The discrete nonlinear Schrödinger equation is a ubiquitous and fundamental equation in nonlinear physics and mathematics. It describes properties of chemical, condensed matter, optical, and other systems where self-trapping mechanisms are present. The latter arise from either strong interaction with the environment or genuine nonlinear properties of the medium. The continuous nonlinear Schrödinger equation is one of the few nonlinear partial differential equations that are analytically solvable. The discrete version we are discussing here cannot be solved analytically in general except for some special cases. One of these is the nonlinear dimer, a system that has only two sites and where an excitation may tunnel from one to the next. The analytical solution is expressed through elliptic functions, the Jacobian functions, or the Weierstrass elliptic function. In the case where both nonlinear sites have the same energy and the excitation is localized initially on one site, we find that there is a self-trapping transition for a specific value of the nonlinearity parameter. For this value the elliptic function time evolution becomes hyperbolic, signaling a true change in the excitation dynamics.