---
tag: SS62J715AZ
kind: concept
name: Bäcklund transform
msc: [37K35, 35Q51, 35Q53]
specializes: []
sources:
  - url: https://en.wikipedia.org/wiki/B%C3%A4cklund_transform
same_as:
  wikidata: Q3537515
  wikipedia: Bäcklund transform
---

## Summary

A Bäcklund transform is a system of first-order partial differential equations that links two functions $u$ and $v$.
The system can be solved for $v$ exactly when $u$ solves one [[partial differential equation|GC5BDZ6QBC]].
Then $v$ solves a second equation.
When the two equations are the same, it is an auto-Bäcklund transform, and it turns known solutions into new ones.
It is one of the main tools of the theory of integrable equations and [[solitons|63D4AXYCT4]].

## Explanations

### L1 human

Think of a machine that takes one solution of an equation and outputs another solution.
The machine is itself a small system of equations of first order, which is easier to solve than the original equation.
The simplest example is the pair of Cauchy–Riemann equations $u_x = v_y$, $u_y = -v_x$.
If $u$ is harmonic, these equations give a harmonic conjugate $v$, so they carry one solution of Laplace's equation to another.
For the [[sine-Gordon equation|802QSF9NM3]], applying the machine to the zero solution gives a kink, and applying it again gives solutions with two kinks.