Bäcklund transforms have their origins in differential geometry: the first nontrivial example is the transformation of pseudospherical surfaces introduced by L. Bianchi and A.V. Bäcklund in the 1880s. This is a geometrical construction of a new pseudospherical surface from an initial such surface using a solution of a linear differential equation. Pseudospherical surfaces can be described as solutions of the sine-Gordon equation, and hence the Bäcklund transformation of surfaces can be viewed as a transformation of solutions of the sine-Gordon equation.
A Bäcklund transform which relates solutions of the same equation is called an invariant Bäcklund transform or auto-Bäcklund transform. If such a transform can be found, much can be deduced about the solutions of the equation especially if the Bäcklund transform contains a parameter. However, no systematic way of finding Bäcklund transforms is known.
The prototypical example of a Bäcklund transform is the Cauchy-Riemann system
which relates the real and imaginary parts u and v of a holomorphic function. This first order system of partial differential equations has the following properties.
Thus, in this case, a Bäcklund transformation of a harmonic function is just a conjugate harmonic function. The above properties mean, more precisely, that Laplace's equation for u and Laplace's equation for v are the integrability conditions for solving the Cauchy-Riemann equations.
These are the characteristic features of a Bäcklund transform. If we have a partial differential equation in u, and a Bäcklund transform from u to v, we can deduce a partial differential equation satisfied by v.
This example is rather trivial, because all three equations (the equation for u, the equation for v and the Bäcklund transform relating them) are linear. Bäcklund transforms are most interesting when just one of the three equations is linear.
Suppose that u is a solution of the sine-Gordon equation
Then the system
By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation.
A Bäcklund transform can turn a non-linear partial differential equation into a simpler, linear, partial differential equation.
For example, if u and v are related via the Bäcklund transform
then v is a solution of the much simpler equation, , and vice versa.
We can then solve the (non-linear) Liouville equation by working with a much simpler linear equation.