A Variational Characterisation of KdV Multisolitons
Speaker: Thierry LaurensDate of Talk: October 7, 2022
Upstream link: UCLA Participating Analysis Seminar
This talk was concerned with a question that comes out of physics: the main question of interest is, “What do waves travelling down a canal look like?” The answer this question admits doesn’t really have a good physics-based answer, as far as I can tell: there’s no intuition or law of physics or anything that addresses it. But as a mathematician, this isn’t a foreign question, and many people have found complete classifications of many things (e.g. finite simple groups).
Thierry was nice enough to have a conversation about his talk and career with me. He started out as a mathematical physicist, but he found that a lot of the questions that interested him the most were mathematical in nature, just like the one in his talk! This led him to gravitate towards math rather than physics.
1. A Question Posed by a Physicist
A long time ago, people were fascinated by waves in canals or something (they got bored easily during the triassic era). Somewhere along the line, someone used physics to develop the following model of a wave travelling down a one-dimensional ultrathin canal of water:
\[\frac{\mathrm du}{\mathrm dt}=\frac{\partial^3u}{\partial x^3}+6u\frac{\partial u}{\partial x}.\]
Here, \(u\) is a function of two real variables - \(x\) and \(t\) - and it represents the height of the wave. This is called the Korteweg-de Vries equation, or KdV equation for short. I am not sure why the differential operators are different. The question posed, and thus the question discussed in the talk, was what do the solutions to this differential equation look like?
2. An Answer Given by a Mathematician
A while ago, people discovered a class of solutions called solitons, which resembles a single “pulse” of water travelling straight down the canal. Going by physical intuition, one would expect that you could get solutions that are superpositions of multiple pulses: maybe some kid jumps in the canal and sends waves going in either direction. Unfortunately, the KdV equations are nonlinear in \(u\), so you can’t just add a couple solitons and get another solution.
Fortunately, there are solutions that very closely resemble a sum of \(N\) distinct solitons, and these are called \(N\)-solitons. What a surprise! These for a subset of the solutions, but not all solutions look like \(N\)-solitons: one can come up with funny-looking algebro-geometric solutions to the KdV equations.
So some solutions look like \(N\)-solitons, and some don’t: how can one characterise the difference? This part was a little hazy to me, but it turns out to be related to certain “conserved quantities”, which are in some sense time-invariant quantities associated to each solution. My own (likely incorrect) interpretation of this idea is the “energy” of the solution, and \(N\)-solitons are local minimisers of these quantities. In loose terms, \(N\)-solitons represent “physically viable” solutions that minimise their energy.
3. “Polished Talks”
This isn’t a mathematical or academic takeaway so much as it as a takeaway about giving talks. I’ve read a lot of advice online about giving talks. Thierry’s talk was one of the best I’ve heard or attended, and it’s because it was “polished”.
Most importantly, the level of detail was perfect: things were phrased precisely enough so that experts could piece things together formally, but things were also phrased loosely enough so plebians like me could comfortably follow along. Moreover, Thierry was prepared with lots of visual or graphical examples. This may not be applicable in every talk, but since this particular topic is so intertwined with physics and intuition, it would be wrong to omit numerical examples.
Preparing for a talk myself let me understand how incredibly difficult it is to actually accomplish this. Papers are often written with such precision and detail that they easily become opaque to people like me, and it takes significant effort to unpack the arguments and organise them in a way that’s suitable for a listening audience.
Anyways, the last takeaway can be summed up as: “Oh my lord this guy’s a good speaker; I hope I can get on that level”.