Bäcklund transform

Remarks

Example: the Cauchy–Riemann system $v_x = -u_y$, $v_y = u_x$ is an auto-Bäcklund transform of Laplace's equation $u_{xx} + u_{yy} = 0$. Indeed $(v_x)_y = -u_{yy}$ and $(v_y)_x = u_{xx}$, so the system is compatible exactly when $u_{xx} + u_{yy} = 0$. Then $v_{xx} + v_{yy} = -u_{yx} + u_{xy} = 0$ for $C^2$ functions, so $v$ is harmonic too.

Example: for a constant $a \ne 0$, the system $$\Big(\frac{v - u}{2}\Big)_x = a \sin\frac{u + v}{2}, \qquad \Big(\frac{u + v}{2}\Big)_y = \frac{1}{a} \sin\frac{v - u}{2}$$ is an auto-Bäcklund transform of the sine-Gordon equation $\varphi_{xy} = \sin \varphi$. With $s = (u + v)/2$ and $d = (v - u)/2$, one gets $d_{xy} = \cos s \sin d$ and $s_{xy} = \cos d \sin s$, so $u_{xy} = \sin(s - d) = \sin u$ and $v_{xy} = \sin(s + d) = \sin v$.

Non-example: the system $v_x = u_x$, $v_y = u_y$ is compatible for every $C^2$ function $u$. So it imposes no equation on $u$, and it is not a Bäcklund transform between two equations.