It is well known that the sine-Gordon equation in laboratory (or light-cone) coordinates, \begin{align} \ddot\phi-\phi''=\sin\phi,\\ \dot\varphi'=\sin\varphi, \end{align} respectively, is an integrable system [1]. However, I have been searching in the literature for the bi-Hamiltonian structure of the sine-Gordon equation, and I've found mathematical literature that I really do not understand (I am a theoretical physicist). Just like the KdV case, which is easily to find in the literature (Lenard, 1964; Drazin & Johnson, 1983; Das, 1989, etc.) that it is a bi-Hamiltonian system, \begin{align} \dot u=uu'+u'''=\partial_x\frac{\delta H_0}{\delta u}=\mathcal{D}\frac{\delta H_1}{\delta u}, \end{align} where \begin{align} H_0=\int dx\;\left(\frac{1}{6}u^3-\frac{1}{2}u'^2\right),\;\, \mathcal{D}=\partial_x^3+\frac{1}{3}\left(\partial_x u+u\partial_x\right),\;\, H_1=\int dx\;\left(\frac{1}{2}u^2\right). \end{align}
I'm asking if somebody knows an article, reference or its own knowledge, where the bi-Hamiltonian structure of the sine-Gordon equation could be read explicitely.