KdV Two-Soliton Collision | mysimulator.uk

Soliton Parameters

Wavenumber k₁ (soliton 1) 1.30

Wavenumber k₂ (soliton 2) 0.80

Initial separation 18

Playback

Time speed 1.0×

Live Data

Speed c₁

1.69

Speed c₂

0.64

Time t

0.0

Peak amplitude

0.845

Peaks visible

How it works

The Korteweg–de Vries (KdV) equation models weakly nonlinear, weakly dispersive shallow-water waves — one of the classic integrable PDEs from the article's section on integrable systems:

u_t + 6 u u_x + u_xxx = 0

Its exact two-soliton solution comes from Hirota's bilinear method. Writing τ-function f and u = 2 ∂²ₓ ln f:

θᵢ = kᵢx − kᵢ³t + δᵢ,   Eᵢ = exp(θᵢ)
f = 1 + E₁ + E₂ + A₁₂·E₁E₂,   A₁₂ = [(k₁−k₂)/(k₁+k₂)]²
u(x,t) = 2(f·f_xx − f_x²) / f²

This simulator evaluates that closed-form expression directly at every point of the bar field each frame — no finite-difference stepping, so it never goes numerically unstable. In isolation each soliton is a sech² bump:

u ≈ (kᵢ²/2) sech²(θᵢ/2),  speed cᵢ = kᵢ²,  amplitude = kᵢ²/2

k₁, k₂ — set each soliton's wavenumber; speed and height both scale with k².
Initial separation — how far apart the two solitons start; the taller one (larger k) always starts on the left and, if faster, catches the other.
Time speed — multiplies the simulation clock; Play/Pause freezes it, Reset returns to t = 0.

The remarkable part is what the formula proves: after the faster soliton overtakes and passes through the slower one, both re-emerge with their original shape and speed exactly restored — only their positions are shifted (a "phase shift"), unlike a linear wave collision which would leave interference debris. This elastic, particle-like behavior is why solitons matter across nonlinear optics, plasma physics and fluid dynamics.

Wave amplitude u(x,t) Still-water baseline

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.