By Padraic Gidney, The University of Sydney
For my summer project, I investigated the behaviour of a particular kind of solution – what’s called a periodic wave train solution – to a particular family of partial differential equations (or PDEs) – among whose members are such famous equations as the Fisher-KPP equation and the non-linear Klein Gordon equation.
A periodic wave train is just a repeating wave that moves at a constant speed in a particular direction. One simple example would be a sine wave that drifts continuously either to the right or to the left.
The question I was considering in my project was: what happens to such wavetrains if we add a little bit of noise to them, and then evolve them under the family of PDEs we’re interested in? (if we don’t add noise, then the wave trains – which are already solutions to our PDEs – just remain as they are when we evolve in time).
Now this turns out to be a very complicated question. For a start, there are an infinite number of wave train solutions to the PDEs we’re interested in. What’s more, there are an infinite number of ways that we might add noise to each wave train, and an infinite number of behaviours which the new, ‘noisy’ wave trains might display when we turn on time.
We shouldn’t be too daunted by all these infinities. The first of them turns out to not really be a problem at all, since – although there are, indeed, an infinite number of wave train solutions to consider – we find that we can still get a neat ordering of them if we just arrange them by their energy and the speed at which they travel (e.g. we would label the wave train with an energy of one and speed of two ‘W12’).
The third infinity – the fact that there are an infinite number of ways for a wave train to evolve – is the real kicker, but we can get around this as well by using a nifty little method called spectral analysis. The idea with spectral analysis is that every function can be broken down into a series of component parts, or “modes” – smaller, much more simple functions, which can be thought of as the sort of building blocks, or atoms, of the functional world. Once we know what modes make up a particular function, e.g. one of our noisy wave train solutions, we know exactly how it will evolve under our PDE, so the problem of figuring out what behaviours our noisy wave trains might display really boils down to figuring out what modes they’re made up of.
But this brings us to our second infinity: the fact that, for each of our original (noiseless) wave trains, there are an infinite number of different ways in which we might add noise. How are we to grapple with this? Do we have to consider each different case separately, or can we say something more general about the way our initial wave train will evolve when we add noise to it?
This is where we get a real stroke of luck. It turns out that, not only can we break individual functions down into their component parts, but we can actually break whole equations down into their component parts as well (technically we’re decomposing operators, not equations, but the distinction is irrelevant for our purposes). When we decompose an equation in this way, what we get out are not the modes of any one particular solution, but rather the full set of modes that any solution could potentially be constructed from – think of it as a buffet of modes, with each solution having to pick from the choices available. We call this buffet of modes the “spectrum” of our equation, and once we know what it looks like, we essentially know all of the behaviours which solutions to that equation can display.
The reason all this is so useful is that, for each of our wave train solutions, there is an associated equation, E(N), which describes how that wave train would evolve under our PDE if we added a particular noise function, N, to it. Finding the spectrum of E(N) in the way described above tells us all of the different modes – all of the different behaviours, in other words – that a noisy version of our initial wave train could take on.
But then we’re done! We have a way to neatly order all of the different wave train solutions to our family of PDEs (i.e. by their energy and speed), and for each wave train solution we can discover – by looking at the associated spectrum – the different ways it might evolve if we added noise to it.
When we actually perform these calculations – i.e. actually find the spectra of our different wave train solutions – it turns out that nearly all of them are what we call “spectrally unstable”. This means that, when we evolve them under time, they lose their original form, dissolving and dissipating into something entirely different.
This might sound a bit disappointing to the untrained ear, but it’s actually the most exciting result we could hope for. Stable waves are boring: they stay the same, or nearly the same, as we evolve in time; the really interesting, non-linear behaviour – the bubbling, broiling, and blowing up – only emerges from unstable waves, which – by the looks of things – are exactly what we’ve got.
Padraic Gidney was one of the recipients of a 2015/16 AMSI Vacation Research Scholarship.