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We discuss disordered nonlinear wave equations with a Hamiltonian structure. There is a large amount of numerical work suggesting a rather universal law for the spreading of initially localized wavepackets in such systems:  this law states that the width w of such a wavepacket grows with time as  t to the power 1/6.

In our work we argue that this law does not capture the long time behaviour, by combining two approaches.  On the one hand, we prove a theorem showing that in the thermal state, the decorrelation time grows faster than polynomial in the nonlinearity parameter. When translated (in a way that is not mathematically rigorous) to the setup with a wavepacket, this suggests that the spreading should be slower than a power law in time. On the other hand, we perform numerics for the thermal decorrelation time (i.e. precisely for the quantity for which we have a theorem).  This numerics suggests that the decorrelation time grows as the nonlinearity parameter to the fourth power. When translated in the same way to the spreading setup, this suggests that the spreading should indeed follow the numerically observed 1/6 law.  We conclude from this that state-of-the-art numerics does likely not capture the long-time behaviour of the wave equations correctly, and that the spreading  of the wavepacket at long times is slower than a power law.