# Jaron’s Blog

## Kinky Physics: Animating Sine-Gordon Solitons

Recently in one of my classes, we studied the Sine-Gordon theory, which is characterized by the following Lagrangian density in 2 spacetime dimensions:
\[\mathcal L=\frac12\partial^\mu\partial_\mu\phi-V(\phi)\]
Here \(\phi\) is a scalar field and \(V(\phi)=\alpha(1-\cos\beta\phi)\). This equation can be thought of as the continuum extension of a discrete system consisting of pendula connected at their bases by torsion springs. This theory is interesting because it possesses many of the heuristic properties that we would expect of a system comprising particles and antiparticles. The particle-like solutions are the soliton, or ‘kink’, solutions to the Sine-Gordon equations of motion.

Our instructor couldn’t find a satisfactory video of Sine-Gordon solitons in action, so I whipped up some *Mathematica* code to visualize the solutions. Here is an example of a kink propagating:
There are also exact analytical forms for arbitrarily many interacting kinks and antikinks. First, we have two kinks colliding and bouncing off each other:
Next, we have a kink and an antikink collide and propagate through each other:
Finally, there exists a bound state between a kink and an antikink. This ‘breather’ state looks like:
Of course, we don’t need analytic solutions to visualize something in *Mathematica*! Here, for instance, is the result of a numeric simulation of a few kinks interacting in fixed boundary conditions:
If you’d like to play with the *Mathematica* code, you can download the notebook here. Reading through the code and looking at the examples should be sufficient to understanding how to use it.