Hubbard-like interactions and emergent dynamical regimes between modulational instability and self-trapping

Abstract We investigate the modulational instability of uniform wave packets governed by a discrete third-order nonlinear Schrödinger equation in finite square lattices, modeling light propagation in two-dimensional nonlinear waveguide arrays. We analyze how initially stable uniform distributions evolve into self-trapped (localized) regimes and the influence of a refractive index detuning selectively applied along the diagonal waveguides on this transition. This detuning effectively emulates the effect of on-site Hubbard-like interactions U in photonic analogs of interacting particles in a one-dimensional lattice. While for $$U = 0$$ U = 0 the system exhibits the known direct transition from stable uniform states to asymptotically localized profiles, we show that $$U > 0$$ U > 0 induces an emergence of intermediate dynamical regimes. These regimes include coherent breathing modes that can be either confined along diagonal or off-diagonal waveguides, as well as chaotic-like propagation patterns. At higher nonlinearities, we identify distinct self-trapped regimes characterized by diagonal or off-diagonal localized modes, depending on the strength of U . The critical nonlinear strengths separating the existing regimes are shown in the phase diagram, underscoring the competing trends imposed by the Hubbard-like interaction on the optical field.