  Nonlinear topological insulators have garnered substantial recent attention
as they have both enabled the discovery of new physics due to interparticle
interactions, and may have applications in photonic devices such as topological
lasers and frequency combs. However, due to the local nature of nonlinearities,
previous attempts to classify the topology of nonlinear systems have required
significant approximations that must be tailored to individual systems. Here,
we develop a general framework for classifying the topology of nonlinear
materials in any discrete symmetry class and any physical dimension. Our
approach is rooted in a numerical $K$-theoretic method called the spectral
localizer, which leverages a real-space perspective of a system to define local
topological markers and a local measure of topological protection. Our
nonlinear spectral localizer framework yields a quantitative definition of
topologically non-trivial nonlinear modes that are distinguished by the
appearance of a topological interface surrounding the mode. Moreover, we show
how the nonlinear spectral localizer can be used to understand a system's
topological dynamics, i.e., the time-evolution of nonlinearly induced
topological domains within a system. We anticipate that this framework will
enable the discovery and development of novel topological systems across a
broad range of nonlinear materials.
