17. Dispersive waves and solitons

Corrections

At 1:05, "looks a little bit like the wave equation" should be the heat equation.

At 4:42, "for a fixed x" should be t.

At 10:35, \( e^{3t} \) (etc.) should be \( e^{3x} \) (etc.).

At 14:45, "it's virtually impossible" isn't quite true. For example, the convolution of the Airy function with a Gaussian can be computed; see this question on MathOverflow. Using this, the solution to \( u_t = u_{xxx} \) with initial value \( u(x,0)=\exp(-x^2) \) turns out to be $$ u(x,t) = \frac{\sqrt{\pi}}{(3t)^{1/3}} \, \exp \left( \frac{x}{12t} + \frac{1}{864 t^2} \right) \, \operatorname{Ai} \left( \frac{x + \frac{1}{48t}}{(3t)^{1/3}} \right) \qquad (\text{for } t > 0 ) . $$ The pictures below show the graph of \( u(x,t) \) as a function of \( x \), for \( t \in \bigl\{ 0, \frac{1}{10}, \frac{1}{2}, 1, 10 \bigr\} \):

At 1:12:58, I wrote \( c_1 = 1 \), but that should of course be \( c_1 = 2 \) ("speed 2", as I said).

And at the very end, "this was the last video of the course" is (obviously) not true anymore!

External videos

• Ripples in a pond. Nice footage, with explanations, of the dispersive waves seen when you drop a small object into a pond of water. This includes also capillary waves, which I didn't mention in my video; these are waves with very short wavelength (less than a few cm), where surface tension (not gravity) is the dominant effect.
• If you want to learn more about the method of stationary phase and such things, check out this marvellous lecture series with Steven Strogatz on YouTube: Asymptotics and perturbation methods. Stationary phase is covered in Lecture 6.
• KdV Solitons. Animations of KdV soliton solutions by Stephen Anco and Mark R. Willoughby, Brock University, Ontario, Canada. (Part of a series of pages about Solitons & Nonlinear Wave Equations, which you might find interesting too.)

Sidansvarig: Hans Lundmark
Senast uppdaterad: 2026-05-04