The Korteweg-de Vries equation

pi = 3.141592653589793
N = 256
L = 40.0
DX = L / N
X = linspace(-0.5 * L, 0.5 * L - DX, N)
K = zeros(N)
for j to N {
    K[j] = 2.0 * pi * j / L
    if j >= N / 2.0 { K[j] = 2.0 * pi * (j - N) / L }
}
K[N / 2.0] = 0.0
ZERO = zeros(N)
IK = complex(ZERO, K)

C = 2.0
X0 = -6.0

#| The soliton of speed C at time t.
def soliton(t) {
    s = 1.0 / cosh(0.5 * sqrt(C) * (X - C * t - X0))
    return 0.5 * C * s * s
}

#| The nonlinear rate, -3 d/dx (u^2), with the derivative taken in Fourier space.
def nonlinear_rate(u) {
    d = complex(u * u, ZERO)
    fft(d)
    d = d * IK
    ifft(d)
    return -3.0 * re(d)
}

#| The dispersive step, exact: every mode turns by exp(i k^3 dt).
def disperse(u, dt) {
    v = complex(u, ZERO)
    fft(v)
    theta = K * K * K * dt
    v = v * complex(cos(theta), sin(theta))
    ifft(v)
    return re(v)
}

#| March the soliton to t_end in `steps` Strang steps.
def march(t_end, steps) {
    dt = t_end / steps
    u = soliton(0.0)
    for s to steps {
        u = disperse(u, 0.5 * dt)
        k1 = nonlinear_rate(u)
        k2 = nonlinear_rate(u + 0.5 * dt * k1)
        k3 = nonlinear_rate(u + 0.5 * dt * k2)
        k4 = nonlinear_rate(u + dt * k3)
        u = u + (dt / 6.0) * (k1 + 2.0 * k2 + 2.0 * k3 + k4)
        u = disperse(u, 0.5 * dt)
    }
    return u
}

#| Where the hump is: its center of mass. A sum over a periodic grid is spectrally accurate
#| for a pulse that has decayed before the edges, so this locates it far below the spacing
#| (a parabola through the top three points is off by 2e-4 here, and no time step fixes that).
def center(u) {
    return sum(X * u) / sum(u)
}

T_END = 2.0
u = march(T_END, 1600.0)
d = u - soliton(T_END)
shape_error = sqrt(sum(d * d) * DX)
mass_drift = abs(sum(u) - sum(soliton(0.0))) * DX
speed = (center(u) - X0) / T_END
d1 = march(T_END, 200.0) - soliton(T_END)
d2 = march(T_END, 400.0) - soliton(T_END)
order = log(sqrt(sum(d1 * d1)) / sqrt(sum(d2 * d2))) / log(2.0)
show shape_error
show mass_drift
show speed
show order

figure("A soliton of speed 2: it moves, and nothing else happens", "x", "u")
plot(X, soliton(0.0), "t = 0")
plot(X, u, "t = 2, computed")
plot(X, soliton(T_END), "t = 2, exact")
style("t = 2, exact", "dash")

test "the soliton keeps its shape" {
    assert_lt(shape_error, 1.0e-5)
}
test "it moves at c, twice its height" {
    assert_near(speed, C, 1.0e-4)
}
test "mass is conserved to rounding: a spectral derivative has exactly zero mean" {
    assert_lt(mass_drift, 1.0e-12)
}
test "the splitting is second order" {
    assert_near(order, 2.0, 0.1)
}


# Two solitons: the taller one starts behind, catches the shorter, and passes through it.
# Hirota's exact two-soliton solution is u = 2 (log tau)_xx with
# tau = 1 + e^eta1 + e^eta2 + A e^(eta1 + eta2), eta_i = k_i (x - x_i) - k_i^3 t and
# A = ((k1 - k2) / (k1 + k2))^2; soliton i has speed k_i^2 and height k_i^2 / 2.
# After the collision each is where it would be, shifted: the taller ahead, the shorter behind.
K1 = 1.6
K2 = 1.0
X1 = -13.0
X2 = -5.0
T2_END = 8.0
T2_STEPS = 1600
# a row of the waterfall every this many steps, every other grid point
T2_EVERY = 40
X_STRIDE = 2

#| Hirota's exact two-soliton solution on the grid X at time t.
#@ 5cc5e18a
def two_solitons(t) {
    A = (K1 - K2) * (K1 - K2) / ((K1 + K2) * (K1 + K2))
    e1 = exp(K1 * (X - X1) - K1 * K1 * K1 * t)
    e2 = exp(K2 * (X - X2) - K2 * K2 * K2 * t)
    e12 = A * e1 * e2
    tau = 1.0 + e1 + e2 + e12
    d1 = K1 * e1 + K2 * e2 + (K1 + K2) * e12
    d2 = K1 * K1 * e1 + K2 * K2 * e2 + (K1 + K2) * (K1 + K2) * e12
    return 2.0 * (d2 * tau - d1 * d1) / (tau * tau)
}

n_rows = T2_STEPS / T2_EVERY + 1
n_cols = N / X_STRIDE
waterfall = zeros(n_rows, n_cols)
w = two_solitons(0.0)
dt2 = T2_END / T2_STEPS
for s to T2_STEPS + 1 {
    if mod(s, T2_EVERY) == 0 {
        for c to n_cols {
            waterfall[s / T2_EVERY, c] = round(1.0e4 * w[c * X_STRIDE]) / 1.0e4
        }
    }
    if s < T2_STEPS {
        w = disperse(w, 0.5 * dt2)
        q1 = nonlinear_rate(w)
        q2 = nonlinear_rate(w + 0.5 * dt2 * q1)
        q3 = nonlinear_rate(w + 0.5 * dt2 * q2)
        q4 = nonlinear_rate(w + dt2 * q3)
        w = w + (dt2 / 6.0) * (q1 + 2.0 * q2 + 2.0 * q3 + q4)
        w = disperse(w, 0.5 * dt2)
    }
}
dw = w - two_solitons(T2_END)
collision_error = max(abs(dw))
show collision_error

figure("Two solitons collide and pass through each other", "x", "t", -0.5 * L, 0.5 * L - DX, 0.0, T2_END)
origin("lower")
plot2d(waterfall, "u")

test "after the collision the two solitons are Hirota's exact solution, phase shifts included" {
    # the splitting's second-order error at dt = 0.005 is 3.2e-4; at dt = 0.0025 it is 7.8e-5
    assert_lt(collision_error, 5.0e-4)
}

# SPDX-License-Identifier: MIT OR Apache-2.0
# SPDX-FileCopyrightText: 2026 The Viv authors