18 research outputs found
Consistency of the B\"acklund transformation for the spin Calogero-Moser system
We prove the consistency of the B\"{a}cklund transformation (BT) for the spin
Calogero-Moser (sCM) system in the rational, trigonometric, and hyperbolic
cases. The BT for the sCM system consists of an overdetermined system of ODEs;
to establish our result, we construct and analyze certain functions that
measure the departure of this overdetermined system from consistency and show,
under mild assumptions, that these functions are identically zero and that this
allows for a unique solution to the initial value problem for the
overdetermined system.Comment: 14 page
Periodic solutions of the non-chiral intermediate Heisenberg ferromagnet equation described by elliptic spin Calogero-Moser dynamics
We present a class of periodic solutions of the non-chiral intermediate
Heisenberg ferromagnet (ncIHF) equation, which was recently introduced by the
authors together with Langmann as a classical, continuum limit of an
Inozemtsev-type spin chain. These exact analytic solutions are constructed via
a spin-pole ansatz written in terms of certain elliptic functions. The
dynamical parameters in our solutions solve an elliptic spin Calogero-Moser
(CM) system subject to certain constraints. In the course of our construction,
we establish a novel B\"acklund transformation for this constrained elliptic
spin CM system.Comment: 37 pages, 3 figure
Observation of PT phase transition in a simple mechanical system
If a Hamiltonian is PT symmetric, there are two possibilities: Either the
eigenvalues are entirely real, in which case the Hamiltonian is said to be in
an unbroken-PT-symmetric phase, or else the eigenvalues are partly real and
partly complex, in which case the Hamiltonian is said to be in a
broken-PT-symmetric phase. As one varies the parameters of the Hamiltonian, one
can pass through the phase transition that separates the unbroken and broken
phases. This transition has recently been observed in a variety of laboratory
experiments. This paper explains the phase transition in a simple and intuitive
fashion and then describes an extremely elementary experiment in which the
phase transition is easily observed.Comment: 9 pages, 9 figure
Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation
We construct elliptic multi-soliton solutions of the spin non-chiral
intermediate long-wave (sncILW) equation with periodic boundary conditions.
These solutions are obtained by a spin-pole ansatz including a dynamical
background term; we show that this ansatz solves the periodic sncILW equation
provided the spins and poles satisfy the elliptic -type spin Calogero-Moser
(sCM) system with certain constraints on the initial conditions. The key to
this result is a B\"{a}cklund transformation for the elliptic sCM system which
includes a non-trivial dynamical background term. We also present solutions of
the sncILW equation on the real line and of the spin Benjamin-Ono equation
which generalize previously obtained solutions by allowing for a non-trivial
background term.Comment: 33 page