A general solution to the sine-gordon equation (with only time dependence)

Question

I have a non-linear equation similar to the sine-gordon equation, but it is ordinary:

$\frac{d^2x}{dt^2} - g \sin(x) = 0$

where $g$ is some positive constant. I'm looking for a general solution of this equation but cannot find one. I know there is a solitonic solution (for some specific boundary conditions) but I would like to have the other solutions as well.

I'll appreciate any help in solving the equation or finding a known source for the solution.

thank you

Answer 1

If you let $\omega_0^2=g$ and replace $x$ by $\theta-\pi$ you get the standard pendulum equation $$ \frac{d \theta^2}{d t^2} + \omega_0^2 \theta = 0 $$ If the initial total energy $\frac1 2 \dot \theta ^2 + \omega_0^2(1 - \cos\theta) < 2 \omega_0^2$, then the solution is bounded and can be written in terms of Jacobi elliptic functions. The derivation is long and tedious, and can be found in http://home.ie.cuhk.edu.hk/~wkshum/wordpress/?p=725

See also the links in http://en.wikipedia.org/wiki/Pendulum_%28mathematics%29#Physical_interpretation_of_the_imaginary_period

Answer 2

As given by user44197, the general solution is given by $$\pm 2 \text{am}\left(\frac{1}{2} \sqrt{\left(c_1-2 g\right) \left(t+c_2\right){}^2}|-\frac{4 g}{c_1-2 g}\right)$$ in which $am$ gives the amplitude for Jacobi elliptic functions.