In this work we want to describe a discrete version of the well–known Adomian decomposition method (ADM) applied to the discrete nonlinear Schrödinger equation (NLS). The ADM was introduced by Adomian [3], [11] in the early 1980s to solve nonlinear ordinary and partial differential equations. The discrete nonlinear Schrödinger equation is omnipresent [8] in applied sciences, e.g. describing the...

We investigate the convergence properties of a three-dimensional quantum lattice Boltzmann scheme for the Dirac equation. These schemes were constructed as discretizations of the Dirac equation based on operator splitting to separate the streaming along the three coordinate axes, but their output has previously only been compared against solutions of the Schrödinger equation. The Schrödinger eq...

A numerical scheme for the one dimensional stationary Schrödinger-Poisson model is described. The scheme is used to simulate a Resonant Tunneling Diode and provides an important reduction of the simulation time. The improvement is twofold. First the grid spacing in the position variable is made coarser by using oscillating interpolation functions derived from the WKB asymptotics. Then the discr...

It is shown that, for a large class of statistical mixtures, the Wigner-Poisson (or Hartree) system can be reduced to an effective Schrödinger-Poisson system, in which the Schrödinger equation contains a new nonlinearity. For the case of a zero-temperature one-dimensional electron gas, this additional nonlinearity is of the form Ψ 4. In the long wavelength limit, the results obtained from the e...

The dynamical equations describing the evolution of a self-gravitating fluid of cold dark matter (CDM) can be written in the form of a Schrödinger equation coupled to a Poisson equation describing Newtonian gravity. It has recently been shown that, in the quasi-linear regime, the Schrödinger equation can be reduced to the exactly solvable free-particle Schrödinger equation. The free-particle Sc...

We apply the proper orthogonal decomposition (POD) to the nonlinear Schrödinger (NLS) equation to derive a reduced order model. The NLS equation is discretized in space by finite differences and is solved in time by structure preserving symplectic midpoint rule. A priori error estimates are derived for the POD reduced dynamical system. Numerical results for one and two dimensional NLS equations...

In this paper, one-dimensional (1D) nonlinear Schrödinger equation iut − uxx +mu+ g(u, ū) ū = 0, with Periodic Boundary Conditions is considered; m / ∈ 1 12Z is a real parameter and the nonlinearity g(u, ū)= ∑ j,l,j+l 4 ajlu j ū , aj l = alj ∈ R, a22 = 0 is a real analytic function in a neighborhood of the origin. The KAM machinery is adapted to fit the above equation so as to construct small-a...