Ballistic Transport for the Schrödinger Operator with Limit-Periodic or Quasi-Periodic Potential in Dimension Two

Optimized Periodic Coulomb Potential in Two Dimension

The 1/r Coulomb potential is calculated for a two dimensional system with periodic boundary conditions. Using polynomial splines in real space and a summation in reciprocal space we obtain numerically optimized potentials which allow us efficient calculations of any periodic (long-ranged) potential up to high precision. We discuss the parameter space of the optimized potential for the periodic ...

Quasi-periodic Solutions for One-Dimensional Nonlinear Lattice Schrödinger Equation with Tangent Potential

In this paper, we construct time quasi-periodic solutions for the nonlinear lattice Schrödinger equation iq̇n + (qn+1 + qn−1) + tanπ(nα̃+ x)qn + |qn|qn = 0, n ∈ Z, where α̃ satisfies a certain Diophantine condition and x ∈ R/Z. We prove that for sufficiently small, the equation admits a family of small-amplitude time quasi-periodic solutions for “most” of x belonging to R/Z.

Modulational Instability for Nonlinear Schrödinger Equations with a Periodic Potential

We study the stability properties of periodic solutions to the Nonlinear Schrödinger (NLS) equation with a periodic potential. We exploit the symmetries of the problem, in particular the Hamiltonian structure and the U(1) symmetry. We develop a simple sufficient criterion that guarantees the existence of a modulational instability spectrum along the imaginary axis. In the case of small amplitud...

On nonperturbative localization with quasi - periodic potential

Finite-Difference Time-Domain Simulation of Light Propagation in 2D Periodic and Quasi-Periodic Photonic Structures

Ultra-short pulse is a promising technology for achieving ultra-high data rate transmission which is required to follow the increased demand of data transport over an optical communication system. Therefore, the propagation of such type of pulses and the effects that it may suffer during its transmission through an optical waveguide has received a great deal of attention in the recent years. We...