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Soliton Canal

A shallow canal whose water surface obeys the Korteweg-de Vries equation, solved live by a real Fourier pseudo-spectral integrator. A soliton is a single smooth hump that travels at constant speed without spreading, because nonlinear steepening exactly cancels dispersive spreading. Taller solitons move faster (speed = twice amplitude), so a tall one launched behind a short one catches up, passes through it, and both emerge with their original shapes and only a shift in position. Launch your own by clicking the water; or pick the contrast preset where an ordinary lump, not a soliton, just fans out into ripples.

Figure 1. Height field of the KdV equation ut+6uux+uxxx=0u_t + 6 u u_x + u_{xxx} = 0 lofted into a 3D reflective canal. Method: Fourier pseudo-spectral derivatives with integrating-factor RK4; WebGL2 Fresnel water.

WHAT TO TRY

  • Watch the hump travel without changing shape: in the KdV equation nonlinear steepening exactly balances dispersive spreading, so the soliton holds together, as Russell first saw on a canal in 1834.
  • Launch a taller soliton: it is narrower and travels faster, the height-speed-width relation unique to KdV solitary waves.
  • Send two solitons of different speeds: the faster one overtakes the slower, they pass through each other, and both emerge unchanged except for a shift, the defining collision of solitons.