Low temperature dynamics of the one-dimensional discrete nonlinear Schr？dinger equation

We study equilibrium time correlations for the discrete nonlinear Schr\"odinger equation on a one-dimensional lattice and unravel three dynamical regimes. There is a high temperature regime with density and energy as the only two conserved fields. Their correlations have zero velocity and spread diffusively. In the low temperature regime umklapp processes are rare with the consequence that phase differences appear as an additional (almost) conserved field. In an approximation where all umklapp is suppressed, while the equilibrium state remains untouched, one arrives at an anharmonic chain. Using the method of nonlinear fluctuating hydrodynamics we establish that the DNLS equilibrium time correlations have the same signature as a generic anharmonic chain, in particular KPZ broadening for the sound peaks and L\'evy 5/3 broadening for the heat peak. In the, so far not sharply defined, ultra-low temperature regime the integrability of the dynamics becomes visible. As an illustration we simulate the completely integrable Ablowitz-Ladik model and confirm ballistic broadening of the time correlations.