A dispersion-free optical soliton

Akira Hasegawa and Fred Tappert showed in 1973 that the same balance of nonlinearity and dispersion behind Russell’s water wave – now governed by the focusing nonlinear Schrodinger equation instead of KdV,

\[i\partial_t\psi + \tfrac12\partial_x^2\psi + |\psi|^2\psi = 0,\]

– lets light pulses in an optical fiber propagate as solitons, self-correcting against the pulse-spreading dispersion that otherwise limits every long-distance optical line. nls_bright_soliton() constructs the exact envelope solution,

\[\psi(x,t) = A\,\mathrm{sech}\big(A(x-x_0-vt)\big)\, e^{\,i\left[v(x-x_0) + (A^2-v^2)t/2\right]},\]

and nls_evolve() propagates it via split-step Fourier integration (here at rest, \(v=0\), amplitude \(A=1\)), showing the envelope \(|\psi|\) is unchanged after “fiber” propagation where an ordinary pulse would visibly spread; the same run can be watched frame by frame with nls_evolve_frames().

import matplotlib.pyplot as plt
import numpy as np

from physicskit.fields import animate_field_1d, nls_bright_soliton, nls_evolve, nls_evolve_frames, plot_field_1d

A fiber-optic bright soliton (focusing NLS: g > 0)

N, L = 1024, 80.0
x = np.linspace(-L / 2, L / 2, N, endpoint=False)
psi0 = nls_bright_soliton(x, t=0.0, A=1.0)

Propagate it down 2000 dt of “fiber”

psi = nls_evolve(psi0, x, dt=0.001, steps=2000, g=1.0)

Nonlinearity (self-phase modulation) exactly cancels dispersion: the envelope \(|\psi(x,t)|\) is unchanged, unlike an ordinary pulse governed by dispersion alone.

shape_error = np.max(np.abs(np.abs(psi) - np.abs(psi0)))

fig, ax = plot_field_1d(x, np.abs(psi0), label="t = 0")
plot_field_1d(x, np.abs(psi), ax=ax, label="t = 2 (fiber units)")
ax.set_title(f"envelope shape error: {shape_error:.1e}")
fig.tight_layout()

print(f"envelope |psi| shape error after propagation: {shape_error:.2e}")
print(f"peak envelope amplitude: t=0 -> {np.abs(psi0).max():.4f}, after -> {np.abs(psi).max():.4f}")

envelope |psi| shape error after propagation: 1.18e-04
peak envelope amplitude: t=0 -> 1.0000, after -> 0.9999

Animating the dispersion-free envelope

nls_evolve_frames() records the same propagation as a sequence of snapshots, showing the shape-preserving envelope travel down the “fiber” rather than comparing only two instants.

frames, times = nls_evolve_frames(psi0, x, dt=0.001, steps_per_frame=50, n_frames=40, g=1.0)

anim = animate_field_1d(x, frames, times, ylabel="|psi(x, t)|")
plt.show()

To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:

anim.save("nls_optical_soliton.gif", writer="pillow", fps=20)