Plane Waves, Traveling Waves, and Solitons
Summary
We end the semester with some basic linear wave theory and solitons.
We obtain linear dispersion relation for the plane wave solutions of several PDEs:
- heat equation
- wave equation
- Klein-Gordon equation
- Airy’s equation
and introduce notions of dispersion and dissipation. Inline with this, we cover phase velocity for plane waves.
We then discuss what a soliton is along with a brief history of integrability in context of nonlinear wave equations, from John Scott Russell’s observation of the great wave of translation in 1984 to the FPUT recurrence at Los Alamos in 1950s, followed by Zabusky and Kruskal’s discovery of a soliton in 1964, and the seminal paper of Gardner, Greene, Kruskal, and Miura in 1967.
We end the semester with calculating the famous $1$-soliton solution $$ u(x,t)=\frac{\sigma}{2}\mathrm{sech}\left( \frac{\sqrt{\sigma}}{2}(x-\sigma t - c)\right)^2, $$ $\sigma>0$, $c\in\mathbb{R}$, of the Korteweg-de Vries equation $$ u_t + 6 u u_x + u_{xxx}=0. $$
