Arbitrarily Weak Nonlinearity Can Destroy the Anderson Localization

Whether the Anderson localization can survive from the weak enough nonlinear interaction is still an open question. In this Letter, we study the effect of nonlinear interaction on disordered chain based on the wave turbulence theory. It is found that the equipartition time $T_{eq}$ is inversely proportional to the square of the nonlinearity strength $\lambda$, i.e., $T_{eq}\propto\lambda^{-2}$, in thermodynamic limit. This result has two fundamentally important consequences. First, the Anderson localized modes can not survive from arbitrarily weak nonlinearity. Secondly, contrary to popular belief, disorder can lead to a more fast thermalization in the weak nonlinear region, which is due to the emergence of three-wave resonance.