Bäcklund transform

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.

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.

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.

The prototypical example of a Bäcklund transform is the Cauchy–Riemann system which relates the real and imaginary parts

This first order system of partial differential equations has the following properties.

, we can deduce a partial differential equation satisfied by

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 where a is an arbitrary parameter, is solvable for a function v which will also satisfy the sine-Gordon equation.

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 where a is an arbitrary parameter, and if u is a solution of the Liouville equation