We classify reductions of the sine-Gordon equation \ (ₔₕ=\) of the form \ ( (u, v) =f ( (u, v) ) \) under the condition that \ (ᵤᵥ\) and \ (ₔₕ\) are functions of \ (\) alone. This condition guarantees that the substitution yields an ordinary differential equation for \ (f\). We prove that such a \ (\) must be, up to a smooth reparameterisation, a function of a bilinear expression \ (u+ v+ uv+\). Consequently, the reduced ODE is either the pendulum equation (traveling waves) or an Amsler-type equation. The proof is self-contained and rigorous.
Anton Kalmykov (Tue,) studied this question.