Editing Bäcklund transform (section)

In [[mathematics]], '''Bäcklund transforms''' or '''Bäcklund transformations''' (named after the Swedish mathematician [[Albert Victor Bäcklund]]) relate [[partial differential equation]]s and their solutions. They are an important tool in [[soliton theory]] and [[integrable system]]s. 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.

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.