What is meant by an auto-Bäcklund transformation?

The sine-Gordon equation in light-cone coordinates is

$$\frac{\partial^{2} \varphi}{\partial \xi \partial \tau}=\sin \varphi$$

where $\xi=\frac{1}{2}(x-t), \tau=\frac{1}{2}(x+t)$ and $\varphi$ is to be understood modulo $2 \pi$. Show that the pair of equations

$$\partial_{\xi}\left(\varphi_{1}-\varphi_{0}\right)=2 \epsilon \sin \left(\frac{\varphi_{1}+\varphi_{0}}{2}\right), \quad \partial_{\tau}\left(\varphi_{1}+\varphi_{0}\right)=\frac{2}{\epsilon} \sin \left(\frac{\varphi_{1}-\varphi_{0}}{2}\right)$$

constitute an auto-Bäcklund transformation for (1).

By noting that $\varphi=0$ is a solution to (1), use the transformation (2) to derive the soliton (or 'kink') solution to the sine-Gordon equation. Show that this solution can be expressed as

$$\varphi(x, t)=4 \arctan \left[\exp \left(\pm \frac{x-c t}{\sqrt{1-c^{2}}}+x_{0}\right)\right]$$

for appropriate constants $c$ and $x_{0}$.

[Hint: You may use the fact that $\int \operatorname{cosec} x \mathrm{~d} x=\log \tan (x / 2)+$ const.]

The following function is a solution to the sine-Gordon equation:

$$\varphi(x, t)=4 \arctan \left[c \frac{\sinh \left(x / \sqrt{1-c^{2}}\right)}{\cosh \left(c t / \sqrt{1-c^{2}}\right)}\right] \quad(c>0) .$$

Verify that this represents two solitons travelling towards each other at the same speed by considering $x \pm c t=$ constant and taking an appropriate limit.