  Nanomagnets are the building blocks of many existing and emergent spintronic
technologies. Magnetization dynamics of nanomagnets is often dominated by
nonlinear processes, which have been recently shown to have many surprising
features and far-reaching implications for applications. Here we develop a
theoretical framework uncovering the selection rules for multimagnon processes
and discuss their underlying mechanisms. For its technological relevance, we
focus on the degenerate three-magnon process in thin elliptical nanodisks to
illustrate our findings. We parameterize the selection rules through a set of
magnon interaction coefficients which we calculate using micromagnetic
simulations. We postulate the selection rules and investigate how they are
altered by perturbations, that break the symmetry of static magnetization
configuration and spatial spin-wave profiles and that can be realized by
applying off-symmetry-axis or nonuniform magnetic fields. Our work provides the
phenomenological understanding into the mechanics of magnon interaction as well
as the formalism for determining the interaction coefficients from simulations
and experimental data. Our results serve as a guide to analyze magnon processes
inherently present in spin-torque devices for boosting their performance or to
engineer a specific nonlinear response of a nanomagnet used in neuromorphic or
quantum magnonic application.
