Theorem about simple classification of reductions of the Sine-Gordon equation to an ordinary differential equation.

We classify reductions of the sine-Gordon equation \(\omega_{uv}=\sin\omega\) of the form \(\omega(u,v)=f(\phi(u,v))\) under the condition that \(\phi_u\phi_v\) and \(\phi_{uv}\) are functions of \(\phi\) alone. This condition guarantees that the substitution yields an ordinary differential equation for \(f\). We prove that such a \(\phi\) must be, up to a smooth reparameterisation, a function of a bilinear expression \(\alpha u+\beta v+\gamma uv+\delta\). Consequently, the reduced ODE is either the pendulum equation (traveling waves) or an Amsler-type equation. The proof is self-contained and rigorous.