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We study the propagation of coherent waves in a nonlinearly-induced randompotential, and find regimes of self-organized criticality and other regimeswhere the nonlinear equivalent of Anderson localization prevails. The regime ofself-organized criticality leads to power-law decay of transport [Phys. Rev.Lett. 121, 233901 (2018)], whereas the second regime exhibits exponentialdecay.

From power law to Anderson localization in nonlinear Schr\"odinger equation with nonlinear randomness