This is my master thesis, as submitted to my university. One of the fascinating qualities of spin waves (or magnons), which are the elementary excitations in magnetically ordered substances, is their nonlinear behavior at moderate excitation powers. This makes spin waves not only an attractive model system to study general nonlinear systems, but it also provides a way to utilize nonlinear dynamics in possible technical applications. In a ferromagnetic nano disk which is magnetized in the vortex state, the spin-wave modes meet strict boundary conditions and therefore inherit a discrete spectrum. When driven with a large enough excitation field, they can decay into other spin-wave modes within well-defined channels due to a nonlinear process called three-magnon scattering. The aim of this thesis is to explore this phenomenon within nonlinear spin-wave theory and by means of micromagnetic simulations. For this purpose, first the linear dynamics are mapped out and the possible scattering channels are predicted. The stability of these channels with respect to static external fields is studied. Within this context, exotic spin-wave modes which arise in a broken cylindrical symmetry are found. Moreover, a model to predict the temporal evolution of the spin-wave modes is developed within the classical Hamiltonian formalism for nonlinear spin-wave dynamics. Together with micromagnetic simulations, this model is then applied in order to study the power-dependence of three-magnon scattering as well as to uncover a phenomenon called stimulated three-magnon scattering, which may allow for an integration of this nonlinear process in magnonic circuits. The results are compared with Brillouin light-scattering experiments which were conducted prior to this thesis or were motivated by it.