The Korteweg-de Vries equation
u_t + uu_x + \epsilon^2u_{xxx} = 0describes one dimensional wave phenomena in the long wave-length limit, for instance waves in shallow water.
Solitons and almost periodic solutions
From nlin/0512066.
2-soliton solution
Solitons are stable wave packets which propagate without changing shape. Their shape is not even altered after collision with another soliton, but there is a change in the phase of the solitons. The video above shows the 2-soliton solution of KdV.
Hyperelliptic solution, genus 2
Almost periodic solutions to the KdV equation can be obtained in terms of multi-dimensional theta functions on hyperelliptic surfaces. The video above shows a periodic solution on a surface of genus 2.
Hyperelliptic solution, genus 6
Almost periodic solutions to the KdV equation can be obtained in terms of multi-dimensional theta functions on hyperelliptic surfaces. The video above shows an almost periodic solution on a surface of genus 6.
See nlin/0512066 for more information and references.
Small dispersion limit
KdV solution for hump-like initial data in the small dispersion limit, from 1202.0962.
Small-dispersion KdV
In the small dispersion limit ( ℇ ≪ 1 ), KdV solutions develop a zone of rapid modulated oscillations near shocks of the dispersionless equation, the Hopf equation. The video shows the solution for ℇ = 0.01 for the initial data
-sech^2(x).
See Small dispersion KdV (below) for the video of the detailed oscillatory zone for this case.
Small-dispersion KdV (Zoom)
The video shows the oscillatory zone of Small dispersion KdV (above) in detail.
KdV + asymptotic solution (Whitham)
The small dispersion limit can be asymptotically described in the following way: before a shock of the dispersionless equation, the KdV solution is approximately given by the solution of the Hopf equation. For values of t beyond breakup of the Hopf solution, Gurevitch and Pitaevski suggested the following picture which was rigorously proven by Lax-Levermore and Venakides. An oscillatory zone is identified where the KdV solution is approximately given as the exact elliptic solution of KdV where the branch points of the elliptic surface depend via the Whitham equations on the physical coordinates. Outside this zone, the KdV solution is approximated via the corresponding Hopf solution. The video shows the KdV solution Small dispersion KdV (blue), as above, and the corresponding asymptotic solution (magenta). The difference of these solutions can be seen in Difference KdV-asymptotic solution (Whitham) (below).
Difference KdV-asymptotic solution (Whitham)
The video shows the difference between the KdV solution and the corresponding asymptotic solution (as above). The oscillatory zone is shown in red. It can be seen that the approximation is worst near breakup and at the boundary of the oscillatory zone. There are always KdV oscillations outside the oscillatory zone.
See 1202.0962 for further details and references.