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Abstract

Mean field and beyond mean field model calculations of Bose–Einstein condensates trapped in optical lattices have shown that initially homogeneous condensates can evolve into self-trapped, strongly localized states in the presence of weak boundary dissipation, a phenomenon called self-localization. A dynamical phase transition from extended to localized states can be observed when the effective nonlinearity exceeds a critical threshold ${{\rm{\Lambda }}}_{\mathrm{eff}}^{{\rm{c}}}$. We investigate this phase transition to self-localization in the mean field approximation of the discrete nonlinear Schrödinger equation. We quantitatively characterize the properties of the discrete breathers, i.e. the nonlinear localized solutions, at the phase transition. This leads us to propose and numerically verify an analytical lower bound ${{\rm{\Lambda }}}_{\mathrm{eff}}^{{\rm{L}}}$ for the critical nonlinearity based on the idea of self-induced Anderson localization.