Abstract: We address the issue of how disorder together with nonlinearity affect energy relaxation in the lattice φ⁴ system. The absence of nonlinearity leads such a model to only supporting fully localized Anderson modes whose energies will not relax. However, through exploring the time decay behavior of each Anderson mode's energy-energy correlation, we find that adding nonlinearity, three distinct relaxation details can occur. (i) A small amount of nonlinearity causes a rapid exponential decay of the correlation for all modes. (ii) In the intermediate value of nonlinearity, this exponential decay will turn to power-law with a large scaling exponent close to -1. (iii) Finally, all Anderson modes' energies decay in a power-law manner but with a quite small exponent, indicating a slow long-time tail decay. Obviously, the last two relaxation details support a new localization mechanism. As an application, we show that these are relevant to the nonmonotonous nonlinearity dependence of thermal conductivity. Our results thus provide new information for understanding the combined effects of disorder and nonlinearity on energy relaxation.

Key words: classical transport, localized modes, Anderson localization