ROGUE

What Is Rogue?

A rogue wave is a surface wave that exceeds twice the surrounding significant wave height — rare, sudden, and often fatal to ships. The Peregrine soliton is the exact analytic solution to the nonlinear Schrödinger equation (NLS) that describes this event: found by Howell Peregrine in 1983, it is a rational (non-oscillatory) function that rises from a uniform background amplitude of 1 to exactly 3× at the point (x=0, t=0), then returns to the background — appearing from nowhere, disappearing without a trace.Rogue renders this solution in three interactive views, using the exact formula with no numerical solver required.

Three views expose different aspects of the same physics. WAVE animates the actual ocean surface including the carrier wave oscillations — you see the individual water crests, and among them a single impossibly large wave appears and vanishes. ENVELOPE strips out the carrier to show the amplitude profile directly, including the two exact zeros that flank the peak. SPACETIME renders the entire x–t amplitude history as a colormap, revealing the iconic funnel-shaped signature that fluid dynamics researchers call the “Peregrine eye”.

The Formula

The Peregrine soliton is a solution to the normalized nonlinear Schrödinger equation with background amplitude one. Every number in the formula is exact:

Equation:   i·ψ_t + (1/2)·ψ_xx + |ψ|²·ψ = 0

Background: ψ₀(x,t) = exp(it)          |ψ₀| = 1 everywhere

Peregrine:  ψ_P(x,t) = [1 − 4(1+2it)/(1+4x²+4t²)] · exp(it)

In Cartesian components (D = 1 + 4x² + 4t²):
  Re[ψ_P] = [(4x²+4t²−3)·cos t + 8t·sin t] / D
  Im[ψ_P] = [(4x²+4t²−3)·sin t − 8t·cos t] / D

Amplitude:  |ψ_P(0, 0)| = 3    (the ×3 rogue event at origin)
Zeros:      |ψ_P(±√3/2, 0)| = 0  (amplitude completely vanishes)
Background: |ψ_P| → 1 as x²+t² → ∞

The surface elevation including the carrier wave (wavenumber k₀) is:

η(x,t) = Re[ψ_P(x,t)] · cos(k₀·x) − Im[ψ_P(x,t)] · sin(k₀·x)

The Physics: Modulation Instability

A uniform ocean surface is unstable. Tiny perturbations at the right wavelength — too long to be dispersed quickly, short enough to feel nonlinear self-focusing — grow exponentially. This is the Benjamin-Feir instability(1967), also called modulation instability in the NLS context.

The Peregrine soliton is the limiting case of this instability: a perturbation whose spatial period has been stretched to infinity and whose temporal period has been compressed to a single event. It represents the extreme limit — a sideband with zero wavenumber that grows without spatial repetition, produces its maximum amplitude exactly once, and then decays. Benjamin and Feir predicted rogue wave formation in 1967; Peregrine found the analytic solution in 1983; the soliton was first observed in an optical fiber by Kibler et al. in 2010 and in water waves by Chabchoub et al. in 2011.

MI growth rate (wavenumber k):    γ(k) = k·√(1 − k²/4)
  γ = 0   for k = 0 and k = 2  (stability boundaries)
  γ_max at k = √2  →  max growth rate = 1

The Peregrine limit:  k → 0  (infinite spatial period)
  — the perturbation extends across the entire ocean
  — grows as a rational (not exponential) function of t
  — peak amplitude = 1 + 2 = 3 (background + 2 from gain)
  — two flanking zeros mark energy stolen from the sea

The Two Zeros

The most striking feature of the Peregrine soliton is not its maximum but its zeros: at t = 0 and x = ±√3/2 ≈ ±0.866, the wave amplitude is exactly zero. The sea is perfectly flat at those two points while a wall of water three times the normal height stands at x = 0 between them.

These zeros are not coincidental. The complex field ψ_P must change sign (from negative to positive) as it crosses the peak at x = 0 when t = 0, and a sign change in a complex field requires it to pass through zero. Energy has been drained from the two flanking regions to feed the peak — the holes in the sea are the source of the rogue wave. In the ENVELOPE view, the two zeros are visible as sharp dips to zero that flank the bright central spike.

The Spacetime Eye

The SPACETIME view renders the full x–t plane coloured by |ψ_P(x,t)|. The pattern has a distinctive funnel or eye shape: a bright gold spot at the origin (the ×3 peak), two dark corridors where amplitude dips below 1 (the zeros and their neighborhood), and a uniform dark background everywhere else. The boundary where |ψ_P| = 1 traces an asymptotic hyperbolic curve as x²+t² → ∞.

This pattern appears in scientific literature on optical fiber rogue waves, water tank experiments, and plasma physics. The moving amber cursor shows the current cross-section being rendered in the WAVE and ENVELOPE views.

Amplitude contour |ψ_P| = 1 satisfies:
  64t² = 32(x²+t²−1/4)  →  t² = (x²+t²)/2 − 1/8

The two zeros at (±√3/2, 0) are phase singularities:
  arg(ψ_P) jumps by π across each zero  (Dirac string in 2D).

Real-World Rogue Waves

True oceanic rogue waves — observed by satellite altimeters, pressure sensors, and eyewitness accounts — follow statistics inconsistent with linear Gaussian seas. The Draupner wave (North Sea, 1995) measured 25.6 m with a surrounding significant wave height of ~12 m, an amplitude ratio of ~2.1. The Andrea wave (North Sea, 2007) reached a height of 21.1 m. Ship logs from the 1970s–1990s described waves that “appeared from nowhere” — matches to the Peregrine phenomenology of sudden appearance and disappearance.

The NLS is a universal envelope equation that applies wherever carrier waves self-interact: deep-water gravity waves, optical fibers, plasma waves, Bose-Einstein condensates, and even Langmuir waves in ionospheric plasma. Rogue events have been observed and confirmed in all of these systems.

All five systems are governed by the same NLS envelope equation — same mathematics, same Peregrine soliton, different physical realizations.

Why No Numerical Solver?

Unlike most wave simulations — which discretize a PDE on a grid, apply finite-difference or spectral operators, and accumulate floating-point error with each time step — Rogue evaluates the Peregrine soliton formula directly at each pixel and frame. The exact analytic solution means the output is always mathematically correct: no numerical diffusion, no stability constraint on the time step, and no accumulation of round-off error. A web browser computing an exact formula per-pixel runs at 60 fps with zero libraries and zero server requests. This is the entire computational approach.

Implementation

Formula:   exact analytic Peregrine solution — no numerical PDE
Canvas:    responsive, up to 920×340 px (HiDPI: scaled by devicePixelRatio)
Modes:
  WAVE      — surface displacement η = Re[ψ_P]·cos(k₀x) − Im[ψ_P]·sin(k₀x)
              carrier wavenumber k₀ = 2.0 (≈6 cycles across x∈[−10,10])
              amplitude scale = H/7 px per unit; crest color encodes local amp
  ENVELOPE  — |ψ_P(x,t)| profile with reference lines at 1.0 and 3.0
              zeros at x=±√3/2 annotated when t ≈ 0
  SPACETIME — full x∈[−6,6] × t∈[−4,4] pixel buffer (W×H), precomputed once
              custom amber/gold colormap; moving time cursor at current t
Animation: tNorm advances at speed×0.06 units/ms, range [−5, +5], loops
HUD:       real-time t and max amplitude; ROGUE EVENT badge at amp > 2.5×
Zero dependencies, zero backend. Pure JS + Canvas 2D.