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 R\mathbb{R} at L2L^2 regularity

We prove global existence and stability of solution to the mass-critical stochastic nonlinear Schr\"odinger equation in $d=1$ at $L^2$ 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 d≥3d \geq 3 with small multiplicative noise

We give decay estimates of the solution to the linear Schr\"odinger equation in dimension $d \geq 3$ 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

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 $\mathbb{Z}$-basis for the algebras generated by all generalized cluster variables.Comment: 11 page

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 3×33\times 3 Lax pairs on the interval

We present an approach for analyzing initial-boundary value problems which is formulated on the finite interval ($0\le x\le L$, where $L$ is a positive constant) for integrable equations whose Lax pairs involve $3\times 3$ 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 $3\times 3$ Riemann-Hilbert problem. The relevant jump matrices are explicitly given in terms of the three matrix-value spectral functions $s(k)$,$S(k)$ and $S_L(k)$, which in turn are defined in terms of the initial values, boundary values at $x=0$ and boundary values at $x=L$, 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 $\mathbb{ZP}$-bases in quantum cluster algebras of finite and affine types. Under the specialization $q$ and coefficients to $1$, 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