A topologically protected Sine-Gordon kink

Yakov Frenkel and Tatiana Kontorova modeled a crystal dislocation as a chain of atoms in a periodic substrate potential – a chain of coupled pendula; in the continuum limit this becomes the Sine-Gordon equation

\[\partial_t^2 u - \partial_x^2 u + \sin u = 0.\]

Its kink solution,

\[u(x,t) = 4\arctan\!\left(e^{\,\gamma(x-x_0-vt)}\right), \qquad \gamma = \frac{1}{\sqrt{1-v^2}},\]

a single \(2\pi\) twist propagating at speed \(v<1\) along the chain, is a topological soliton: no local, continuous deformation can untwist it. sine_gordon_kink() builds this exact traveling-wave solution and sine_gordon_evolve() propagates it with independent leapfrog finite differences, confirming the twist survives intact; the same run can be watched frame by frame with sine_gordon_evolve_frames().

import matplotlib.pyplot as plt
import numpy as np

from physicskit.fields import animate_field_1d, plot_field_1d, sine_gordon_evolve, sine_gordon_evolve_frames, sine_gordon_kink

A kink launched at v = 0.5 (natural units, linear wave speed = 1)

N, L, v, x0 = 4000, 200.0, 0.5, -50.0
x = np.linspace(-L / 2, L / 2, N)
dx = x[1] - x[0]
dt = 0.4 * dx
u_prev = sine_gordon_kink(x, -dt, v, x0)
u0 = sine_gordon_kink(x, 0.0, v, x0)

Propagate with leapfrog finite differences on the discrete chain

steps = 1000
u, _ = sine_gordon_evolve(u0, u_prev, x, dt, steps)

The \(2\pi\) twist survives intact, at the position predicted by the exact kink solution – it cannot be untwisted by the local dynamics, only shifted.

expected = sine_gordon_kink(x, steps * dt, v, x0)
max_dev = np.max(np.abs(u - expected))

fig, ax = plot_field_1d(x, u0, label="t = 0")
plot_field_1d(x, u, ax=ax, label=f"t = {steps * dt:.0f}")
ax.set_title(f"max deviation from exact kink: {max_dev:.2e}")
fig.tight_layout()

print(f"field before the kink (x -> -L/2): {u[0]:.4f} (expect 0)")
print(f"field after the kink  (x -> +L/2): {u[-1]:.4f} (expect 2*pi = {2 * np.pi:.4f})")
print(f"max deviation from exact traveling kink: {max_dev:.2e}")

field before the kink (x -> -L/2): 0.0000 (expect 0)
field after the kink  (x -> +L/2): 6.2832 (expect 2*pi = 6.2832)
max deviation from exact traveling kink: 3.23e-03

Animating the traveling kink

sine_gordon_evolve_frames() records the same leapfrog propagation as a sequence of snapshots, so the topologically protected twist can be watched traveling down the chain rather than compared only at the start and end.

frames, times = sine_gordon_evolve_frames(u0, u_prev, x, dt, steps_per_frame=steps // 40, n_frames=40)

anim = animate_field_1d(x, frames, times, ylabel="u(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("sine_gordon_kink.gif", writer="pillow", fps=20)