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We investigate the spatial nonlinear localization of light on a quasi-plane-wave background with a harmonic perturbation induced by modulation instability in a quadratic nonlinear optical medium. In particular, we demonstrate experimentally the excitation of deterministic Akhmediev breathers and thus the growth-decay dynamics of modulation instability in a LiNbO_{3} slab waveguide. The results should stimulate new interest in modulation instability, extreme events, turbulence, recurrence, and supercontinuum generation in quadratic nonlinear optics.