Numerical simulations of the Klein-Gordon field with nonlinear interaction

WARNING: the codes found here have been written for Octave 2.1

Here is a collection of the scripts for the numerical simulation of the Klein-Gordon field subject to certain nonlinear interactions. The complex-valued U(1)-invariant Klein-Gordon field could be thought of as a string in three dimensions, drawn to its equilibrium position by a smooth family of harmonic oscillators.
The nonlinear interaction is represented by one or more nonlinear oscillators located at discrete points (or by a smooth family of nonlinear oscillators or by a nonlinear term in the equation). The scripts illustrate the global attraction to the solitary waves: an arbitrary initial data of finite energy approaches solitary waves (in the case of a string in 3D, these are given by rotating solitary profiles).

All the scripts are written for Octave and could be run with, e.g., 


octave one-oscillator.m
The visualization is done with the aid of GNUPlot. You need to have both Octave and GNUPlot installed.

Remarks

The Klein-Gordon field on the real line is approximated by the finite Klein-Gordon field on a finite interval, with the absorbing boundary conditions (chosen as du/dx=-du/dt at the right end and du/dx=du/dt at the left end).

Both Octave and GNUPlot are freely available software packages (Octave is GPL-licensed while GNUPlot is a freeware). Both are readily available for most common operating systems such as Linux.

Related links

Written by Andrew Comech.
The author was partially supported by National Science Foundation under N.S.F. Research Grants DMS-0200880, 0434698, 0621257