Bäcklund变换是两个非线性偏微分方程之间的一对变换关系

偏微分方程1：

偏微分方程2：

是Bäcklund变换是求非线性偏微分方程精确解的一种重要的变换。

1876年瑞典数学家巴克隆德发现Sine-Gordon方程的不同解u、v

之间有如下关系：

这就是Sine-Gordon方程的Bäcklund自变换。

将Bäcklund自变换第一式对t取微商，二式对x微商：

消除v即得；

消除u项即得

Bäcklund变换常用于求Sine-Gordon方程、高维广义Burger I型方程、高维广义Burger II型方程的精确解：

In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions. They are an important tool in soliton theory and integrable systems. A Bäcklund transform is typically a system of first order partial differential equations relating two functions, and often depending on an additional parameter. It implies that the two functions separately satisfy partial differential equations, and each of the two functions is then said to be a Bäcklund transformation of the other.