Lagrangian for a non-linear wave equation

Question

I have the following wave equation that I am trying to understand better:

$$\frac{\partial^2 \varphi}{\partial t^2}-\frac{\partial^2 \sin{\varphi}}{\partial x^2}=0.$$

This equation describes an LC ladder except the inductors have been replaced with Josephson junctions, if you want the physical motivation. My goal is to eventually find the Lagrangian which generates this equation. Does anyone know how to do that? If not has anyone seen this particular equation anywhere before?

Answer 1

Here is one proposal in 1+1D:

1. Define a Lagrangian density $$ {\cal L} ~=~ \frac{1}{2} (\partial_t\Phi)^2 +\cos(\partial_x\Phi).\tag{1}$$

2. Then the Euler-Lagrange (EL) equation is $$ \partial^2_t\Phi-\partial_x\sin(\partial_x\Phi)~\approx~0.\tag{2}$$

3. Define OP's field $$\varphi~:=~\partial_x\Phi.\tag{3}$$

4. OP's wave equation follows by differentiating the EL eq. (2) wrt. $x$.