Bäcklund transform

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In mathematics, Bäcklund transforms relate solutions of differential equations. They were introduced by Bianchi and Bäcklund in the 1880s. At that time Bäcklund transforms were defined on pseudospherical surfaces, and the transform itself was again a pseudospherical surface, i.e. Bäcklund transforms are geometric transformations.

Bäcklund transforms are also used to solve partial differential equations, as in soliton theory. They are typically sets of first order partial differential equations relating two functions.

If we have a partial differential equation in u, and a Bäcklund transform from u to v, we can deduce a second partial differential equation satisfied by v.

The Cauchy-Riemann equations are the most straightforward example.

If u satisfies Laplace's equation, and v is related to u via the Bäcklund transform

$u_x=v_y \quad u_y=-v_x$

then, since we must have

u x y = u y x

we see that v is a second solution of Laplace's 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

$\begin{matrix} v_x & = & u_x + a \exp \left[ \frac{u+v}{2} \right] \\ v_y & = & -u_y - \frac{2}{a} \exp \left[ \frac{u-v}{2} \right] \end{matrix} \,\!$

where a is an arbitrary parameter,

and if u is a solution of the Liouville equation

$u_{xy}=\exp u \,\!$

then v is a solution of the much simpler equation, $v x y = 0$ , and vice versa.

We can then solve the (non-linear) Liouville equation by working with a much simpler linear equation.

A Bäcklund transform which relates two different solutions of the same equation, such as the Cauchy-Riemann equations, is called an invariant 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.

No systematic way of finding Bäcklund transforms is known.