Can a wave keep its exact shape forever — and can two waves collide and pass through each other completely unchanged?
▶ Launch the interactive simulationIntegrate the KdV equation u_t + 6u·u_x + u_xxx = 0 (Zabusky–Kruskal conservative leapfrog, periodic box). Launch single solitons u=(c/2)sech²[…] of several heights and track each peak to recover speed c vs amplitude a; then collide a tall soliton with a short one and watch what survives. Conserved ∫u and ∫u² confirm the integrator is faithful — the soliton itself is established by the speed law and the intact pass-through.
YES to both — a soliton, where nonlinear steepening (6u·u_x) exactly balances dispersion (u_xxx). Its speed is PROPORTIONAL to its amplitude, c = 2a, so a taller hump runs faster (the lab recovers the line c=2a); and when the tall soliton catches the short one they merge into a single lump then re-emerge with their ORIGINAL shapes and speeds — only a small phase shift — because the interaction is integrable, not mere superposition. This shape-preserving, pass-through behaviour is exactly what Zabusky & Kruskal found in 1965 explaining the FPUT recurrence, coining the word 'soliton'. (∫u, ∫u² conserved to ~1e-8 — the integrator is faithful; the soliton claim rests on the measured speed law and pass-through, not conservation alone)