641 research outputs found
Wave Solutions of Evolution Equations and Hamiltonian Flows on Nonlinear Subvarieties of Generalized Jacobians
The algebraic-geometric approach is extended to study solutions of
N-component systems associated with the energy dependent Schrodinger operators
having potentials with poles in the spectral parameter, in connection with
Hamiltonian flows on nonlinear subvariaties of Jacobi varieties. The systems
under study include the shallow water equation and Dym type equation. The
classes of solutions are described in terms of theta-functions and their
singular limits by using new parameterizations. A qualitative description of
real valued solutions is provided
Measurement induced chaos with entangled states
The dynamics of an ensemble of identically prepared two-qubit systems is
investigated which is subjected to the iteratively applied measurements and
conditional selection of a typical entanglement purification protocol. It is
shown that the resulting measurement-induced non-linear dynamics of the
two-qubit state exhibits strong sensitivity to initial conditions and also true
chaos. For a special class of initially prepared two-qubit states two types of
islands characterize the asymptotic limit. They correspond to a separable and a
maximally entangled two-qubit state, respectively, and their boundaries form
fractal-like structures. In the presence of incoherent noise an additional
stable asymptotic cycle appears.Comment: 5 pages, 3 figure
A forward-backward splitting algorithm for the minimization of non-smooth convex functionals in Banach space
We consider the task of computing an approximate minimizer of the sum of a
smooth and non-smooth convex functional, respectively, in Banach space.
Motivated by the classical forward-backward splitting method for the
subgradients in Hilbert space, we propose a generalization which involves the
iterative solution of simpler subproblems. Descent and convergence properties
of this new algorithm are studied. Furthermore, the results are applied to the
minimization of Tikhonov-functionals associated with linear inverse problems
and semi-norm penalization in Banach spaces. With the help of
Bregman-Taylor-distance estimates, rates of convergence for the
forward-backward splitting procedure are obtained. Examples which demonstrate
the applicability are given, in particular, a generalization of the iterative
soft-thresholding method by Daubechies, Defrise and De Mol to Banach spaces as
well as total-variation based image restoration in higher dimensions are
presented
Non-Analytic Vertex Renormalization of a Bose Gas at Finite Temperature
We derive the flow equations for the symmetry unbroken phase of a dilute
3-dimensional Bose gas. We point out that the flow equation for the interaction
contains parts which are non-analytic at the origin of the frequency-momentum
space. We examine the way this non-analyticity affects the fixed point of the
system of the flow equations and shifts the value of the critical exponent for
the correlation length closer to the experimental result in comparison with
previous work where the non-analyticity was neglected. Finally, we emphasize
the purely thermal nature of this non-analytic behaviour comparing our approach
to a previous work where non-analyticity was studied in the context of
renormalization at zero temperature.Comment: 21 pages, 4 figure
On the integrability of stationary and restricted flows of the KdV hierarchy.
A bi--Hamiltonian formulation for stationary flows of the KdV hierarchy is
derived in an extended phase space. A map between stationary flows and
restricted flows is constructed: in a case it connects an integrable
Henon--Heiles system and the Garnier system. Moreover a new integrability
scheme for Hamiltonian systems is proposed, holding in the standard phase
space.Comment: 25 pages, AMS-LATEX 2.09, no figures, to be published in J. Phys. A:
Math. Gen.
On a Camassa-Holm type equation with two dependent variables
We consider a generalization of the Camassa Holm (CH) equation with two
dependent variables, called CH2, introduced by Liu and Zhang. We briefly
provide an alternative derivation of it based on the theory of Hamiltonian
structures on (the dual of) a Lie Algebra. The Lie Algebra here involved is the
same algebra underlying the NLS hierarchy. We study the structural properties
of the CH2 hierarchy within the bihamiltonian theory of integrable PDEs, and
provide its Lax representation. Then we explicitly discuss how to construct
classes of solutions, both of peakon and of algebro-geometrical type. We
finally sketch the construction of a class of singular solutions, defined by
setting to zero one of the two dependent variables.Comment: 22 pages, 2 figures. A few typos correcte
Entangled States and Entropy Remnants of a Photon-Electron System
In the present paper an example of entanglement between two different kinds
of interacting particles, photons and electrons is analysed. The initial-value
problem of the Schroedinger equation is solved non-perturbatively for the
system of a free electron interacting with a quantized mode of the
electromagnetic radiation. Wave packets of the dressed states so obtained are
constructed in order to describe the spatio-temporal separation of the
subsystems before and after the interaction. The joint probability amplitudes
are calculated for the detection of the electron at some space-time location
and the detection of a definite number of photons. The analytical study of the
time evolution of entanglement between the initially separated electron wave
packet and the radiation mode leads to the conclusion that in general there are
non-vanishing entropy remnants in the subsystems after the interaction. On the
basis of the simple model to be presented here, the calculated values of the
entropy remnants crucially depend on the character of the switching-on and off
of the interaction.Comment: 12 pages, 2 figure
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