Bäcklund Transformation

A Bäcklund transformation is a system of differential relations, called Bäcklund relations, that maps a solution of one differential equation to a solution of another, possibly identical, one. Bäcklund transformations occur for both ordinary differential equations and partial differential equations, but their classical use in soliton theory chiefly concerns nonlinear partial differential equations. In the latter setting, the Bäcklund relations are usually first-order partial differential equations whose compatibility conditions imply that the transformed function satisfies the target equation.

Starting with a known solution, the defining relations can therefore be solved to generate additional solutions. Many Bäcklund transformations contain a parameter, and repeated transformations can lead to superposition principles and families of soliton solutions.