Bäcklund transform

The Cauchy–Riemann equations

The prototypical example of a Bäcklund transform is the Cauchy–Riemann system

${\displaystyle u_x=v_y,u_y=-v_x,}$

which relates the real and imaginary parts ${\displaystyle u}$ and ${\displaystyle v}$ of a holomorphic function. This first order system of partial differential equations has the following properties.

1. If ${\displaystyle u}$ and ${\displaystyle v}$ are solutions of the Cauchy–Riemann equations, then ${\displaystyle u}$ is a solution of the Laplace equation
   ${\displaystyle u_{xx}+u_{yy}=0}$
   (i.e., a harmonic function), and so is ${\displaystyle v}$. This follows straightforwardly by differentiating the equations with respect to ${\displaystyle x}$ and ${\displaystyle y}$ and using the fact that
   ${\displaystyle u_{xy}=u_{yx},v_{xy}=v_{yx}.}$
2. Conversely if ${\displaystyle u}$ is a solution of Laplace's equation, then there exist functions ${\displaystyle v}$ which solve the Cauchy–Riemann equations together with ${\displaystyle u}$.

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 ${\displaystyle u}$ and Laplace's equation for ${\displaystyle v}$ 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 ${\displaystyle u}$, and a Bäcklund transform from ${\displaystyle u}$ to ${\displaystyle v}$, we can deduce a partial differential equation satisfied by ${\displaystyle v}$.

This example is rather trivial, because all three equations (the equation for ${\displaystyle u}$, the equation for ${\displaystyle v}$ and the Bäcklund transform relating them) are linear. Bäcklund transforms are most interesting when just one of the three equations is linear.

The sine-Gordon equation

Suppose that u is a solution of the sine-Gordon equation

${\displaystyle u_{xy}=\sin u.}$

Then the system

${\displaystyle \begin{array}{cc}{v}_x& =u_x+2a\sin (\frac{v+u}{2})\\ v_y& =-u_y+\frac{2}{a}\sin (\frac{v-u}{2})\end{array}}$

where a is an arbitrary parameter, is solvable for a function v which will also satisfy the sine-Gordon equation. This is an example of an auto-Bäcklund transform.

By using a matrix system, it is also possible to find a linear Bäcklund transform for solutions of sine-Gordon equation.

The Liouville 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

${\displaystyle \begin{array}{cc}{v}_x& =u_x+2a\exp (\frac{u+v}{2})\\ v_y& =-u_y-\frac{1}{a}\exp (\frac{u-v}{2})\end{array}}$

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

${\displaystyle u_{xy}=\exp u}$

then v is a solution of the much simpler equation, ${\displaystyle v_{xy}=0}$, and vice versa.

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