Here we present supplementary online video material concerning the following publications:

**Philipp Heinen**1, **Aleksandr N. Mikheev**1,2, **Christian-Marcel Schmied**1, and **Thomas Gasenzer**1,2

1Kirchho*ff*-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany

2Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

**Philipp Heinen****1****, Aleksandr N. Mikheev1,2, and Thomas Gasenzer1,2 **

1Kirchho*ff*-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany

2Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany

*N* expansion at next-to-leading order. The resulting kinetic equation is analysed for possible scaling solutions in space and time that are characterised by a set of universal scaling exponents and encode self-similar transport to low momenta. Assuming the momentum occupancy distribution to exhibit a scaling form we can determine the exponents by identifying the dominating contributions to the scattering integral and power counting. If the field exhibits strong variations across many wells of the cosine potential, the scattering integral is dominated by the scattering of many quasiparticles such that the momentum of each single participating mode is only weakly constrained. Remarkably, in this case, in contrast to wave turbulent cascades, which correspond to local transport in momentum space, our results suggest that kinetic scattering here is dominated by rather non-local processes corresponding to a spatial containment in position space. The corresponding universal correlation functions in momentum and *position* space corroborate this conclusion. Numerical simulations performed in accompanying work yield scaling properties close to the ones predicted here.

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All videos by Philipp Heinen

The videos show the approach of the sine-Gordon system towards a non-thermal fixed point. The evolution shown is computed in the non-relativistic limit, where the evolution is governed by a non-linear Schrödinger equation with Bessel function nonlinearity, cf. Eq. (7) in the first of the above papers.

**Video 1**: Sine-Gordon system evolving close to a non-thermal fixed point, for *m*/*Q* = 20, *F*_{0} = 10^{3}. The **upper panels** show the 2D spatial distribution of the field amplitude (**left**) and phase (**right**). The **lower panels** depict the momentum spectrum *f*(*t*,**p**) = ⟨*ψ*^{+}(*t*,**p**)*ψ*(*t*,**p**)⟩ (left) and the density-density correlator D(*t*,**p**) = ⟨*ρ*(*t*,**p**)* ρ*(

The initial state, in each case, corresponds to a box-like even occupancy of all momentum modes up to a maximum cutoff, with random phases in each mode and random Gaussian noise added to all modes, including the empty modes at large momenta. The video corresponds to the evolution, 4 snapshots of which are shown in Fig. 1 of the first paper above. The time is indicated, in units of *Q*^{–1}, at the top of each video.

**Video 2**: Sine-Gordon system evolving close to a non-thermal fixed point, for *m*/*Q* = 80, *F*_{0} = 10^{3}. The **upper panels** show the 2D spatial distribution of the field amplitude (**left**) and phase (**right**). The **lower panels** depict the momentum spectrum *f*(*t*,**p**) = ⟨*ψ*^{+}(*t*,**p**)*ψ*(*t*,**p**)⟩ (left) and the density-density correlator D(*t*,**p**) = ⟨*ρ*(*t*,**p**)* ρ*(

The initial state, in each case, corresponds to a box-like even occupancy of all momentum modes up to a maximum cutoff, with random phases in each mode and random Gaussian noise added to all modes, including the empty modes at large momenta. The time is indicated, in units of *Q*^{–1}, at the top of each video.

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