Exact solution to a classical spontaneously broken scalar theory

As promised in my preceding post I said that a classical spontaneously broken scalar theory can be exactly solved. This is true as I will show. Consider the equation

You can check by yourself that the exact solution is given by

being the v.e.v. of the field and an elliptical Jacobi function. As always the following dispersion relation must be true

giving a consistent classical solution. When one goes to see the spectrum of the theory, the Fourier series of the Jacobi dn function has a zero mass excitation, the Goldstone boson.

Update: A proper full solution is given by

being an integration constant.