63,294 research outputs found
Fano Interference in Two-Photon Transport
We present a general input-output formalism for the few-photon transport in
multiple waveguide channels coupled to a local cavity. Using this formalism, we
study the effect of Fano interference in two-photon quantum transport. We show
that the physics of Fano interference can manifest as an asymmetric spectral
line shape in the frequency dependence of the two-photon correlation function.
The two-photon fluorescence spectrum, on the other hand, does not exhibit the
physics of Fano interference
Long-time asymptotic for the derivative nonlinear Schr\"odinger equation with decaying initial value
We present a new Riemann-Hilbert problem formalism for the initial value
problem for the derivative nonlinear Schr\"odinger (DNLS) equation on the line.
We show that the solution of this initial value problem can be obtained from
the solution of some associated Riemann-Hilbert problem. This new
Riemann-Hilbert problem for the DNLS equation will lead us to use nonlinear
steepest-descent/stationary phase method or Deift-Zhou method to derive the
long-time asymptotic for the DNLS equation on the line.Comment: 41 page
The Ostrovsky-Vakhnenko equation on the half-line: a Riemann-Hilbert approach
We analyze an initial-boundary value problem for the Ostrovsky-Vakhnenko
equation on the half-line. This equation can be viewed as the short wave model
for the Degasperis-Procesi (DP) equation. We show that the solution u(x,t) can
be recovered from its initial and boundary values via the solution of a 3\times
3 vector Riemann-Hilbert problem formulated in the complex plane of a spectral
parameter z.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1204.5252,
arXiv:1311.0495 by other author
Global well-posedness for the defocusing mass-critical stochastic nonlinear Schr\"odinger equation on at regularity
We prove global existence and stability of solution to the mass-critical
stochastic nonlinear Schr\"odinger equation in at regularity. Our
construction starts with the existence of solution to the truncated subcritical
problem. With the presence of truncation, we construct the solution to the
critical equation as the limit of subcritical solutions. We then obtain uniform
bounds on the solutions to the truncated critical problems that allow us to
remove truncation in the limit.Comment: 37 pages. The material presented in this article is a re-orgasination
of arXiv:1803.03257 and part of arXiv:1807.04402 of the authors. The other
part of arXiv:1807.04402, which has not been covered in the current article,
will be re-written as another independent one (forthcoming). Only the current
article and the forthcoming one will be submitted for journal publicatio
Decay of the stochastic linear Schr\"odinger equation in with small multiplicative noise
We give decay estimates of the solution to the linear Schr\"odinger equation
in dimension with a small noise which is white in time and colored
in space. As a consequence, we also obtain certain asymptotic behaviour of the
solution. The proof relies on the bootstrapping argument used by
Journ\'e-Soffer-Sogge for decay of deterministic Schr\"odinger operators.Comment: 15 page
Hall algebras associated to triangulated categories
By counting with triangles and the octohedral axiom, we find a direct way to
prove the formula of To\"en in \cite{Toen2005} for a triangulated category with
(left) homological-finite condition.Comment: 12 pages. Final version, to appear in Duk
Leading-order temporal asymptotics of the Fokas-Lenells Equation without solitons
We use the Deift-Zhou method to obtain, in the solitonless sector, the
leading order asymptotic of the solution to the Cauchy problem of the
Fokas-Lenells equation as t\ra+\infty on the full-line.Comment: 47 pages. arXiv admin note: substantial text overlap with
arXiv:solv-int/9701001 by other author
Initial-boundary value problem for integrable nonlinear evolution equations with Lax pairs on the interval
We present an approach for analyzing initial-boundary value problems which is
formulated on the finite interval (, where is a positive
constant) for integrable equations whose Lax pairs involve
matrices. Boundary value problems for integrable nonlinear evolution PDEs can
be analyzed by the unified method introduced by Fokas and developed by him and
his collaborators. In this paper, we show that the solution can be expressed in
terms of the solution of a Riemann-Hilbert problem. The relevant
jump matrices are explicitly given in terms of the three matrix-value spectral
functions , and , which in turn are defined in terms of the
initial values, boundary values at and boundary values at ,
respectively. However, these spectral functions are not independent, they
satisfy a global relation. Here, we show that the characterization of the
unknown boundary values in terms of the given initial and boundary data is
explicitly described for a nonlinear evolution PDE defined on the interval.
Also, we show that in the limit when the length of the interval tends to
infity, the relevant formulas reduce to the analogous formulas obtained for the
case of boundary value problems formulated on the half-line.Comment: arXiv admin note: substantial text overlap with arXiv:1304.4586; text
overlap with arXiv:1108.2875 by other author
The multiplication theorem and bases in finite and affine quantum cluster algebras
We prove a multiplication theorem for quantum cluster algebras of acyclic
quivers. The theorem generalizes the multiplication formula for quantum cluster
variables in \cite{fanqin}. We apply the formula to construct some
-bases in quantum cluster algebras of finite and affine types.
Under the specialization and coefficients to , these bases are the
integral bases of cluster algebra of finite and affine types (see \cite{CK1}
and \cite{DXX}).Comment: 20 pages, the integral bases of cluster algebra of affine types are
replace
The cluster character for cyclic quivers
We define an analogue of the Caldero-Chapoton map (\cite{CC}) for the cluster
category of finite dimensional nilpotent representations over a cyclic quiver.
We prove that it is a cluster character (in the sense of \cite{Palu}) and
satisfies some inductive formulas for the multiplication between the
generalized cluster variables (the images of objects of the cluster category
under the map). Moreover, we construct a -basis for the algebras
generated by all generalized cluster variables.Comment: 11 page
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