13 research outputs found
Stability of pulses on optical fibers with phase-sensitive amplifiers
Pulse stability is crucial to the effective propagation of information in a soliton-based optical communication system. It is shown in this paper that pulses in optical fibers, for which attenuation is compensated by phase-sensitive amplifiers, are stable over a large range of parameter values. A fourth-order nonlinear diffusion model due to Kath and co-workers is used. The stability proof invokes a number of mathematical techniques, including the Evans function and Grillakis' functional analytic approach
Second-order corrections to mean field evolution for weakly interacting Bosons. I
Inspired by the works of Rodnianski and Schlein and Wu, we derive a new
nonlinear Schr\"odinger equation that describes a second-order correction to
the usual tensor product (mean-field) approximation for the Hamiltonian
evolution of a many-particle system in Bose-Einstein condensation. We show that
our new equation, if it has solutions with appropriate smoothness and decay
properties, implies a new Fock space estimate. We also show that for an
interaction potential , where is
sufficiently small and , our program can be easily
implemented locally in time. We leave global in time issues, more singular
potentials and sophisticated estimates for a subsequent part (part II) of this
paper
Impurity and quaternions in nonrelativistic scattering from a quantum memory
Models of quantum computing rely on transformations of the states of a
quantum memory. We study mathematical aspects of a model proposed by Wu in
which the memory state is changed via the scattering of incoming particles.
This operation causes the memory content to deviate from a pure state, i.e.
induces impurity. For nonrelativistic particles scattered from a two-state
memory and sufficiently general interaction potentials in 1+1 dimensions, we
express impurity in terms of quaternionic commutators. In this context, pure
memory states correspond to null hyperbolic quaternions. In the case with point
interactions, the scattering process amounts to appropriate rotations of
quaternions in the frequency domain. Our work complements a previous analysis
by Margetis and Myers (2006 J. Phys. A 39 11567--11581).Comment: 16 pages, no figure
Stability of pulses on optical fibers with phase-sensitive amplifiers
Pulse stability is crucial to the effective propagation of information in a soliton-based optical communication system. It is shown in this paper that pulses in optical fibers, for which attenuation is compensated by phase-sensitive amplifiers, are stable over a large range of parameter values. A fourth-order nonlinear diffusion model due to Kutz and co-workers is used. The stability proof invokes a number of mathematical techniques, including the Evans function and Grillakis' functional analytic approach