  We demonstrate, both theoretically and experimentally, the concept of
non-linear second-order topological insulators, a class of bulk insulators with
quantized Wannier centers and a bulk polarization directly controlled by the
level of non-linearity. We show that one-dimensional edge states and
zero-dimensional corner states can be induced in a trivial crystal insulator
made of evanescently coupled resonators with linear and nonlinear coupling
coefficients, simply by tuning the excitation intensity. This allows global
external control over topological phase transitions and switching to a phase
with non-zero bulk polarization, without requiring any structural or
geometrical changes. We further show how these non-linear effects enable
dynamic tuning of the spectral properties and localization of the topological
edge and corner states. Such self-induced second-order topological insulators,
which can be found and implemented in a wide variety of physical platforms
ranging from electronics to microwaves, acoustics, and optics, hold exciting
promises for reconfigurable topological energy confinement, power harvesting,
data storage, and spatial management of high-intensity fields.
