D06 — Solitons and KdV Inverse Scattering
| Field | Value |
|---|---|
| Domain | Dynamical Systems |
| System | Korteweg-de Vries equation on ℝ |
| Group | ℝ (translation symmetry) |
| H^k tier | H⁰ |
| ISA | Origami (β → ∞) |
| Status | Validated |
| Opcodes | ORBIT · SPLIT · SPLAT · LABEL |
| Paper | Paper 512 |
Physical system
The KdV equation ∂_t u + 6u ∂_x u + ∂_x³ u = 0 governs shallow water waves, ion-acoustic plasma waves, and lattice dynamics (Toda lattice continuum limit). Its key feature: localised initial conditions decompose into a finite number of solitons — stable, particle-like waves that travel at constant velocity and collide elastically, emerging from collisions unchanged in shape and speed. The soliton velocities are the eigenvalues of the Lax operator L = −∂_x² + u.
Target category
Symp_∞ — the infinite-dimensional symplectic manifold of Schwartz-class initial data for KdV, with action-angle variables given by the inverse scattering transform (IST). Objects at each soliton sector: the n-soliton manifold M_n ≅ ℝⁿ × 𝕋ⁿ (velocities × phases).
Interpretation functor
F: C → Symp_∞ defined by:
| Opcode | F(opcode) |
|---|---|
| ORBIT | Soliton propagation: u(x,t) = u(x − ct); a single soliton is a tropical fixed point of the KdV flow |
| SPLIT | Soliton emergence: a smooth initial hump decomposes into n separate solitons (eigenvalue decomposition of L) |
| SPLAT | Soliton collision and recombination: two solitons pass through each other and re-emerge — SPLIT then SPLAT |
| LABEL | Lax eigenvalue κₙ: encodes soliton velocity cₙ = 4κₙ² and amplitude 2κₙ² |
ISA programme
SCATTER: LABEL[κₙ | Lu = −κₙ²u, n=1…N] -- solve Lax eigenvalue problem (direct scattering)
EVOLVE: ORBIT[κₙ(t) = κₙ(0)] -- eigenvalues are constants of motion
PHASE: LABEL[δₙ(t) = 8κₙ³t + δₙ(0)] -- phase evolution (trivial ORBIT)
INVERT: SPLIT + SPLAT -- reconstruct u(x,t) via Marchenko equation
OUTPUT: ORBIT[u(x,t) = −2∂_x² log det(I+K)] -- N-soliton solution
Computable output
- Soliton velocities cₙ = 4κₙ²: extracted exactly from initial data via the Lax spectrum. N solitons emerge from any initial condition with N bound states.
-
N-soliton solution: exact closed-form determinantal formula. For N=2:
u(x,t) = −2∂_x² log[1 + e^{2η₁} + e^{2η₂} + A₁₂²e^{2η₁+2η₂}]
where ηₙ = κₙ(x − 4κₙ²t) − δₙ and A₁₂ = (κ₁−κ₂)/(κ₁+κ₂).
- Phase shift after collision: Δxₙ = (1/κₙ) log|A₁₂| — the soliton is displaced but not destroyed. This is the SPLAT output: two objects enter, two objects exit, total invariants preserved.
- Tropical connection: a soliton is a tropical curve — the graph of a piecewise-linear function in the (x, t) plane. The ORBIT fixed-point condition (∂_t u = 0 in the moving frame) is exactly the tropical polynomial equation max(0, κx) = c for the soliton profile. KdV solitons are the β→∞ limit of dispersive wave packets.
Validation
- N-soliton formula: verified by Gardner, Greene, Kruskal & Miura (1967) — the founding paper of the inverse scattering transform. Exact for all N.
- Elastic collision: observed in shallow water wave experiments (Zabusky & Kruskal 1965 numerical experiment that discovered solitons).
- Toda lattice: N-particle Toda lattice has exact N-soliton solution via the same Lax pair; validated numerically for N up to 10³.
- Tropical limit: the piecewise-linear (tropical) soliton profile is the β→∞ limit of the exact sech² profile — confirmed analytically.