---
tag: 802QSF9NM3
kind: concept
name: Sine-Gordon equation
msc: [35Q53, 35Q51]
specializes: [GC5BDZ6QBC]
sources:
  - url: https://en.wikipedia.org/wiki/Sine-Gordon_equation
same_as:
  wikidata: Q2558473
  wikipedia: Sine-Gordon equation
---

## Summary

The sine-Gordon equation is the nonlinear wave equation $u_{tt} - u_{xx} + \sin u = 0$ for a real function $u(x,t)$.
Its name is a pun on the [[Klein–Gordon equation|AYCCDQWCJF]] $u_{tt} - u_{xx} + u = 0$, which is its linearisation at $u = 0$.
It first appeared in the geometry of surfaces of constant negative Gaussian curvature.
It is completely integrable and has kink solitons that connect the rest states $0$ and $2\pi$.

## Explanations

### L1 human

Picture a long row of pendulums hanging from a horizontal wire, with neighbours joined by springs.
The angle $u$ of the pendulums along the row obeys the sine-Gordon equation: $\sin u$ is the pull of gravity and $u_{xx}$ is the pull of the springs.
A kink is a place where the pendulums make one full turn, and this twist can travel along the row without changing shape.
The static kink is $u(x) = 4 \arctan(e^x)$, which goes from $0$ at $x = -\infty$ to $2\pi$ at $x = +\infty$.