I'm stuck at a seemingly easy problem but I don't know how to approach it (partially due to the shape of the sine-Gordon equation). Let's say that $\omega(u,v)$ is a solution of the sine-Gordon equation $\omega_{uv} = \sin{\omega}$ such that it satisfies the additional condition of 

$$\left(\log\left(\frac{\omega_u}{\omega_v}\right)\right)_{uv} = 0$$

The above equation results in 

$$\frac{\omega_u}{\omega_v} = U(u)\,V(v)$$

How can I identify the nature of the functions $U(u)$ and $V(v)$? I actually expect them to be both constants. I would appreciate any suggestion for that. 

Please let me know if you need additional info!