Exact solutions could not be the right way to understand infrared behavior of Yang-Mills theory. As we know from quantum field theory, we can start with some approximation (weak coupling). With this in mind, it can be proved that the following holds (see http://arxiv.org/abs/0903.2357) for a gauge coupling going formally to infinity

$$
A_\mu^a(x)=\eta_\mu^a\phi(x)+O(1/\sqrt{N}g)
$$

being $\eta_\mu^a$ a set of constants and $\phi(x)$ a solution to the equation

$$
\Box\phi(x)+\lambda\phi(x)^3=0.
$$

provided $\lambda=Ng^2$. This is the content of the so called mapping theorem. The relevant aspect of this theorem is that one can provide a set of exact solutions for the scalar field in the form

$$
\phi(x)=\mu\left(\frac{2}{\lambda}\right)^\frac{1}{4}{\rm sn}(p\cdot x+\theta,i)
$$

being $\mu$ and $\theta$ constants, sn a Jacobi elliptic function and provided the following dispersion relation holds

$$
p^2=\mu^2\left(\frac{\lambda}{2}\right)^\frac{1}{2}.
$$

That is, one has massive classical solutions even if we started from massless equations. So, we can start from these classical approximate solutions to build up an infrared quantum field theory for the Yang-Mills field and displaying in this way a mass gap (see http://arxiv.org/abs/1011.3643).

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