg-de Vries (KdV) equation is proposed. Our approach is
based on the use of the finite-gap equations and the general representation of
these solutions in the form of rational functions of the elliptic Weierstrass
function. The calculation of initial elliptic finite-gap solutions is reduced
to the solution of the finite-band equations with respect to the parameters of
the representation. The time evolution of these solutions is described via the
dynamic equations of their poles, integrated with the help of the finite-gap
equations. The proposed approach is applied by calculating the elliptic 1-, 2-
and 3-gap solutions of the KdV equations
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