  This work explores the possibility of creating and controlling unconventional
nonlinearities by periodic driving, in a broad class of systems described by
the nonlinear Schr\"odinger equation (NLSE). By means of a parent quantum
many-body description, we demonstrate that such driven systems are well
captured by an effective NLSE with emergent nonlinearities, which can be finely
controlled by tuning the driving sequence. We first consider a general class of
two-mode nonlinear systems - relevant to optical Kerr cavities, waveguides and
Bose-Einstein condensates - where we find an emergent four-wave mixing
nonlinearity, which originates from pair-hopping processes in the parent
quantum picture. Tuning this drive-induced nonlinearity is shown to modify the
phase-space topology, which can be detected through relative population and
phase measurements. We then couple individual (two-mode) dimers in view of
designing extended lattice models with unconventional nonlinearities and
controllable pair-hopping processes. Following this general dimerization
construction, we obtain an effective lattice model with drive-induced
interactions, whose ground-state exhibits orbital order, chiral currents and
emergent magnetic fluxes through the spontaneous breaking of time-reversal
symmetry. We analyze these intriguing properties both in the weakly-interacting
(mean-field) regime, captured by the effective NLSE, and in the
strongly-correlated quantum regime. Our general approach opens a route for the
engineering of unconventional optical nonlinearities in photonic devices and
controllable drive-induced interactions in ultracold quantum matter.
