Twin Solitons
y = 2/cosh(x − 2 − t%12) + 2/cosh(x + 6 − t%12)
Open in the app The dials and keys named below are the app's.
What it draws
1/cosh(u) is the hyperbolic secant: one smooth peak of height 1 at u = 0, falling off like a decaying exponential on both flanks. Each term here doubles that to height 2. Each is down to half height at |u| = arccosh(2) ≈ 1.32, so a hump is 2.63 wide across its half-height points.
The first hump is centred where x = 2 + (t mod 12) and the second where x = −6 + (t mod 12). Both slide right at one unit per second and stay 8 apart. The mod 12 wraps the clock, so at Speed 1 the pair is thrown back to its opening positions every 12 seconds, and the ribbon's recent past keeps the seam for a while afterwards.
The sech profile
That shape is not decoration. Take a wave equation in which a nonlinearity steepens a pulse and a dispersion spreads it. When the two balance, the pulse that survives is a secant hyperbolic. The Korteweg–De Vries equation of shallow water has (c/2)·sech², and the bright soliton of the nonlinear Schrödinger equation, the pulse an optical fibre carries, has envelope A·sech(A(x − vt)). Such a pulse holds its shape over enormous distances and survives a collision: two of them pass through each other and come out with their old heights and speeds, shifted only slightly in position.
None of that is solved here. These are two soliton portraits driven along by hand, at the same height and the same speed, so neither can catch the other. Real Korteweg–De Vries solitons are taller when faster and narrower when taller, which is what makes an overtaking collision possible at all.
History
John Scott Russell chased one on horseback along the Union Canal near Edinburgh in 1834 and called it the Wave of Translation. Diederik Korteweg and Gustav de Vries derived its equation in 1895. Norman Zabusky and Martin Kruskal, computing collisions in 1965, found the pulses came out intact and coined the name soliton.
Try
- Drop the wrap and let them run once: 2/cosh(x − 2 − t) + 2/cosh(x + 6 − t), then Rewind to start again.
- Give the follower a speed of its own, 2/cosh(x − 2 − t%12) + 2/cosh(x + 6 − 1.6·(t%12)): it overtakes, and at the crossing the two heights simply add to 4. A true soliton collision does not do that.
- Switch to the shallow-water profile, 2/cosh(x − 2 − t%12)² + 2/cosh(x + 6 − t%12)²: squaring pulls the flanks in, halving the width to 1.76.
- Look from Top: each hump leaves a straight track through the ribbon, and its slant is the speed.