Physics
Solitons (KdV)
Nonlinear waves that keep their shape; taller ones run faster and pass through.
What happens
Waves of the Korteweg-de Vries equation that refuse to spread: nonlinearity steepens them exactly as fast as dispersion smears them, so each soliton keeps its shape forever. Taller ones run faster; stage the collision and watch the tall one pass through the short one and emerge intact, only shifted.
Equations used in the module
Equations u_t + 6u u_x + u_xxx = 0 u = (c/2) sech²(√c(x−ct)/2); c = 2A κ_i = √(A_i/2); κ₁ > κ₂ Δx₁ = ln((κ₁+κ₂)/(κ₁−κ₂))/κ₁ Δx₂ = −ln((κ₁+κ₂)/(κ₁−κ₂))/κ₂
