A More Detailed Look at My Project
This page contains extra information about the program I created, for the benefit of collaborators and other experts in the field. I suggest you see My Project at a Glance for an overview of my program before reading this page.
The Maths of My Project
The non-linear partial differential equation representing a potential that I have been mostly focusing on is
As is evident, this equation is a second order differential equation and is best solved by first being separated into two first order differential equations. These are much simpler to solve.
So, we now have
Then, I would use a numerical method to solve the now simplified equation. This type of method involves going back to the first principles of calculus and replacing differentials such as dy/dx with (y2-y1)/(x2-x1) and so on. So, our first first order differential equation can now be written as
This particular type of numerical method is called the Vernier Leapfrog Method. It is characterised by the way in which first order derivatives are defined. Instead of it being the change in height over the distance dx, it is instead defined as the change in height over 2dx. The second order derivative is then defined from this.
Now the equation is rewritten in this numerical way, solution by computer is possible.
My Program
My program relies on an update method to solve the set of first order differential equations shown above. The values of 'pi' can be found at each point in time over a time loop. Then, phi can be found by updating 'pi' in another loop. From here, phi can be plotted against time to see if an Oscillon emerges.
Future work
From reading literature on the specific topic of Oscillons, I found that the majority of researchers use a similar method for searching for Oscillons. A partial differential equation representing a known potential is solved with various initial conditions, to create a number of different systems. These systems are then studied individually to see if an Oscillon emerges. Although this method works, it is labour intensive and can be a bit 'hit and miss', even if there is a range of initial conditions for which you think a Oscillon may occur. This is because until you program the initial conditions and allow the model to evolve, you do not know if an Oscillon will emerge. Even more frustratingly, you cannot know that an Oscillon will exist to outlive the system until you run the program for a long period of time and record the lifetime of the Oscillon. This is a lengthy process and can often end in disappointment.
I propose a different method of finding Oscillons within potentials. I would like to explore the possibility of modelling an ideal Oscillon (which lasts for an acceptable time period) and then working backwards to see which potential (if any) would be suitable to sustain it. To my knowledge, this approach has not yet been tested and so I don't know if it will work very well, but I'd like to try!
References
- 'Computational Physics' (2001), J.M.Thijssen
- 'd-dimensional oscillating scalar field lumps and the dimensionality of space' (2004), Marcelo Gleiser
- 'Oscillons in Scalar Field Theories: Applications in Higher Dimensions and Inflation' (2006), Marcelo Gleiser
- 'Numerical Investigations of Oscillons in 2 Dimensions' (2006), Mark Hindmarsh and Petja Salmi
- 'An Introduction to Computational Physics' (2006), Tao Pang
- 'Oscillons and Domain Walls' (2007), Mark Hindmarsh and Petja Salmi