Asgard
Beowulf Day Wintersemester 2000/2001: Project description

Soliton's Dissipation Through Emission Of Radiation

Researchers

Project Description

Introduction

The aim is to propagate the ground state wave function of the following nonlinear Hartree-equation in an external harmonic potential:

is the wave function, is an external potential, is a convolution potential (Coulomb or Yukawa), * denotes convolution and g is a coupling constant. There are so far no analytical solutions. Therefore we use a numerical approach.

Why is the NHE interesting?
Let's consider a system of bosons with weak attractive two-body interactions.
It is proven that in the limit, where the interaction strength vanishes, the dynamics of the system can be described by the NHE (1).

The solutions are solitary waves looking like approximate -functions. Physically, such solutions describe bound compounds ("snow balls") of many bosons. In reference [1] it is shown that, under the influence of a slowly varying external potential, the center of mass of such a compound moves along a trajectory which is a solution of Newton's equations of motion corrected by a very small friction term.

The friction causes that the compound loses energy radiating waves.

Calculation of the Ground State

First the Ground State (minimizer) is calculated in two dimensions. The following eigenvalue-equation is solved:

Here there is no external potential . We consider to be either a Coulomb or a Yukawa potential.

The solution has a rotational invariance, is smooth and has a global maximum (see Fig.1).

Fig 1: Radial part of the calculated minimizer and its 3-D representation.

Propagation in a harmonic potential

Now the oscillation in 2-D of the minimizer in an external harmonic potential is studied.
Thereby the decrease of the amplitude and the energy spectrum are observed.

Tool: C++ Code of W. Aschbacher.

Problem: Absorbing Boundaries
Solution: Negative imaginary potential on some boundary strip:

Parameters to determine: a, b, , number of absorbing layers.

Run simulations on Asgard with different values of the parameters:

Absorption of a sequence of Gaussian wave packets with different initial momenta.
Absorption of the radiation stemming from a minimizer with vanishing initial momentum.
Propagation of the Minimizer in an external harmonic potential.

The following picture shows the results obtained so far.

Fig 2: The amplitude of the oscillation of the minimizer in an external harmonic potential well as a function of time for a 2-D grid with 256 grid points in each direction. The y-axis shows the quotient of the actual amplitude (= distance from the minimum of the potential) and the initial amplitude.

We observe a decrease of the amplitude as a function of the time steps. We expect that if the simulation time is much greater, one should be able to see the minimizer oscillating into the minimum of the potential.

CLICK ON THIS LINK TO SEE A NICE FILM OF THE RESULTS.

Publications

Gianluca Interlandi, Soliton's Dissipation Through Emission Of Radiation, Diploma Thesis, September 2000.

References

[1] Jürg Fröhlich, Tai-Peng Tsai, Horng-Tzer Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation.

Written by: Gianluca Interlandi <gianluca@student.ethz.ch>