4,030,095 research outputs found
Inverse problem for an inhomogeneous Schr\"odinger equation
An inverse problem is considered for an inhomogeneous Schr\"odinger equation.
Assuming that the potential vanishes outside a finite interval and satisfies
some other technical assumptions, one proves the uniqueness of the recovery of
this potential from the knowledge of the wave function at the ends of the above
interval for all energies. An algorithm is given for the recovery of the
potential from the above data
Global well-posedness and scattering for the defocusing energy-critical nonlinear Schr\"odinger equation in
We obtain global well-posedness, scattering, uniform regularity, and global
spacetime bounds for energy-space solutions to the defocusing
energy-critical nonlinear Schr\"odinger equation in . Our
arguments closely follow those of Colliander-Keel-Staffilani-Takaoka-Tao,
though our derivation of the frequency-localized interaction Morawetz estimate
is somewhat simpler. As a consequence, our method yields a better bound on the
-norm
PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation
We discuss several PT-symmetric deformations of superderivatives. Based on
these various possibilities, we propose new families of complex PT-symmetric
deformations of the supersymmetric Korteweg-de Vries equation. Some of these
new models are mere fermionic extensions of the former in the sense that they
are formulated in terms of superspace valued superfields containing bosonic and
fermionic fields, breaking however the supersymmetry invariance. Nonetheless,
we also find extensions, which may be viewed as new supersymmetric Korteweg-de
Vries equation. Moreover, we show that these deformations allow for a
non-Hermitian Hamiltonian formulation and construct three charges associated to
the corresponding flow.Comment: 10 page
Discrete--time ratchets, the Fokker--Planck equation and Parrondo's paradox
Parrondo's games manifest the apparent paradox where losing strategies can be
combined to win and have generated significant multidisciplinary interest in
the literature. Here we review two recent approaches, based on the
Fokker-Planck equation, that rigorously establish the connection between
Parrondo's games and a physical model known as the flashing Brownian ratchet.
This gives rise to a new set of Parrondo's games, of which the original games
are a special case. For the first time, we perform a complete analysis of the
new games via a discrete-time Markov chain (DTMC) analysis, producing winning
rate equations and an exploration of the parameter space where the paradoxical
behaviour occurs.Comment: 17 pages, 5 figure
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