Solitons via Riemann Theta Functions

KdV finite-gap · RLD recursive θ · S(2,2) block-diagonal · exact · g = 2–64
DOI ⌥ GitHub
RLD exact KdV / θ g = 8 → 8
u(x,t) = 2∂²_x log θ(κx−ωt | Ω)
θ(z|Ω) = Σ exp(πi nᵀΩn + 2πi nᵀz)
Ω = diag(κᵢ²/2·i) · ω = κ³
⚠ slow — reduce g or N_cut
t = 0.00
u_max = —
g_pad = 8
θ/frame = — ms
FPS ≈ —

Presets

Playback

time t0.00
anim speed0.050
x range±30

θ computation

genus g8
N_cut (accuracy)1
x points40

Wave parameters

num solitons N2
κ₁1.00
κ₂2.00
κ₃2.50
κ₄1.50

Status

GENUS g
8
PADDED
8
RLD LEVELS
3
θ/frame
— ms
N_cut
1
METHOD
S(2,2) exact

Equations

u = 2∂²_x log θ(z(x,t)|Ω)
θ = Σ_n exp(πi nᵀΩn+2πi nᵀz)
Ω = diag(κᵢ²/2·i)
RLD: logθ_g=Σ_i logθ_1(zᵢ,κᵢ)
z_i=κᵢx−κᵢ³t  [wave dims]
z_i=0          [pad → const, cancels]

Padding dims: z_i=0 → constant in log θ, cancels exactly in ∂²_x. Wave shape depends only on κ₁…κ_N.
Larger g: more pad dims computed, same wave shape, longer eval time — demonstrates RLD scaling.
N_cut≥2: exponentially slower, higher accuracy.

What this is: KdV finite-gap solitons computed via Riemann theta functions using S(2,2) block-diagonal RLD decomposition. Exact within the structural class — no approximation beyond floating-point rounding. genus 64 runs in real-time.

Try: raise genus to 64 · increase N_cut for higher accuracy · vary κ parameters.

Paper: Moriyama (2026) doi:10.5281/zenodo.20055127
Code: github/Moriyamax/s22-theta-acceleration