In this paper we establish mixed norm estimates of interactive Schrödinger waves and apply them to study smoothing properties and global well-posedness of the nonlinear Schrödinger equations with mass critical nonlinearity.

The Aharonov-Bohm (AB) effect in non-commutative quantum mechanics (NCQM) is studied. First, by introducing a shift for the magnetic vector potential we give the Schrödinger equations in the presence of a magnetic field on NC space and NC phase space, respectively. Then by solving the Schrödinger equations, we obtain the Aharonov-Bohm (AB) phase on NC space and NC phase space, respectively. PAC...

Abstract We consider Schrödinger equations with variable coefficients, and it is supposed to be a long-range type perturbation of the flat Laplacian on Rn. We characterize the wave front set of solutions to Schrödinger equations in terms of the initial state. Then it is shown that the singularity propagates following the classical flow, and it is formulated in a semiclassical setting. Methods a...

This paper presents a high order local discontinuous Galerkin time-domain method for solving time dependent Schrödinger equations. After rewriting the Schrödinger equation in terms of a first order system of equations, a numerical flux is constructed to preserve the conservative property for the density of the particle described. Numerical results for a model square potential scattering problem...

We study matrix generalizations of derivative nonlinear Schrödinger-type equations, which were shown by Olver and Sokolov to possess a higher symmetry. We prove that two of them are ‘C-integrable’ and the rest of them are ‘S-integrable’ in Calogero’s terminology. PACS numbers: 02.10Jf, 02.30.Jr, 03.65.Ge, 11.30.-j Submitted to: Inverse Problems † E-mail address: [email protected]...

Nonlinear evolution equations are of tremendous interest in both theory and applications. In this talk we introduce parallel algorithms for numerical simulations of CMKdV, NLS and and CNLS equations in 1+1 and 1+2 dimensions. The parallel methods are implemented on multiprocessor system. Numerical experiments have shown that these methods give accurate results and considerable speedup. This tal...

These notes were written as a guideline for a short talk; hence, the references and the statements of the theorems are often not given in full details. The point of these notes is to summarize the different directions that the study of dispersive equations has taken in the last ten years. I would like to mention here a few recent textbooks that treat different parts of this subject. They could ...

Interaction and correlation effects in quantum dots play a fundamental role in defining both their equilibrium and transport properties. Numerical methods are commonly employed to study such systems. In this paper we investigate the numerical calculation of quantum transport of electrons in spherical centered defect InGaAs/AlGaAs quantum dot (SCDQD). The simulation is based on the imaginary time...

A generalized inverse scattering method has been developed for arbitrary n dimensional Lax equations. Subsequently, the method has been used to obtain N soliton solutions of a vector higher order nonlinear Schrödinger equation, proposed by us. It has been shown that under suitable reduction, vector higher order nonlinear Schrödinger equation reduces to higher order nonlinear Schrödinger equatio...

In this paper we use a similarity transformation connecting some families of Nonlinear Schrödinger equations with time-varying coefficients with the autonomous cubic nonlinear Schrödinger equation. This transformation allows one to apply all known results for that equation to the original non-autonomous case with the additional dynamics introduced by the transformation itself. In particular, us...